T ESTING FOR A U NIT R OOT IN A R ANDOM C OEFFICIENT PANEL D ATA M ODEL : S UPPLEMENT Joakim Westerlund∗

Rolf Larsson

University of Gothenburg

Uppsala University

Sweden

Sweden

September 27, 2010

Abstract In this supplement, we provide the proofs of Lemmas A.2 and A.5 in Westerlund and Larsson (2010).

1 Results The autoregressive parameter ρi is modeled as ρi = 1 +

ci , Nη Tδ

where ci ∼ iid(µc , σc2 ) with µmc = E(cim ) < ∞ for all m ≥ 1. We further assume that ci is independent of the increment of the standard Brownian motion W (r ). Define W1 (r ) = W (r ) + ci τ

Z r 0

W (s)ds,

U (r ) = W (r ) − rW (1), Z (r ) = W1 (r ) − where τ =

T 1− δ Nη .

Z 1 0

W1 (s)ds,

This is the same notation used in the main paper with the dependence

upon i in W (r ), W1 (r ), U (r ) and Z (r ) suppressed. Lemmas A.2 and A.5 can be stated in the following way. Lemma A.2. Under the above conditions, ∗ Corresponding

author: Department of Economics, University of Gothenburg, P. O. Box 640, SE405 30 Gothenburg, Sweden. Telephone: +46 31 786 5251, Fax: +46 31 786 1043, E-mail address: [email protected].

1

( a)

Z 1 0

E(W1 (r )2 )dr =

1 1 1 + µ1c τ + µ2c τ 2 , 2 3 12

Z 1

2 1 1 E(W1 (r )4 )dr = 1 + µ1c τ + µ2c τ 2 + µ3c τ 3 + µ4c τ 4 , 3 3 21 0 "µZ ¶2 # 1 7 13 97 101 101 (c) E W1 (r )2 dr + µ1c τ + µ2c τ 2 + µ3c τ 3 + µ4c τ 4 . = 12 15 180 630 5040 0

(b)

Lemma A.5. Under the above conditions, " ¶2 # µZ r Z 1 dr = E c2i U (r )2 Z (s) ds ( a) 0

(b)

(c) (d)

Z 1 0

Z 1 0

Z 1 0

1 1 827 µ2c + µ3c τ + µ4c τ 2 , 420 560 907200

0

"

µZ c4i

E

¶4 #

r 0

"

Z (s) ds

dr =

µZ

¶3 #

c3i U (r )

E

r 0

·

µZ

E ci U (r )

r

3 0

1 1 µ4c + µ5c τ + O(µ6c τ 2 ), 1890 1260

dr =

Z (s) ds

97 7 µ4c τ − µ5c τ 2 + O(µ6c τ 3 ), 4320 103950

¶¸ Z (s) ds

dr = −

1 µ2c τ. 140

2 Proofs Proof of Lemma A.2. Consider (a). Note that Z 1

1 E(W (r )2 )dr = , 2 0 µ ¶ Z 1 Z r Z 1Z r Z 1Z r 1 E W (r ) W (s)ds dr = E(W (r )W (s))dsdr = sdsdr = 6 0 0 0 0 0 0 "µZ ¶ # Z Z Z Z 1

0

E

r

0

2

W (s)ds

dr =

=

1

r

r

r =0

s =0

t =0

E(W (s)W (t))dsdtdr

Z 1 Z r µZ t r =0

t =0

s =0

s+

Z r s=t

¶ t dsdtdr =

1 , 12

giving Z 1 0

E(W1 (r )2 )dr =

=

Z 1 0

" E W (r )2 + 2ci τW (r )

1 1 1 + µ1c τ + µ2c τ 2 . 2 3 12

This completes the proof of (a). 2

Z r 0

µZ W (s)ds + c2i τ 2

r 0

¶2 # W (s)ds

dr

Consider (b). Clearly, Z 1 0

Z 1µ

4

W1 (r ) dr =

0

Z 1

=

0

W (r ) + ci τ

Z r 0

W (r )4 dr + 4ci τ

+ 6( ci τ ) + ( ci τ )

4

Z 1

2

0

W (r )

Z 1 µZ r 0

0

2

¶4 W (s) ds Z 1

µZ

W (r )3

0 r 0

dr µZ r 0

¶ W (s) ds dr

¶2 W (s) ds

dr + 4(ci τ )

3

Z 1

¶4

W (s) ds

0

µZ W (r )

¶3

r

W (s) ds

0

dr

dr.

Suppose that r ≥ s ≥ v ≥ w. Then, h i E [W (r ) W (s) W (v) W (w)] = E W (s)2 W (v) W (w) h h i i = E (W (s) − W (v))2 W (v) W (w) + E W (v)3 W (w) h i h i = E (W (s) − W (v))2 E W (w)2 h i h i h i + 3E (W (v) − W (w))2 E W (w)2 + E W (w)4

= (s − v) w + 3 (v − w) w + 3w2 = w (s + 2v) . It follows that µZ 1 ¶ Z 1 ³ Z 1 ´ E W (r )4 dr = E W (r )4 dr = 3r2 dr = 1, 0 0 0 ·Z 1 µZ r ¶ ¸ Z 1 Z r Z ³ ´ E W (r )3 W (s) ds dr = E W (r )3 W (s) dsdr = 0

r =0

0

3 , 8

= "Z E

1 0

W (r )2

µZ

r 0

#

¶2 W (s) ds

= 2

dr

s =0

= 2

Z 1 Z r Z s r =0

s =0

t =0

Z 1 Z r Z s r =0

s =0

t =0

1 r =0

E

= 6 = 6

1 0

µZ W (r )

r

t (r + 2s) dtdsdr =

1 6

0

#

¶3 W (s) ds

Z 1 Z r Z s Z t r =0 Z 1

s =0 Z r

t =0 Z s

u =0 Z t

r =0

s =0

t =0

u =0

dr

E (W (r ) W (s) W (t) W (u)) dudtdsdr u (s + 2t) dudtdsdr =

3

s =0

3srdsdr

³ ´ E W (r )2 W (s) W (t) dtdsdr

and "Z

Z r

1 . 12

By the same argument, "Z µZ 1

E

0

= 24 = 24

#

¶4

r

W (s) ds

0

dr

Z 1 Z r Z s Z t

Z u

r =0 Z 1

s =0 Z r

t =0 Z s

u =0 Z t

v =0 Z u

r =0

s =0

t =0

u =0

v =0

E (W (s) W (t) W (u) W (v)) dvdudtdsdr v (t + 2u) dvdudtdsdr =

1 , 21

suggesting µZ E

1 0

¶

1 1 1 3 = 1 + 4µ1c τ + 6µ2c τ 2 + 4µ3c τ 3 + µ4c τ 4 8 6 12 21 3 1 1 = 1 + µ1c τ + µ2c τ 2 + µ3c τ 3 + µ4c τ 4 , 2 3 21

W1 (r )4 dr

which completes the proof of (b). Finally, consider (c). Note that Z 1 0

2

W1 (r ) dr =

=

Z 1µ 0

Z 1 0

W (r ) + ci τ

¶2

Z r

W (s) ds

0

2

W (r ) dr + 2ci τ

Z 1

dr

W (r )

0

Z r 0

W (s) dsdr + (ci τ )

2

Z 1 µZ r 0

0

¶2 W (s) ds

giving µZ

1 0

2

¶2

µZ

=

W1 (r ) dr

1

+ 2( ci τ )

2

+ 4( ci τ )

2

Z 1 0

2

W (r ) dr

µZ Z 1 0

"Z

"µZ E

1 0

2

1

4

W (r ) dr

1 0

+ 4( c i τ )3

Here is

+ 4ci τ

W (r ) dr

0

+ ( ci τ )

¶2

2

W (r )

W (r ) µZ

0

r 0

¶2 #

=

0

W (r )2 dr

0

Z 1 µZ r 0

Z r

Z r

Z 1

0

0

W (s) ds

Z 1 Z 1

= 2

4

0

W (s) dsdr

dr

s =0

Z 1 µZ r 0

0

¶2 W (s) ds

.

³ ´ E W (r )2 W (s)2 dsdr

s =0

Z 1 Z r r =0

Z r

¶2

#2 dr

Z 1 Z r r =0

W (r )

W (s) dsdr

¶2

= 2

0 ¶2

W (s) ds

W (s) dsdr

r =0

Z 1

s =0

³ ´ E W (r )2 W (s)2 dsdr s (r + 2s) dsdr =

7 , 12

dr

dr,

µZ E

W (r ) dr

r =0

s =0

t =0

r =0

s =0

t =0

r =0

s =r

t =0

r =0

s =r

t =r

+ +

r =0 Z 1

s =0 t =0 Z 1 Z r

r =0 Z 1

s =r Z 1

t =0 Z s

r =0

s =r

t =r

"Z

1 0

2

W (r ) dr

= 2 + 2 = 2 + 2 + 2 + 2 = 2

3trdtdsdr r (t + 2r ) dtdsdr =

Z 1 µZ r 0

+ 2 + 2 + 2

r =0 s =0 t =0 u =0 Z 1 Z 1 Z s Z t

0

13 , 60 #

¶2 W (s) ds

dr

³ ´ E W (r )2 W (t) W (u) dudtdsdr

r =0

s =0

t =0

u =0

r =0

s =0

t =0

u =0

r =0

s =r

t =0

u =0

r =0

s =0

t =0

u =0

r =0

s =r

t =0

u =0

r =0

s =r

t =r

u =0

r =0

s =r

t =r

u =r

Z 1 Z r Z s Z t Z 1 Z 1 Z s Z t

Z 1 Z r Z s Z t Z 1 Z 1 Z r Z t Z 1 Z 1 Z s Z r Z 1 Z 1 Z s Z t

Z 1 Z r Z s Z t r =0 Z 1

W (s) dsdr

t (r + 2s) dtdsdr

Z 1 Z 1 Z s Z s

= 2

0

³ ´ E W (r )2 W (s) W (t) dtdsdr

Z 1 Z r Z s

=

W (r )

¶

³ ´ E W (r )2 W (s) W (t) dtdsdr

Z 1 Z 1 Z s

+

0

Z r

³ ´ E W (r )2 W (s) W (t) dtdsdr

Z 1 Z 1 Z r

+

Z 1

³ ´ E W (r )2 W (s) W (t) dtdsdr

Z 1 Z r Z s

=

=

0

2

Z 1 Z 1 Z s

=

E

1

s =0 t =0 u =0 Z 1 Z r Z t

r =0

s =r

t =0

u =0

r =0 Z 1

s =r Z 1

t =r Z s

u =0 Z t

r =0

s =r

t =r

u =r

Z 1 Z 1 Z s Z r

³ ´ E W (r )2 W (t) W (u) dudtdsdr ³ ´ E W (r )2 W (t) W (u) dudtdsdr ³ ´ E W (r )2 W (t) W (u) dudtdsdr ³ ´ E W (r )2 W (t) W (u) dudtdsdr ³ ´ E W (r )2 W (t) W (u) dudtdsdr ³ ´ E W (r )2 W (t) W (u) dudtdsdr ³ ´ E W (r )2 W (t) W (u) dudtdsdr u (r + 2t) dudtdsdr u (r + 2t) dudtdsdr 3urdudtdsdr r (u + 2r ) dudtdsdr =

5

37 , 360

"µZ E

=

1

W (r )

0

¶2 #

Z r

W (s) dsdr

0

Z 1 Z r Z 1 Z t

= 2

r =0 s =0 t =0 u =0 Z 1 Z r Z r Z t

= 2 + 2 + 2 = 2 + 2 + 2

E (W (r ) W (s) W (t) W (u)) dudtdsdr

r =0 Z 1

s =0 Z r

t =0 Z s

u =0 Z t

r =0 Z 1

s =0 Z r

t =0 u =0 Z r Z s

r =0

s =0

t=s

u =0

r =0 Z 1

s =0 Z r

t=s Z s

u=s Z t

r =0 Z 1

s =0 Z r

t =0 u =0 Z r Z s

r =0 Z 1

s =0 Z r

t=s Z r

u =0 Z t

r =0

s =0

t=s

u=s

Z 1 Z r Z r Z t

E (W (r ) W (s) W (t) W (u)) dudtdsdr E (W (r ) W (s) W (t) W (u)) dudtdsdr E (W (r ) W (s) W (t) W (u)) dudtdsdr E (W (r ) W (s) W (t) W (u)) dudtdsdr u (s + 2t) dudtdsdr u (t + 2s) dudtdsdr s (t + 2u) dudtdsdr =

1 12

and "Z E

=

1 0

W (r )

Z r 0

W (s) dsdr

Z 1 Z r Z 1 Z t

= 2 = 2 + 2

Z t

0

0

#

¶2 W (s) ds

dr

E (W (r ) W (s) W (u) W (v)) dvdudtdsdr

r =0 s =0 t =0 u =0 Z 1 Z r Z 1 Z t

v =0 Z u

r =0 Z 1

s =0 Z r

t =0 Z r

u =0 Z t

v =0 Z u

r =0

s =0

t =0

u =0

r =0

s =0

t =r

u =0

Z 1 Z r Z 1 Z t

Z 1 µZ r

v =0

Z u

v =0

E (W (r ) W (s) W (u) W (v)) dvdudtdsdr E (W (r ) W (s) W (u) W (v)) dvdudtdsdr E (W (r ) W (s) W (u) W (v)) dvdudtdsdr,

6

where

= + = + + + = + + +

Z 1 Z r Z r Z t

Z u

r =0 Z 1

s =0 Z r

t =0 Z s

v =0 Z u

r =0 Z 1

s =0 Z r

t =0 u =0 v =0 Z r Z t Z u

r =0

s =0

r =0 Z 1

s =0 Z r

t =0 u =0 v =0 Z r Z s Z u

r =0 Z 1

s =0 Z r

t=s Z r

u =0 v =0 Z t Z s

r =0 Z 1

s =0 Z r

t=s Z r

u=s Z t

v =0 Z u

r =0 Z 1

s =0 Z r

t=s Z s

u=s Z t

v=s Z u

r =0

s =0

t =0

u =0

r =0 Z 1

s =0 Z r

t=s Z r

u =0 v =0 Z t Z s

r =0 Z 1

s =0 Z r

t=s Z r

u=s Z t

v =0 Z u

r =0

s =0

t=s

u=s

v=s

t=s

u =0 Z t

u =0

Z 1 Z r Z s Z t

Z 1 Z r Z r Z s

v =0

Z u

v =0

Z u

E (W (r ) W (s) W (u) W (v)) dvdudtdsdr E (W (r ) W (s) W (u) W (v)) dvdudtdsdr E (W (r ) W (s) W (u) W (v)) dvdudtdsdr E (W (r ) W (s) W (u) W (v)) dvdudtdsdr E (W (r ) W (s) W (u) W (v)) dvdudtdsdr E (W (r ) W (s) W (u) W (v)) dvdudtdsdr E (W (r ) W (s) W (u) W (v)) dvdudtdsdr v (s + 2u) dvdudtdsdr v (s + 2u) dvdudtdsdr v (u + 2s) dvdudtdsdr s (u + 2v) dvdudtdsdr =

7

41 5040

and

= +

Z 1 Z r Z 1 Z t

Z u

r =0 Z 1

s =0 Z r

t =r Z 1

u =0 Z s

v =0 Z u

r =0 Z 1

s =0 Z r

t =r Z 1

u =0 v =0 Z r Z s

r =0

s =0

t =r

u=s

v =0

r =0 Z 1

s =0 Z r

t =r Z 1

u=s Z t

v=s Z s

r =0 Z 1

s =0 Z r

t =r Z 1

u =r Z t

v =0 Z r

r =0 Z 1

s =0 Z r

t =r Z 1

u =r Z t

v=s Z u

r =0 Z 1

s =0 Z r

t =r Z 1

u =r Z s

v =r Z u

r =0

s =0

t =r

u =0

v =0

r =0 Z 1

s =0 Z r

t =r Z 1

u=s Z r

v =0 Z u

r =0 Z 1

s =0 Z r

t =r Z 1

u=s Z t

v=s Z s

r =0 Z 1

s =0 Z r

t =r Z 1

u =r Z t

v =0 Z r

r =0 Z 1

s =0 Z r

t =r Z 1

u =r Z t

v=s Z u

r =0

s =0

t =r

u =r

v =r

Z 1 Z r Z 1 Z r Z u

+ + + + =

E (W (r ) W (s) W (u) W (v)) dvdudtdsdr E (W (r ) W (s) W (u) W (v)) dvdudtdsdr E (W (r ) W (s) W (u) W (v)) dvdudtdsdr E (W (r ) W (s) W (u) W (v)) dvdudtdsdr E (W (r ) W (s) W (u) W (v)) dvdudtdsdr E (W (r ) W (s) W (u) W (v)) dvdudtdsdr E (W (r ) W (s) W (u) W (v)) dvdudtdsdr v (s + 2u) dvdudtdsdr

Z 1 Z r Z 1 Z r Z s

+ + + + +

v (u + 2s) dvdudtdsdr s (u + 2v) dvdudtdsdr v (r + 2s) dvdudtdsdr s (r + 2v) dvdudtdsdr s (v + 2r ) dvdudtdsdr =

1 , 84

giving "Z E

1 0

W (r )

Z r 0

W (s) dsdr

Z 1 µZ r 0

0

¶2 W (s) ds

# dr =

101 . 2520

Also, " E

=

Z 1 µZ r 0

0

¶2 W (s) ds

#2  dr 

Z 1 Z r Z r Z 1 Z u Z u

= 2 = 8

E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr

r =0 s =0 t =0 u =0 Z 1 Z r Z r Z r

v =0 x =0 Z u Z u

r =0 Z 1

s =0 Z r

t =0 Z s

u =0 Z r

v =0 Z u

x =0 Z v

r =0

s =0

t =0

u =0

v =0

x =0

E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr,

8

where

= + + + + +

Z 1 Z r Z s Z r

Z u Z v

r =0 Z 1

s =0 Z r

t =0 Z s

u =0 Z t

v =0 Z u

r =0 Z 1

s =0 Z r

t =0 Z s

u =0 v =0 x =0 Z s Z t Z v

r =0

s =0

t =0

u=t

v =0

x =0

r =0 Z 1

s =0 Z r

t =0 Z s

u=t Z r

v=t Z t

x =0 Z v

r =0 Z 1

s =0 Z r

t =0 Z s

u=s Z r

v =0 x =0 Z s Z v

r =0 Z 1

s =0 Z r

t =0 Z s

u=s Z r

v=t Z u

x =0 Z v

r =0

s =0

t =0

u=s

v=s

x =0

x =0 Z v

Z 1 Z r Z s Z s Z u Z v

E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr

with

=

=

Z 1 Z r Z s Z t

Z u Z v

r =0 Z 1

s =0 Z r

t =0 Z s

u =0 Z t

v =0 Z u

x =0 Z v

r =0

s =0

t =0

u =0

v =0

x =0

Z 1 Z r Z s Z s Z t

Z v

r =0 Z 1

s =0 Z r

t =0 Z s

u=t Z s

v =0 Z t

x =0 Z v

r =0

s =0

t =0

u=t

v =0

x =0

Z 1 Z r Z s Z s Z u Z v

= + = +

=

r =0 Z 1

s =0 Z r

t =0 Z s

u=t Z s

v=t Z u

x =0 Z t

r =0 Z 1

s =0 Z r

t =0 Z s

u=t Z s

v=t Z u

x =0 Z v

r =0

s =0

t =0

u=t

v=t

x =t

r =0 Z 1

s =0 Z r

t =0 Z s

u=t Z s

v=t Z u

x =0 Z v

r =0

s =0

t =0

u=t

v=t

x =t

Z 1 Z r Z s Z s Z u Z t

Z 1 Z r Z s Z r Z t

Z v

r =0 Z 1

s =0 Z r

t =0 Z s

u=s Z r

v =0 Z t

x =0 Z v

r =0

s =0

t =0

u=s

v =0

x =0

E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr x (t + 2v) dxdvdudtdsdr =

11 , 40 320

E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr x (t + 2v) dxdvdudtdsdr =

1 , 4032

E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr x (v + 2t) dxdvdudtdsdr t (v + 2x ) dxdvdudtdsdr =

1 , 2016

E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr x (t + 2v) dxdvdudtdsdr =

9

1 , 4032

Z 1 Z r Z s Z r Z s Z v

= + = +

r =0 Z 1

s =0 Z r

t =0 Z s

u=s Z r

v=t Z s

x =0 Z t

r =0 Z 1

s =0 Z r

t =0 Z s

u=s Z r

v=t Z s

x =0 Z v

r =0 Z 1

s =0 Z r

t =0 Z s

u=s Z r

v=t Z s

x =t Z t

r =0 Z 1

s =0 Z r

t =0 Z s

u=s Z r

v=t Z s

x =0 Z v

r =0

s =0

t =0

u=s

v=t

x =t

E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr x (v + 2t) dxdvdudtdsdr t (v + 2x ) dxdvdudtdsdr =

1 2016

and Z 1 Z r Z s Z r Z u Z v

= + + = + +

r =0 Z 1

s =0 Z r

t =0 Z s

u=s Z r

v=s Z u

x =0 Z t

r =0 Z 1

s =0 Z r

t =0 Z s

u=s Z r

v=s Z u

x =0 Z s

r =0 Z 1

s =0 Z r

t =0 Z s

u=s Z r

v=s Z u

x =t Z v

r =0

s =0

t =0

u=s

v=s

x =s

r =0 Z 1

s =0 Z r

t =0 Z s

u=s Z r

v=s Z u

x =0 Z s

r =0 Z 1

s =0 Z r

t =0 Z s

u=s Z r

v=s Z u

x =t Z v

r =0

s =0

t =0

u=s

v=s

x =s

Z 1 Z r Z s Z r Z u Z t

E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr x (s + 2t) dxdvdudtdsdr t (s + 2x ) dxdvdudtdsdr t ( x + 2s) dxdvdudtdsdr =

1 , 1344

which can be used to obtain Z 1 Z r Z s Z r r =0

s =0

t =0

giving

Z u Z v

u =0

v =0

" E

x =0

E (W (s) W (t) W (v) W ( x )) dxdvdudtdsdr =

Z 1 µZ r 0

0

¶2 W (s) ds

101 , 40 320

#2  101 dr  = . 5040

Thus, by combining these results, "µZ ¶2 # 1 7 13 97 41 101 2 = + µ1c τ + µ2c τ 2 + µ3c τ 3 + µ4c τ 4 , E W1 (r ) dr 12 15 180 1260 5040 0 which establishes (c), and hence proof of the lemma is complete.

Proof of Lemma A.5. 10

¥

Consider (a). From

Z r 0

we obtain " E U (r )2

µZ

r 0

"

Z (s) ds =

= E W (r )

2

µZ

+ r E W (r )

2

µZ

1

+ r E W (1)

µZ

r 0

"

+ r 4 E W (1)2

0

W1 (s) ds − r

µZ

1 0

0

0

− 2rE W (r )

0

W1 (s) ds,

2

¶2 # W1 (s) ds Z r 0

"

W1 (s) ds µZ

− 2rE W (r ) W (1)

W1 (s) ds Z r

Z 1

·

¶2 #

+ 4r2 E W (r ) W (1) 2

r

W1 (s) ds

·

2

µZ

¶2 #

r

0

"

Z 1

Z (s) ds

0

" 2

0

W1 (s) ds − r

¶2 #

= E (W (r ) − rW (1))2 "

Z r

W1 (s) ds

· 3

− 2r E W (1)

W1 (s) ds

W1 (s) ds ¶2 #

W1 (s) ds µZ

W1 (s) ds − 2r3 E W (r ) W (1)

0

¶2 #

0

0

¸

"

¸

Z 1

r

Z 1

2

Z r

¶2 #

0

W1 (s) ds

Z 1 0

1 0

¶2 # W1 (s) ds ¸

W1 (s) ds

W1 (s) ds

= Arrrr − 2rArr1r + r2 Arr11 − 2rA1rrr + 4r2 A1r1r − 2r3 A1r11 + r2 A11rr − 2r3 A111r + r4 A1111 , where

· Arstu = E W (r ) W (s)

Z t 0

W1 (v) dv

Z u 0

¸ W1 (w) dw .

Assume that r ≥ s and t ≥ u. Then, · ¶ Z tµ Z v Arstu = E W (r ) W (s) W ( v ) + ci τ W ( x ) dx dv 0 0 ¶ ¸ Z t Z u Z uµ Z w × W ( w ) + ci τ W (y) dy dw = f rsvw dvdw, 0

where

v =0

0

· µ Z f rsvw = E W (r ) W (s) W (v) + ci τ

w =0

¶µ ¶¸ Z w W ( x ) dx W ( w ) + ci τ W (y) dy 0 0 · ¸ Z w = E [W (r ) W (s) W (v) W (w)] + ci τE W (r ) W (s) W (v) W (y) dy 0 · ¸ Z v + ci τE W (r ) W (s) W (w) W ( x ) dx 0 · ¸ Z v Z w 2 + ( ci τ ) E W (r ) W ( s ) W ( x ) dx W (y) dy . v

0

0

11

From the proof of Lemma A.2 we know that if r ≥ s ≥ v ≥ w, E [W (r ) W (s) W (v) W (w)] = w (s + 2v). By using this result, · Z E W (r ) W ( s ) W ( v )

=

Z w 0

=

Z v 0

Z v 0

0

w 0

+

y (s + 2v) dy +

=

Z v Z0 w s

Z v 0

W (y) dy Z w

Z w

0

w 0

Z w v

¸ W (y) dy

E [W (r ) W (s) W (v) W (y)] dy +

Z s v

+ =

Z0 r s

Z v 0

E [W (r ) W (s) W (y) W (v)] dy

E [W (r ) W (y) W (s) W (v)] dy y (s + 2v) dy +

Z s v

and if w ≥ r ≥ s ≥ v, · Z E W (r ) W ( s ) W ( v )

=

E [W (r ) W (s) W (y) W (v)] dy

¢ 1 ¡ 2 v 2w + 2sw − sv , 2

v (s + 2y) dy +

Z w s

¢ 1 ¡ = − v s2 − 4sw + vs − w2 , 2

Z v

1 2 w (s + 2v) . 2

W (y) dy

v (s + 2y) dy =

v

y (s + 2v) dy =

¸

E [W (r ) W (s) W (v) W (y)] dy +

if r ≥ w ≥ s ≥ v, · Z E W (r ) W ( s ) W ( v )

=

¸

E [W (r ) W (s) W (v) W (y)] dy =

Similarly, if r ≥ s ≥ w ≥ v, · Z E W (r ) W ( s ) W ( v )

=

w

w 0

¸ W (y) dy

E [W (r ) W (s) W (v) W (y)] dy + E [W (r ) W (y) W (s) W (v)] dy + y (s + 2v) dy +

Z s v

v (y + 2s) dy

Z s Z vw

v (s + 2y) dy +

¢ 1 ¡ = − v sv − 2rw − 4sw + r2 + s2 . 2

12

r

E [W (r ) W (s) W (y) W (v)] dy

Z r s

E [W (y) W (r ) W (s) W (v)] dy v (y + 2s) dy +

Z w r

v (r + 2s) dy

We also need, for r ≥ s ≥ v ≥ w, · Z v Z E W (r ) W ( s ) W ( x ) dx

= + + = =

Z w Z x x =0 Z w

y =0 Z w

x =0 Z v

y= x Z w

x = w y =0 Z w Z x x =0

y =0

0

w 0

¸ W (y) dy

E [W (r ) W (s) W ( x ) W (y)] dydx E [W (r ) W (s) W (y) W ( x )] dydx E [W (r ) W (s) W ( x ) W (y)] dydx y (s + 2x ) dydx +

Z w Z w x =0

y= x

x (s + 2y) dydx +

Z v x =w

Z w y =0

y (s + 2x ) dydx

¢ 1 2¡ w 3sv − sw + 3v2 , 6

for r ≥ v ≥ s ≥ w, · E W (r ) W ( s )

= + + + = +

Z w Z x x =0 Z w x =0

Z s

0

W ( x ) dx

Z w 0

¸ W (y) dy

E [W (r ) W (s) W ( x ) W (y)] dydx

y =0 Z w

E [W (r ) W (s) W (y) W ( x )] dydx

y= x

Z w

x = w y =0 Z v Z w

E [W (r ) W (s) W ( x ) W (y)] dydx

E [W (r ) W ( x ) W (s) W (y)] dydx

x =s Z w

y =0 Z x

x =0 Z s

y =0 Z w

x =w

y =0

= −

Z v

y (s + 2x ) dydx + y (s + 2x ) dydx +

Z w Z w x =0 Z v

y= x Z w

x =s

y =0

¢ 1 2¡ w −12sv + 2sw + 3s2 − 3v2 , 12

13

x (s + 2y) dydx y ( x + 2s) dydx

for v ≥ r ≥ s ≥ w, · Z E W (r ) W ( s )

= + + + + = + +

Z w Z x x =0 Z w

y =0 Z w

x =0 Z s

y= x Z w

y =0 Z w

x =r

y =0

x =0 Z s

y =0 Z w

0

W (y) dy

E [W (r ) W (s) W ( x ) W (y)] dydx

E [W (r ) W ( x ) W (s) W (y)] dydx E [W ( x ) W (r ) W (s) W (y)] dydx Z w Z w

y (s + 2x ) dydx +

x = w y =0 Z v Z w y =0

W ( x ) dx

¸

E [W (r ) W (s) W (y) W ( x )] dydx

Z w Z x

x =r

0

Z w

E [W (r ) W (s) W ( x ) W (y)] dydx

x = w y =0 Z r Z w x =s Z v

v

y (s + 2x ) dydx +

x (s + 2y) dydx

x =0 Z r

y= x Z w

x =s

y =0

y (r + 2s) dydx = −

y ( x + 2s) dydx

¢ 1 2¡ w −6rv − 12sv + 2sw + 3r2 + 3s2 , 12

for r ≥ v ≥ w ≥ s, · E W (r ) W ( s )

= + + + + + = + +

Z s

Z x

x =0 Z s

y =0 Z s

x =0 Z s

y= x Z w

x =0 y = s Z v Z s x =s Z v

y =0 Z x

x =s Z v

y=s Z w

x =s Z s

y= x Z x

x =0 Z s

y =0 Z w

x =0 y = s Z v Z x x =s

= −

y=s

Z v 0

W ( x ) dx

Z w 0

¸ W (y) dy

E [W (r ) W (s) W ( x ) W (y)] dydx E [W (r ) W (s) W (y) W ( x )] dydx E [W (r ) W (y) W (s) W ( x )] dydx E [W (r ) W ( x ) W (s) W (y)] dydx E [W (r ) W ( x ) W (y) W (s)] dydx E [W (r ) W (y) W ( x ) W (s)] dydx y (s + 2x ) dydx + x (y + 2s) dydx + s ( x + 2y) dydx +

Z s

Z s

x =0 y = x Z v Z s x = s y =0 Z v Z w x =s

y= x

x (s + 2y) dydx y ( x + 2s) dydx s (y + 2x ) dydx

¢ 1 ¡ 3 s 2v − 12v2 w + 3sv2 − 6vw2 + 3sw2 , 12 14

for v ≥ r ≥ w ≥ s, · E W (r ) W ( s )

= + + + + + + + = + + + =

Z s

Z x

x =0 Z s

y =0 Z s

x =0 Z s

y= x Z w

x =0 y = s Z r Z s x =s Z r

y =0 Z x

x =s

y=s

x =s Z v

y= x Z s

x =r Z v

y =0 Z x

x =r Z s

y=s Z x

x =0 Z s

y =0 Z w

Z r Z w

x =0 y = s Z r Z x x =s Z v

y=s Z s

x =r

y =0

Z v 0

W ( x ) dx

Z w 0

¸ W (y) dy

E [W (r ) W (s) W ( x ) W (y)] dydx E [W (r ) W (s) W (y) W ( x )] dydx E [W (r ) W (y) W (s) W ( x )] dydx E [W (r ) W ( x ) W (s) W (y)] dydx E [W (r ) W ( x ) W (y) W (s)] dydx E [W (r ) W (y) W ( x ) W (s)] dydx E [W ( x ) W (r ) W (s) W (y)] dydx E [W ( x ) W (r ) W (y) W (s)] dydx y (s + 2x ) dydx + x (y + 2s) dydx + s ( x + 2y) dydx + y (r + 2s) dydx +

Z s

Z s

x =0 y = x Z r Z s x = s y =0 Z r Z w x =s y= x Z v Z x x =r

y=s

x (s + 2y) dydx y ( x + 2s) dydx s (y + 2x ) dydx

s (r + 2y) dydx

¢ 1 ¡ s −12r3 + 12r2 w + 3sr2 + 6rv2 − 6srv + 6rw2 + 4v3 − 3sw2 , 12

15

and for v ≥ w ≥ r ≥ s, · Z E W (r ) W ( s )

= + + + + + + + + + + + = + + + + +

Z s

Z x

x =0 Z s

y =0 Z s

x =0 Z s

y= x Z r

x =0 Z s

y=s Z w

x =0 y =r Z r Z s x =s

y =0

x =s Z r

y=s Z r

x =s Z r

y= x Z w

x =s Z v

y =r Z s

x =r Z v

y =0 Z r

x =r Z v

y=s Z x

x =r Z v

y =r Z w

x =r Z s

y= x Z x

x =0 Z s

y =0 Z r

Z r Z x

x =0 y = s Z r Z s x =s

y =0

x =s Z v

y= x Z s

x =r Z v

y =0 Z x

x =r

y =r

Z r Z r

v 0

W ( x ) dx

Z w 0

¸ W (y) dy

E [W (r ) W (s) W ( x ) W (y)] dydx E [W (r ) W (s) W (y) W ( x )] dydx E [W (r ) W (y) W (s) W ( x )] dydx E [W (y) W (r ) W (s) W ( x )] dydx E [W (r ) W ( x ) W (s) W (y)] dydx E [W (r ) W ( x ) W (y) W (s)] dydx E [W (r ) W (y) W ( x ) W (s)] dydx E [W (y) W (r ) W ( x ) W (s)] dydx E [W ( x ) W (r ) W (s) W (y)] dydx E [W ( x ) W (r ) W (y) W (s)] dydx E [W ( x ) W (y) W (r ) W (s)] dydx E [W (y) W ( x ) W (r ) W (s)] dydx y (s + 2x ) dydx + x (y + 2s) dydx +

x =0 y = x Z s Z w

x =s

s (y + 2x ) dydx +

s (y + 2r ) dydx +

Z s

x =0 y =r Z r Z x

y ( x + 2s) dydx +

y (r + 2s) dydx +

Z s

y=s

Z r Z w

x = s y =r Z v Z r

x =r y = s Z v Z w x =r

y= x

x (s + 2y) dydx x (r + 2s) dydx s ( x + 2y) dydx s (r + 2x ) dydx

s (r + 2y) dydx s ( x + 2r ) dydx

¢ 1 ¡ = − s 3r2 v − 3r2 s + 3r2 w − 3v2 w + v3 + 3rsv + 3rsw − 12rvw . 6

16

This yields, for r ≥ s ≥ v ≥ w, ¡ ¢ 1 1 f rsvw = w (s + 2v) + µ1c τw2 (s + 2v) + µ1c τw 2v2 + 2sv − sw 2 2 ¡ ¢ 1 2 2 2 µ2c τ w 3sv − sw + 3v , + 6 which implies Arrrr =

Z r Z r

Z r Z v

f rrvw dwdv = 2 f rrvw dwdv v =0 w =0 v =0 w =0 µ Z r Z v ¡ ¢ 1 1 = 2 w (r + 2v) + µ1c τw2 (r + 2v) + µ1c τw 2v2 + 2rv − rw 2 2 v =0 w =0 ¶ ¡ ¢ 1 + µ2c τ 2 w2 3rv − rw + 3v2 dwdv 6 ¢ 1 4¡ = r 19µ2c τ 2 r2 + 105µ1c τr + 150 . 180

Hence, A1111 =

¢ 1 ¡ 19µ2c τ 2 + 105µ1c τ + 150 . 180

But we also have, for v ≥ r ≥ s ≥ w, ¡ ¢ 1 f rsvw = w (r + 2s) − µ1c τw sw − 2rv − 4sv + r2 + s2 2 ¡ ¢ 1 1 2 + µ1c τw (r + 2s) − µ2c τ 2 w2 −6rv − 12sv + 2sw + 3r2 + 3s2 , 2 12 or, with s = r, ¡ ¢ 3 1 1 f rrvw = 3wr − µ1c τw rw − 6rv + 2r2 + µ1c τw2 r − µ2c τ 2 w2 r (−9v + w + 3r ) 2 2 6 1 = 3wr + µ1c τrw (3v − r + w) − µ2c τ 2 w2 r (−9v + w + 3r ) , 6 suggesting Arr1r =

Z 1 Z r v =0

w =0

= Arrrr +

f rrvw dwdv = 2

Z 1 Z r v =r

w =0

Z r Z v v =0

w =0

f rrvw dwdv +

Z 1 Z r v =r

w =0

f rrvw dwdv

f rrvw dwdv,

where Z 1 Z r

f rrvw dwdv µ ¶ 1 = 3wr + µ1c τrw (3v − r + w) − µ2c τ 2 w2 r (−9v + w + 3r ) dwdv 6 v =r w =0 ¡ ¢ 1 = − r3 (r − 1) 6µ2c τ 2 r + µ2c τ 2 r2 + 14µ1c τr + 36 + 18µ1c τ 24 v =r Z 1

w =0 Z r

17

such that ¢ 1 4¡ r 19µ2c τ 2 r2 + 105µ1c τr + 150 180 ¡ ¢ 1 3 r (r − 1) 6µ2c τ 2 r + µ2c τ 2 r2 + 14µ1c τr + 36 + 18µ1c τ . 24

Arr1r =

−

Moreover, by using the same steps as for Arrrr , Arr11 =

Z 1 Z 1

= 2 + 2

v =0 Z r

w =0 Z v

f rrvw dwdv

v =0 w =0 Z 1 Z v v =r

= Arr1r +

w =r Z r

f rrvw dwdv +

Z r Z 1 v =0

w =r

f rrvw dwdv +

Z 1 Z r v =r

w =0

f rrvw dwdv

f rrvw dwdv Z 1

v =0

w =r

f rrvw dwdv + 2

Z 1 Z v v =r

w =r

f rrvw dwdv,

where, because of symmetry, Z r Z 1 v =0

w =r

f rrvw dwdv =

Z 1 Z r v =r

= −

w =0

f rrvw dwdv

¡ ¢ 1 3 r (r − 1) 6µ2c τ 2 r + µ2c τ 2 r2 + 14µ1c τr + 36 + 18µ1c τ . 24

Moreover, since for v ≥ w ≥ r ≥ s, ¡ ¢ 1 f rsvw = s (w + 2r ) − µ1c τs r2 − 4rw + sr − w2 2 ¡ ¢ 1 µ1c τs rs − 2wv − 4rv + w2 + r2 − 2 ¡ ¢ 1 − µ2c τ 2 s 3r2 v − 3r2 s + 3r2 w − 3v2 w + v3 + 3rsv + 3rsw − 12rvw 6 ¡ ¢ = s (w + 2r ) − µ1c τs rs − 2rv − 2rw − vw + r2 ¡ ¢ 1 µ2c τ 2 s 3r2 v − 3r2 s + 3r2 w − 3v2 w + v3 + 3rsv + 3rsw − 12rvw − 6 so that ¡ ¢ f rrvw = r (w + 2r ) − µ1c τr 2r2 − 2rv − 2rw − vw ¡ ¢ 1 − µ2c τ 2 r 6r2 v − 3r3 + 6r2 w − 3v2 w + v3 − 12rvw , 6 giving Z 1 Z v v =r

w =r

1 r (r − 1)2 (2µ2c τ 2 + 39µ2c τ 2 r + 6µ2c τ 2 r2 120 + 3µ2c τ 2 r3 + 15µ1c τ + 150µ1c τr + 15µ1c τr2 + 20 + 160r ).

f rrvw dwdv =

18

It follows that ¢ 1 4¡ r 19µ2c τ 2 r2 + 105µ1c τr + 150 180 1 − r (r − 1) (2µ2c τ 2 + 37µ2c τ 2 r − 33µ2c τ 2 r2 + 27µ2c τ 2 r3 + 2µ2c τ 2 r4 60 + 15µ1c τ + 135µ1c τr − 45µ1c τr2 + 55µ1c τr3 + 20 + 140r + 20r2 ),

Arr11 =

and since for 1 ≥ r ≥ v ≥ w, ¡ ¢ 1 1 f 1rvw = w (r + 2v) + µ1c τw2 (r + 2v) + µ1c τw 2v2 + 2rv − rw 2 2 ¡ 2 ¢ 1 2 2 + µ2c τ w 3v + 3rv − rw , 6 we get A1rrr =

Z r Z r

Z r Z v

f 1rvw dwdv = 2 f 1rvw dwdv v =0 w =0 v =0 w =0 µ Z r Z v ¡ ¢ 1 1 = 2 w (r + 2v) + µ1c τw2 (r + 2v) + µ1c τw 2v2 + 2rv − rw 2 2 v =0 w =0 ¶ ¡ ¢ 1 µ2c τ 2 w2 3v2 + 3rv − rw dwdv + 6 ¢ 1 4¡ = r 19µ2c τ 2 r2 + 105µ1c τr + 150 . 180

Moreover, A1r1r =

Z 1 Z r v =0

w =0

f 1rvw dwdv = A1rrr +

Z 1 Z r v =r

w =0

f 1rvw dwdv,

where, since for r ≥ v ≥ s ≥ w, ¡ ¢ 1 1 f rsvw = w (v + 2s) + µ1c τw2 (v + 2s) − µ1c τw s2 − 4sv + sw − v2 2 2 ¡ ¢ 1 2 2 2 − µ2c τ w −12sv + 2sw + 3s − 3v2 12 ¡ ¢ 1 = w (v + 2s) + µ1c τw vw + sw − s2 + 4sv + v2 2 ¡ ¢ 1 − µ2c τ 2 w2 −12sv + 2sw + 3s2 − 3v2 12 suggesting that if 1 ≥ v ≥ r ≥ w, ¡ ¢ 1 f 1rvw = w (v + 2r ) + µ1c τw vw + rw − r2 + 4rv + v2 2 ¡ ¢ 1 − µ2c τ 2 w2 −12rv + 2rw + 3r2 − 3v2 12 19

such that Z 1 Z r v =r

w =0

µ

Z 1 Z r

¡ ¢ 1 w (v + 2r ) + µ1c τw vw + rw − r2 + 4rv + v2 2 v = r w =0 ¶ ¡ ¢ 1 2 2 2 2 dwdv − µ2c τ w −12rv + 2rw + 3r − 3v 12 1 = − r2 (r − 1) (2µ2c τ 2 r + 14µ2c τ 2 r2 + 5µ2c τ 2 r3 72 + 6µ1c τ + 48µ1c τr + 42µ1c τr2 + 18 + 90r ),

f 1rvw dwdv =

which in turn implies ¢ 1 4¡ r 19µ2c τ 2 r2 + 105µ1c τr + 150 180 1 2 − r (r − 1) (2µ2c τ 2 r + 14µ2c τ 2 r2 + 5µ2c τ 2 r3 + 6µ1c τ 72 + 48µ1c τr + 42µ1c τr2 + 18 + 90r ).

A1r1r =

We also have A1r11 =

=

Z 1 Z 1 v =0 Z r

w =0 Z r

v =0

w =0

= A1rrr + 2

f 1rvw dwdv f 1rvw dwdv + 2

Z 1 Z r v =r

w =0

Z 1 Z r v =r

w =0

f 1rvw dwdv + 2

f 1rvw dwdv +

Z 1 Z v v =r

w =r

Z 1 Z 1 v =r

w =r

f 1rvw dwdv

f 1rvw dwdv,

where, for r ≥ v ≥ w ≥ s, ¡ ¢ 1 ¡ ¢ 1 f rsvw = s (v + 2w) + µ1c τs 2w2 + 2vw − sv − µ1c τs w2 − 4wv + sw − v2 2 2 ¡ ¢ 1 − µ2c τ 2 s 2v3 − 12v2 w + 3sv2 − 6vw2 + 3sw2 12 ¡ ¢ 1 = s (v + 2w) + µ1c τs v2 + 6vw − sv + w2 − sw 2 ¡ ¢ 1 − µ2c τ 2 s 2v3 − 12v2 w + 3sv2 − 6vw2 + 3sw2 12 and therefore, with 1 ≥ v ≥ w ≥ r, ¡ ¢ 1 f 1rvw = r (v + 2w) + µ1c τr v2 + 6vw − rv + w2 − rw 2 ¡ ¢ 1 − µ2c τ 2 r 2v3 − 12v2 w + 3rv2 − 6vw2 + 3rw2 , 12 giving Z 1 Z v v =r

w =r

1 r (r − 1)2 (12µ2c τ 2 + 19µ2c τ 2 r + 16µ2c τ 2 r2 + 3µ2c τ 2 r3 120 + 65µ1c τ + 80µ1c τr + 35µ1c τr2 + 80 + 100r ).

f 1rvw dwdv =

20

It follows that A1r11 =

+ + + = − +

¢ 1 4¡ 1 r 19µ2c τ 2 r2 + 105µ1c τr + 150 − r2 (r − 1) (2µ2c τ 2 r + 14µ2c τ 2 r2 180 36 2 3 2 5µ2c τ r + 6µ1c τ + 48µ1c τr + 42µ1c τr + 18 + 90r ) 1 r (r − 1)2 (12µ2c τ 2 + 19µ2c τ 2 r + 16µ2c τ 2 r2 + 3µ2c τ 2 r3 60 65µ1c τ + 80µ1c τr + 35µ1c τr2 + 80 + 100r ) ¢ 1 4¡ r 19µ2c τ 2 r2 + 105µ1c τr + 150 180 1 r (r − 1) (36µ2c τ 2 + 21µ2c τ 2 r + µ2c τ 2 r2 + 31µ2c τ 2 r3 + 16µ2c τ 2 r4 180 195µ1c τ + 75µ1c τr + 105µ1c τr2 + 105µ1c τr3 + 240 + 150r + 150r2 ).

Similarly, A11rr =

Z r Z r v =0

w =0

f 11vw dwdv = 2

Z r Z v v =0

w =0

f 11vw dwdv,

where, for 1 ≥ v ≥ w, ¡ ¢ 1 1 f 11vw = w (1 + 2v) + µ1c τw2 (1 + 2v) + µ1c τw 2v2 + 2v − w 2 2 ¡ ¢ 1 µ2c τ 2 w2 3v2 + 3v − w , + 6 such that

=

µ

¡ ¢ 1 1 w (1 + 2v) + µ1c τw2 (1 + 2v) + µ1c τw 2v2 + 2v − w 2 2 v =0 w =0 ¶ ¡ ¢ 1 µ2c τ 2 w2 3v2 + 3v − w dwdv 6 ¢ 1 3¡ r 9µ2c τ 2 r2 + 10µ2c τ 2 r3 + 45µ1c τr + 60µ1c τr2 + 60 + 90r 180

A11rr = 2

+

Z r Z v

Finally, A111r =

Z 1 Z r v =0

w =0

f 11vw dwdv = A11rr +

Z 1 Z r v =r

w =0

f 11vw dwdv

where Z 1 Z r v =r

w =0

Z 1 Z r

µ

1 w (1 + 2v) + µ1c τw2 (1 + 2v) 2 v =r w =0 ¶ ¡ 2 ¢ 1 ¡ 2 ¢ 1 2 2 + µ1c τw 2v + 2v − w + µ2c τ w 3v + 3v − wv dwdv 2 6 1 2 r (r − 1) (20µ2c τ 2 r + 17µ2c τ 2 r2 + 5µ2c τ 2 r3 + 60µ1c τ = − 144 + 84µ1c τr + 48µ1c τr2 + 144 + 72r ),

f 11vw dwdv =

21

giving ¢ 1 3¡ r 9µ2c τ 2 r2 + 10µ2c τ 2 r3 + 45µ1c τr + 60µ1c τr2 + 60 + 90r 180 1 2 − r (r − 1) (20µ2c τ 2 r + 17µ2c τ 2 r2 + 5µ2c τ 2 r3 + 60µ1c τ + 84µ1c τr 144 + 48µ1c τr2 + 144 + 72r ).

A111r =

By putting everything together, Z 1 0

−2

Z 1 0 1

Z

4

−2

Z 1 0

Z 1

r3 A1r11 dr =

0

r2 A11rr dr =

0 Z 1

r3 A111r dr =

0

Z 1 0

rA1rrr dr =

r2 A1r1r dr =

0 Z 1

Z 1

−2

rArr1r dr =

r2 Arr11 dr =

0

−2

7 1 19 µ2c τ 2 + µ1c τ + , 1260 72 6 401 11 17 − µ2c τ 2 − µ1c τ − , 10 080 45 45 2663 89 31 µ2c τ 2 + µ1c τ + , 90 720 560 140 19 1 5 − µ2c τ 2 − µ1c τ − , 720 6 18 3053 257 22 µ2c τ 2 + µ1c τ + , 45 360 630 35 71 163 11 − µ2c τ 2 − µ1c τ − , 1512 630 30 13 8 161 µ2c τ 2 + µ1c τ + , 12 960 168 63 1889 47 2 − µ2c τ 2 − µ1c τ − , 60 480 252 7 19 7 1 µ2c τ 2 + µ1c τ + , 900 60 6

Arrrr dr =

r4 A1111 dr =

which in turn can be used to obtain " µZ ¶ # Z E U (r )2

r

0

2

=

Z (s) ds

1

0

( Arrrr − 2rArr1r + r2 Arr11 − 2rA1rrr + 4r2 A1r1r − 2r3 A1r11

+ r2 A11rr − 2r3 A111r + r4 A1111 )dr 827 1 1 = µ2c τ 2 + µ1c τ + , 907 200 560 420 or " E c2i U (r )2

µZ

r 0

¶2 # Z (s) ds

=

827 1 1 µ4c τ 2 + µ3c τ + µ2c . 907 200 560 420

This completes the proof of (a).

22

Consider (b). We have µZ r ¶4 µZ r ¶4 µZ r ¶3 Z 1 Z (s) ds = W1 (s) ds − 4r W1 (s) ds W1 (s) ds 0

0

µZ

+ 6r2 − 4r

3

r 0

Z r 0

0

¶2 µ Z W1 (s) ds

0

µZ

W1 (s) ds

1

1 0

0

¶2

W1 (s) ds ¶3

µZ

+r

W1 (s) ds

1

4 0

¶4 W1 (s) ds

,

and therefore, "µZ E

r 0

¶4 # Z (s) ds

with

µZ Brstu = E

r 0

= Brrrr − 4rB1rrr + 6r2 B11rr − 4r3 B111r + r4 B1111 ,

W1 (v) dv

Z s 0

W1 (w) dw

Z t 0

W1 ( x ) dx

Z u 0

¶ W1 (y) dy .

Here is µZ Brstu = E

+ + + + +

r

Z t

¶

Z u

W (w) dw W ( x ) dx W (y) dy 0 0 ¶Z s ¸ Z t Z u µ1c τE W (z) dzdv W (w) dw W ( x ) dx W (y) dy 0 0 0 0 0 ·Z r µZ s Z w ¶Z t ¸ Z u µ1c τE W (v) dv W (z) dzdw W ( x ) dx W (y) dy 0 0 0 0 0 ·Z r µZ t Z x ¶Z u ¸ Z s µ1c τE W (v) dv W (w) dw W (z) dzdx W (y) dy 0 0 0 0 0 ·Z r µ ¶¸ Z s Z t Z uZ y µ1c τE W (v) dv W (w) dw W ( x ) dx W (z) dzdy 0 0 0 0 0 ¡ ¢ O µ2c τ 2 . 0

W (v) dv ·µZ r Z v

Z s 0

Consider the first term in the right-hand side, which we can write as Irstu =

Z r Z s v =0

Z t

w =0

Z u

x =0

y =0

E (W (v) W (w) W ( x ) W (y)) dydxdwdv.

By using the same arguments as before, Irrrr = 4!

Z r Z v

Z w Z x

v =0 Z r

w =0 Z v

x =0 Z w

y =0 Z x

v =0

w =0

x =0

y =0

= 24

E (W (v) W (w) W ( x ) W (y)) dydxdwdv y (w + 2x ) dydxdwdv =

1 6 r , 3

suggesting µZ I1111 = E

1 0

W (v) dv

Z 1 0

W (w) dw 23

Z 1 0

W ( x ) dx

Z 1 0

¶ W (y) dy

1 = . 3

But we also have Z

I1rrr =

= = =

I11rr =

+ = = =

1 6 r +2 3

Z 1 Z v =r

Z

Z

Z

1 r r r 1 6 r + E (W (v) W (w) W ( x ) W (y)) dydxdwdv 3 v = r w =0 x =0 y =0 Z 1 Z r Z w Z x 1 6 r + 3! E (W (v) W (w) W ( x ) W (y)) dydxdwdv 3 v =r w =0 x =0 y =0 Z 1 Z r Z w Z x 1 6 r +6 y (w + 2x ) dydxdwdv 3 v =r w =0 x =0 y =0 1 1 6 1 5 r + r (1 − r ) = r 5 (3 − r ) , 3 2 6

Z 1 Z r

Z r

Z r

v =r w =0 x =0 1 Z r Z r

w =r

x =0

y =0

y =0

E (W (v) W (w) W ( x ) W (y)) dydxdwdv

E (W (v) W (w) W ( x ) W (y)) dydxdwdv Z

Z

Z

Z r

Z r

Z

v r x 1 1 6 E (W (v) W (w) W ( x ) W (y)) dydxdwdv r + r 5 (1 − r ) + 4 3 v =r w =r x =0 y =0 Z 1 Z v Z r Z x 1 6 r + r 5 (1 − r ) + 4 y (w + 2x ) dydxdwdv 3 v =r w =r x =0 y =0 ¢ 1 6 1 1 3¡ r + r5 (1 − r ) + r3 (13r + 2) (1 − r )2 = r 2 + 9r − 6r2 + r3 , 3 18 18

and I111r = Irrrr + 3

+ 3

Z 1 Z 1

=

v =r

=

w =r

x =r

y =0

y =0

E (W (v) W (w) W ( x ) W (y)) dydxdwdv

E (W (v) W (w) W ( x ) W (y)) dydxdwdv

E (W (v) W (w) W ( x ) W (y)) dydxdwdv

Z 1 Z v v =r

Z w Z r

w =r

x =r

y =0

E (W (v) W (w) W ( x ) W (y)) dydxdwdv

1 1 6 3 5 r + r (1 − r ) + r3 (13r + 2) (1 − r )2 3 2 6

+ 6 =

Z

w =0 x =0 r Z r

1 6 3 5 1 r + r (1 − r ) + r3 (13r + 2) (1 − r )2 3 2 6

+ 3! =

v =r

v =r w =r x =0 y =0 1 Z 1 Z 1 Z r

Z

+

Z 1 Z r

Z 1 Z v v =r

w =r

Z w Z r x =r

y =0

y (w + 2x ) dydxdwdv

1 6 3 5 1 1 r + r (1 − r ) + r3 (13r + 2) (1 − r )2 + r2 (2r + 1) (1 − r )3 3 2 6 2 1 2 r (3 − r ) . 6

24

By putting these results together, "µZ ¶4 # r ¢ 1 6 1 1 ¡ E Z (s) ds = r − 4r r5 (3 − r ) + 6r2 r3 2 + 9r − 6r2 + r3 3 6 18 0 1 1 − 4r3 r2 (3 − r ) + r4 + O (µ1c τ ) 6 3 1 4 4 r (1 − r ) + O (µ1c τ ) , = 3 and therefore "µZ ¶4 # Z 1 Z r E Z (s) ds dr = 0

0

1 0

1 4 1 r (1 − r )4 dr + O (µ1c τ ) = + O (µ1c τ ) . 3 1890

We are now going to compute the O (µ1c τ ) term. We begin by considering Brrrr . By using the same technique as in the proof of (a), ·µZ r Z v ¶Z r ¸ Z r Z r E W (z) dzdv W (w) dw W ( x ) dx W (y) dy 0 0 0 0 0 · ¸ Z r Z r Z r Z r Z v = E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =0 x =0 y =0 v =0 z =0 · ¸ Z r Z w Z x Z r Z v = 3! E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =0 x =0 y =0 v =0 z =0 µZ r Z w Z x Z y 1 2 = 6 v ( x + 2y) dvdydxdw w =0 x =0 y =0 v =0 2 Z r Z w Z x Z x ¢ 1 ¡ 2 + y 2v + 2xv − xy dvdydxdw w =0 x =0 y =0 v = y 2 Z r Z w Z x Z w ¢ 1 ¡ + − y x2 − 4xv + yx − v2 dvdydxdw 2 w =0 x =0 y =0 v = x ¶ Z r Z w Z x Z r ¢ 1 ¡ 1 2 2 + − y xy − 2wv − 4xv + w + x dvdydxdw = r7 . 2 8 w =0 x =0 y =0 v = w Hence, because of symmetry, Brrrr =

¡ ¢ 1 6 1 r + µ1c τr7 + O µ2c τ 2 . 3 2

In B1rrr there are two types of terms. The first is ·µZ 1 Z v ¶Z r ¸ Z r Z r E W (z) dzdv W (w) dw W ( x ) dx W (y) dy 0 0 0 0 0 · ¸ Z r Z r Z r Z 1 Z v = E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =0 x =0 y =0 v =0 z =0 · ¸ Z r Z r Z r Z r Z v = E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =0 x =0 y =0 v =0 z =0 · ¸ Z r Z r Z r Z 1 Z v + E W (w) W ( x ) W (y) W (z) dz, dvdydxdw w =0

x =0

y =0

v =r

z =0

25

where the first integral is known, while the second is given by · ¸ Z r Z r Z r Z 1 Z v E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =0 x =0 y =0 v =r z =0 · ¸ Z r Z w Z x Z 1 Z v = 6 E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =0

= 6

Z r

w =0

x =0

y =0

v =r

y =0

¢ 1 ¡ − y xy − 2wv − 4xv + w2 + x2 dvdydxdw 2 v =r

z =0

Z w Z x Z 1 x =0

1 5 r (r + 3) (1 − r ) . 12

= It follows that

·µZ E

1 0

Z v 0

¶Z W (z) dzdv

r 0

W (w) dw

Z r 0

W ( x ) dx

Z r 0

¸ W (y) dy

¢ 1 1 5¡ 1 7 r + r 5 (r + 3) (1 − r ) = r 6 − 4r + r2 . 8 12 24

=

The second type of term in B1rrr is ·µZ r Z v ¶Z 1 ¸ Z r Z r E W (z) dzdv W (w) dw W ( x ) dx W (y) dy 0 0 0 0 0 · ¸ Z 1 Z r Z r Z r Z v = E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =0 x =0 y =0 v =0 z =0 · ¸ Z r Z r Z r Z r Z v = E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =0 x =0 y =0 v =0 z =0 · ¸ Z 1 Z r Z r Z r Z v + E W (w) W ( x ) W (y) W (z) dz dvdydxdw, w =r

x =0

y =0

v =0

Z r Z r

·

z =0

where Z 1

Z r

Z v

¸

E W (w) W ( x ) W (y) W (z) dz dvdydxdw z =0 · ¸ Z v = 2 E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =r x =0 y =0 v =0 z =0 · ¸ Z 1 Z r Z x Z v Z v + 2 E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =r x =0 v =0 y =0 z =0 · ¸ Z 1 Z r Z v Z x Z v + 2 E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =r Z 1

w =r

Z 1

x =0 Z r

v =0

Z r

y =0 v =0 Z x Z y

x =0

y =0

z =0

Z x Z y

1 2 v ( x + 2y) dvdydxdw 2 w =r x =0 y =0 v =0 Z 1 Z r Z x Z v ¢ 1 ¡ 2 + 2 y 2v + 2xv − xy dydvdxdw w =r x =0 v =0 y =0 2 Z 1 Z r Z v Z x ¢ 1 ¡ 13 + 2 − y x2 − 4xv + yx − v2 dydxdvdw = r6 (1 − r ) , 2 72 w =r v =0 x =0 y =0

= 2

26

suggesting E

=

·µZ r Z 0

v 0

¶Z W (z) dzdv

1 0

W (w) dw

Z r 0

W ( x ) dx

Z r 0

¸ W (y) dy

1 7 13 6 1 6 r + r (1 − r ) = r (13 − 4r ) , 8 72 72

which can be used to obtain B1rrr =

=

µ ¶ ¢ ¡ ¢ 1 5 1 5¡ 1 6 2 r (3 − r ) + µ1c τ r 6 − 4r + r + r (13 − 4r ) + O µ2c τ 2 6 24 24 ¡ ¢ ¡ ¢ 1 5 1 r (3 − r ) + µ1c τr5 2 + 3r − r2 + O µ2c τ 2 . 6 8

In B11rr there are also two types of terms, among which the first is given by ·µZ 1 Z v ¶Z 1 ¸ Z r Z r E W (z) dzdv W (w) dw W ( x ) dx W (y) dy 0 0 0 0 0 · ¸ Z 1 Z r Z r Z 1 Z v = E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =0 x =0 y =0 v =0 z =0 · ¸ Z r Z r Z r Z r Z v = E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =0 x =0 y =0 v =0 z =0 · ¸ Z 1 Z r Z r Z r Z v + E W (w) W ( x ) W (y) W (z) dz dvdydxdw w = r x =0 y =0 v =0 z =0 · ¸ Z r Z r Z r Z 1 Z v + E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =0 x =0 y =0 v = r z =0 · ¸ Z 1 Z r Z r Z 1 Z v + E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =r

x =0

y =0

v =r

z =0

where all integrals are known, except the last one, which is given by · ¸ Z 1 Z r Z r Z 1 Z v E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =r x =0 y =0 v = r z =0 · ¸ Z 1 Z r Z x Z w Z v = 2 E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =r x =0 y =0 v = r z =0 · ¸ Z 1 Z r Z x Z 1 Z v + 2 E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =r

Z 1

x =0

Z r

y =0

v=w

z =0

Z x Z w

¢ 1 ¡ = 2 − y x2 − 4xv + yx − v2 dvdydxdw 2 w =r x =0 y =0 v = r Z 1 Z r Z x Z 1 ¢ 1 ¡ + 2 − y xy − 2wv − 4xv + w2 + x2 dvdydxdw 2 w =r x =0 y =0 v = w ¢ 1 3¡ = r 1 + 8r + 3r2 (1 − r )2 , 24

27

such that ·µZ E

= =

1 0

Z v 0

¶Z W (z) dzdv

1 0

W (w) dw

Z r 0

W ( x ) dx

Z r 0

¸ W (y) dy

¢ 1 7 13 6 1 1 ¡ r + r (1 − r ) + r5 (r + 3) (1 − r ) + r3 1 + 8r + 3r2 (1 − r )2 8 72 12 24 ´ 1 3³ r 3 + 18r − 18r2 + 7r3 − r4 . 72

The second type of term is given by ·µZ r Z v ¶Z 1 ¸ Z 1 Z r E W (z) dzdv W (w) dw W ( x ) dx W (y) dy 0 0 0 0 0 · ¸ Z 1 Z 1 Z r Z r Z v = E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =0 x =0 y =0 v =0 z =0 · ¸ Z r Z r Z r Z r Z v = E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =0 x =0 y =0 v =0 z =0 · ¸ Z 1 Z r Z r Z r Z v + 2 E W (w) W ( x ) W (y) W (z) dz dvdydxdw w = r x =0 y =0 v =0 z =0 · ¸ Z 1 Z 1 Z r Z r Z v + E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =r

x =r

y =0

v =0

z =0

where Z 1

·

Z 1 Z r Z r

¸

Z v

E W (w) W ( x ) W (y) W (z) dz dvdydxdw z =0 · ¸ Z v = 2 E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =r x =r y =0 v =0 z =0 · ¸ Z v Z 1 Z w Z r Z r E W (w) W ( x ) W (y) W (z) dz dvdydxdw + 2 w =r Z 1

w =r

= 2 + 2

Z 1

w =r

Z 1

w =r

x =r y =0 Z w Z r

x =r

y =0

v =0 Z y

v=y

Z w Z r Z y x =r

y =0

v =0

x =r

y =0

v=y

Z w Z r Z r

z =0

1 2 v ( x + 2y) dvdydxdw 2 ¢ 1 ¡ 2 y 2v + 2xv − xy dvdydxdw 2

1 4 r (6r + 1) (1 − r )2 24

= which yields

E

= =

·µZ r Z 0

v 0

¶Z W (z) dzdv

1 0

W (w) dw

Z 1 0

W ( x ) dx

1 7 13 6 1 r + r (1 − r ) + r4 (6r + 1) (1 − r )2 8 36 24 ¢ 1 4¡ 2 r 3 + 12r − 7r + r3 , 72

28

Z r 0

¸ W (y) dy

and therefore ¢ 1 3¡ r 2 + 9r − 6r2 + r3 18 µ ¶ ´ ¢ ¡ ¢ 1 3³ 1 4¡ 2 3 4 2 3 + O µ2c τ 2 + 2µ1c τ r 3 + 18r − 18r + 7r − r + r 3 + 12r − 7r + r 72 72 ¡ ¢ ¡ ¢ ¡ ¢ 1 3 1 = r 2 + 9r − 6r2 + r3 + µ1c τr3 1 + 7r − 2r2 + O µ2c τ 2 . 18 12

B11rr =

The first type of term in B111r is ·µZ 1 Z v ¶Z 1 ¸ Z 1 Z r E W (z) dzdv W (w) dw W ( x ) dx W (y) dy 0 0 0 0 0 · ¸ Z 1 Z 1 Z r Z 1 Z v = E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =0 x =0 y =0 v =0 z =0 · ¸ Z v Z r Z r Z r Z r = E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =0 x =0 y =0 v =0 z =0 · ¸ Z 1 Z r Z r Z r Z v + 2 E W (w) W ( x ) W (y) W (z) dz dvdydxdw w = r x =0 y =0 v =0 z =0 · ¸ Z r Z r Z r Z 1 Z v + E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =0 x =0 y =0 v =r z =0 · ¸ Z 1 Z 1 Z r Z r Z v + E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =r x = r y =0 v =0 z =0 · ¸ Z 1 Z r Z r Z 1 Z v + 2 E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =r x =0 y =0 v =r z =0 · ¸ Z 1 Z 1 Z r Z 1 Z v + E W (w) W ( x ) W (y) W (z) dz dvdydxdw, w =r

x =r

y =0

v =r

where Z 1

Z 1 Z r Z 1

z =0

·

Z v

¸

E W (w) W ( x ) W (y) W (z) dz dvdydxdw z =0 · ¸ Z v = 2 E W (w) W ( x ) W (y) W (z) dz dydvdxdw w =r x =r v = r y =0 z =0 · ¸ Z 1 Z w Z v Z r Z v + 2 E W (w) W ( x ) W (y) W (z) dz dydxdvdw w =r v = x x = r y =0 z =0 · ¸ Z 1 Z v Z w Z r Z v + 2 E W (w) W ( x ) W (y) W (z) dz dydxdwdv w =r Z 1

v =r

Z 1

x =r y =0 v =r Z w Z x Z r

w =r

x =r

y =0

z =0

Z w Z x Z r

¢ 1 ¡ 2 y 2v + 2xv − xy dydvdxdw w =r x =r v = r y =0 2 Z 1 Z w Z v Z r ¢ 1 ¡ + 2 − y x2 − 4xv + yx − v2 dydxdvdw 2 w =r v = r x = r y =0 Z 1 Z v Z w Z r ¢ 1 ¡ + 2 − y xy − 2wv − 4xv + w2 + x2 dydxdwdv 2 v =r w = r x = r y =0 ¢ 1 2¡ = r 15 + 32r + 13r2 (1 − r )3 , 72

= 2

29

which yields ·µZ E

1

Z v

0

0

¶Z W (z) dzdv

1 0

W (w) dw

Z 1 0

W ( x ) dx

Z r 0

¸ W (y) dy

1 7 13 6 1 1 r + r (1 − r ) + r5 (r + 3) (1 − r ) + r4 (6r + 1) (1 − r )2 8 36 12 24 ¢ ¢ 1 3¡ 1 ¡ r 1 + 8r + 3r2 (1 − r )2 + r2 15 + 32r + 13r2 (1 − r )3 12 72 ¢ 1 2¡ r 15 − 7r + r2 . 72

= + =

The second type of term is ·µZ r Z v ¶Z 1 ¸ Z 1 Z 1 E W (z) dzdv W (w) dw W ( x ) dx W (y) dy 0 0 0 0 0 · ¸ Z 1 Z 1 Z 1 Z r Z v = E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =0 x =0 y =0 v =0 z =0 · ¸ Z r Z r Z r Z r Z v = E W (w) W ( x ) W (y) W (z) dz dvdydxdw w =0 x =0 y =0 v =0 z =0 · ¸ Z 1 Z r Z r Z r Z v + 3 E W (w) W ( x ) W (y) W (z) dz dvdydxdw w = r x =0 y =0 v =0 z =0 · ¸ Z 1 Z 1 Z r Z r Z v + 3 E W (w) W ( x ) W (y) W (z) dz dvdydxdw w = r x = r y =0 v =0 z =0 · ¸ Z 1 Z 1 Z 1 Z r Z v + E W (w) W ( x ) W (y) W (z) dz dvdydxdw, w =r

x =r

y =r

v =0

z =0

where Z 1

= 6 = 6 =

Z 1 Z 1 Z r

w =r Z 1 w =r

Z 1

w =r

x =r y = r v =0 Z w Z x Z r x =r

y =r

·

E W (w) W ( x ) W (y) W (z) dz dvdydxdw z =0 · ¸ Z v E W (w) W ( x ) W (y) W (z) dz dvdydxdw

v =0

Z w Z x Z r x =r

y =r

¸

Z v

v =0

z =0

1 2 v ( x + 2y) dvdydxdw 2

1 3 r (2r + 1) (1 − r )3 6

implying E

= =

·µZ r Z 0

v 0

¶Z W (z) dzdv

1 0

W (w) dw

Z 1 0

W ( x ) dx

Z 1 0

¸ W (y) dy

1 1 1 7 13 6 r + r (1 − r ) + r4 (6r + 1) (1 − r )2 + r3 (2r + 1) (1 − r )3 8 24 8 6 1 3 r (4 − r ) , 24

30

which can be added to the first type of term, giving µ ¶ ¢ ¡ ¢ 1 2 1 2¡ 1 3 2 B111r = r (3 − r ) + µ1c τ r 15 − 7r + r + r (4 − r ) + O µ2c τ 2 6 24 24 ¡ ¢ 1 2 1 = r (3 − r ) + µ1c τr2 (5 − r ) + O µ2c τ 2 . 6 8 Thus, since B1111 = we obtain "µZ E

r

0

¶4 #

¡ ¢ 1 1 + µ1c τ + O µ2c τ 2 , 3 2

µ ¶ ¡ ¢ 1 6 1 1 5 1 r + µ1c τr7 − 4r r (3 − r ) + µ1c τr5 2 + 3r − r2 3 2 6 8 µ ¶ ¢ ¡ ¢ 1 3¡ 1 2 2 3 3 2 r 2 + 9r − 6r + r + µ1c τr 1 + 7r − 2r + 6r 18 12 ¶ µ ¶ µ ¡ ¢ 1 1 1 1 2 r (3 − r ) + µ1c τr2 (5 − r ) + r4 + µ1c τ + O µ2c τ 2 − 4r3 6 8 3 2 ¡ ¢ 1 4 = r (1 − r )4 (2 + 3µ1c τ ) + O µ2c τ 2 , 6

=

Z (s) ds

which in turn implies Z 1 0

or

Z 1 0

"µZ E

"

0

µZ c4i

E

r

r 0

¶4 # Z (s) ds

dr =

¡ ¢ 1 (2 + 3µ1c τ ) + O µ2c τ 2 , 3780

dr =

¡ ¢ 1 1 µ4c + µ5c τ + O µ6c τ 2 . 1890 1260

¶4 # Z (s) ds

This establishes (b). Finally, consider (c). From µZ

r 0

¶3 Z (s) ds

µZ

=

r 0

¶3 W1 (s) ds

µZ

+ 3r

r

2 0

µZ

− 3r

0

¶ µZ W1 (s) ds

1 0

31

r

¶2 Z W1 (s) ds

W1 (s) ds

1 0

¶2

−r

W1 (s) ds

µZ

1

3 0

¶3 W1 (s) ds

,

we get " E U (r )

µZ

r 0

¶3 #

"

µZ

= E (W (r ) − rW (1))

Z (s) ds

"

µZ

= E W (r ) − 3rE W (r )

W1 (s) ds

0

µZ

+ 3r E W (r )

r 0

µZ

− r 3 E W (r ) "

¶2 Z

r

2

"

1 0

µZ

− rE W (1) + 3r2 E W (1)

µZ

"

r 0

µZ

+ r E W (1)

1 0

W1 (s) ds

1 0

¶2 # W1 (s) ds

W1 (s) ds

r

3

4

¶ µZ ¶3 #

0

− 3r E W (1)

W1 (s) ds

0

W1 (s) ds

µZ

"

#

1

¶3 #

r 0

"

Z (s) ds

W1 (s) ds

µZ

"

0

¶3 #

¶3 #

r 0

"

r

¶2 Z W1 (s) ds

1 0

¶ µZ W1 (s) ds ¶3 #

# W1 (s) ds

1 0

¶2 # W1 (s) ds

W1 (s) ds

= Crrrr − 3rCr1rr + 3r2 Cr11r − r3 Cr111 − rC1rrr + 3r2 C11rr − 3r3 C111r + r4 C1111 , where µ ¶ Z s Z t Z u Crstu = E W (r ) W (w) dw W ( x ) dx W (y) dy 0 0 0 · µZ s Z w ¶Z t ¸ Z u + µ1c τE W (r ) W (z) dzdw W ( x ) dx W (y) dy 0 0 0 0 · µZ t Z x ¶Z u ¸ Z s + µ1c τE W (r ) W (w) dw W (z) dzdx W (y) dy 0 0 0 0 · µ ¶¸ Z s Z t Z uZ y ¡ ¢ + µ1c τE W (r ) W (w) dw W ( x ) dx W (z) dzdy + O µ2c τ 2 . 0

0

0

32

0

Note that µ E W (r )

= 3!

Z r

implying

W (w) dw

0

Z w Z x

w =0 x =0 y =0 r Z w Z x

Z

= 6

Z r

w =0

x =0

y =0

Z r 0

Z r

W ( x ) dx

0

¶ W (y) dy

E (W (r ) W (w) W ( x ) W (y)) dydxdw 1 5 r , 2

y (w + 2x ) dydxdw =

¶ µ Z 1 Z 1 Z 1 1 E W (1) W (w) dw W ( x ) dx W (y) dy = . 2 0 0 0

Similarly, µ E W (1)

= 3!

Z r

Z

= 6

Z r 0

W (w) dw

Z w Z x

w =0 x =0 y =0 r Z w Z x w =0

x =0

y =0

Z r 0

Z r

W ( x ) dx

0

¶ W (y) dy

E (W (1) W (w) W ( x ) W (y)) dydxdw

y (w + 2x ) dydxdw =

1 5 r . 2

Moreover, by using Z 1

= 2 = 2

= 2

= 2

E (W (r ) W (w) W ( x ) W (y)) dydxdw

x =0 Z r

y =0 Z x

w =r Z 1

x =0 Z r

y =0 Z x

w =r

x =0

y =0

Z r

Z r

x =0 Z r

y =0 Z x

w =r Z 1

x =0 Z r

y =0 Z x

w =r

x =0

y =0

Z 1 Z r

E (W (w) W (r ) W ( x ) W (y)) dydxdw y (r + 2x ) dydxdw =

5 4 r (1 − r ) , 6

E (W (1) W (w) W ( x ) W (y)) dydxdw

w =r Z 1

Z 1

= 2

Z r

w =r Z 1

Z 1

= 2

Z r

E (W (1) W (w) W ( x ) W (y)) dydxdw y (w + 2x ) dydxdw =

1 3 r (4r + 1) (1 − r ) , 6

E (W (r ) W (w) W ( x ) W (y)) dydxdw

w =r Z 1

x =r y =0 Z w Z r

w =r Z 1

x =r Z w

y =0 Z r

w =r

x =r

y =0

E (W (w) W ( x ) W (r ) W (y)) dydxdw y ( x + 2r ) dydxdw =

33

1 2 r (8r + 1) (1 − r )2 , 6

Z 1

= 2

Z 1 Z r

E (W (1) W (w) W ( x ) W (y)) dydxdw

w =r Z 1

x = r y =0 Z w Z r

w =r Z 1

x =r Z w

y =0 Z r

w =r

x =r

y =0

= 2

E (W (1) W (w) W ( x ) W (y)) dydxdw y (w + 2x ) dydxdw =

1 2 r (5r + 4) (1 − r )2 , 6

and Z 1

Z 1 Z 1

E (W (r ) W (w) W ( x ) W (y)) dydxdw

w =r x =r y =r Z 1 Z w Z x

= 3!

Z

= 6

w =r x =r y =r 1 Z w Z x w =r

x =r

y =r

E (W (w) W ( x ) W (y) W (r )) dydxdw

r ( x + 2y) dydxdw = r (2r + 1) (1 − r )3

we obtain µ ¶ Z 1 Z r Z r E W (r ) W (w) dw W ( x ) dx W (y) dy

= + =

Z r

Z r

w =r

x =0

=

y =0

0

E (W (r ) W (w) W ( x ) W (y)) dydxdw E (W (r ) W (w) W ( x ) W (y)) dydxdw

1 5 5 4 1 r + r (1 − r ) = r4 (5 − 2r ) , 2 6 6

E W (1)

+

0

w =0 x =0 y =0 Z 1 Z r Z r

µ

=

0

Z r

Z r

Z r

Z 1 0

W (w) dw

Z r

w =0 x =0 y =0 Z 1 Z r Z r w =r

x =0

y =0

Z r 0

W ( x ) dx

Z r 0

¶ W (y) dy

E (W (1) W (w) W ( x ) W (y)) dydxdw E (W (1) W (w) W ( x ) W (y)) dydxdw

¢ 1 5 1 3 1 ¡ r + r (4r + 1) (1 − r ) = r3 1 + 3r − r2 , 2 6 6

34

µ

Z 1

E W (r ) Z r

=

+ 2 Z

+

w =0 Z 1

E W (1)

+ =

Z r

x =0 Z r

y =0 Z r

x =r

y =0

0

W ( x ) dx

0

¶ W (y) dy

E (W (r ) W (w) W ( x ) W (y)) dydxdw

w =r x =0 y =0 1 Z 1 Z r

µ Z r

Z

W (w) dw

Z r

E (W (r ) W (w) W ( x ) W (y)) dydxdw

E (W (r ) W (w) W ( x ) W (y)) dydxdw

1 1 5 5 4 r + r (1 − r ) + r2 (8r + 1) (1 − r )2 2 3 6 ¢ 1 2¡ 2 r 1 + 6r − 5r + r3 , 6

=

+ 2

Z r

w =r

=

=

0

Z 1

Z r

w =0 Z 1

Z 1 0

W (w) dw

Z r

x =0 Z r

y =0 Z r

x =r

y =0

0

W ( x ) dx

Z r 0

¶ W (y) dy

E (W (1) W (w) W ( x ) W (y)) dydxdw

w = r x =0 y =0 1 Z 1 Z r w =r

Z 1

E (W (1) W (w) W ( x ) W (y)) dydxdw

E (W (1) W (w) W ( x ) W (y)) dydxdw

1 1 1 5 1 3 r + r (4r + 1) (1 − r ) + r2 (5r + 4) (1 − r )2 = r2 (4 − r ) . 2 3 6 6

and µ E W (r )

=

Z r

+ 3 + 3 Z

+ = =

Z r

Z 1 0

W (w) dw

Z r y =0 Z r

x =0 Z r

w =r Z 1

x =0 y =0 Z 1 Z r

w =r x =r y =0 1 Z 1 Z 1 x =r

y =r

0

W ( x ) dx

Z 1 0

¶ W (y) dy

E (W (r ) W (w) W ( x ) W (y)) dydxdw

w =0 Z 1

w =r

Z 1

E (W (r ) W (w) W ( x ) W (y)) dydxdw E (W (r ) W (w) W ( x ) W (y)) dydxdw

E (W (r ) W (w) W ( x ) W (y)) dydxdw

1 5 5 4 1 r + r (1 − r ) + r2 (8r + 1) (1 − r )2 + r (2r + 1) (1 − r )3 2 2 2 1 r (2 − r ) . 2

35

Thus, by putting everything together, " µZ r ¶3 # ¢ 1 5 1 1 ¡ E U (r ) Z (s) ds r − 3r r4 (5 − 2r ) + 3r2 r2 1 + 6r − 5r2 + r3 = 2 6 6 0 ¢ 1 1 1 ¡ − r3 r (2 − r ) − r r5 + 3r2 r3 1 + 3r − r2 2 2 6 1 1 − 3r3 r2 (4 − r ) + r4 + O (µ1c τ ) 6 2 = O (µ1c τ ) . Consider the O (µ1c τ ) term. We have ¶Z r ¸ · µZ r Z w Z r W ( x ) dx W (y) dy E W (r ) W (z) dzdw 0 0 0 0 · ¸ Z r Z r Z r Z w = E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z r Z x Z r Z w = 2 E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 µZ r Z x Z y 1 2 = 2 w ( x + 2y) dwdydx x =0 y =0 w =0 2 Z r Z x Z x ¢ 1 ¡ 2 + y 2w + 2xw − xy dwdydx x =0 y =0 w = y 2 ¶ Z r Z x Z r ¢ 1 ¡ 2 13 6 2 − y x − 4xw + yx − w dwdydx = r , 72 x =0 y =0 w = x 2 which implies Crrrr =

¡ ¢ 1 5 13 r + µ1c τr6 + O µ2c τ 2 , 2 24

and therefore, C1111 =

¡ ¢ 1 13 + µ1c τ + O µ2c τ 2 . 2 24

Similarly, · µZ r Z E W (1)

¸ W (y) dy 0 0 0 0 · ¸ Z r Z r Z r Z w = E W (1) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z r Z x Z r Z w 13 6 = 2 E W (1) W ( x ) W ( y ) W (z) dz dwdydx = r , 72 x =0 y =0 w =0 0 w

¶Z

W (z) dzdw

giving C1rrr =

r

W ( x ) dx

Z r

¡ ¢ 1 5 13 r + µ1c τr6 + O µ2c τ 2 . 2 24

36

Let us now consider Cr1rr , which contains two types of terms. The first one is · µZ 1 Z w ¶Z r ¸ Z r E W (r ) W (z) dzdw W ( x ) dx W (y) dy 0 0 0 0 · ¸ Z r Z r Z 1 Z w = E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z r Z r Z r Z w = E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z r Z r Z 1 Z w + E W (r ) W ( x ) W ( y ) W (z) dz dwdydx, x =0

y =0

w =r

0

where Z r

= 2 = 2 =

·

Z r Z 1

x =0 Z r x =0

Z r

x =0

y =0 w =r Z x Z 1

¸

Z w

E W (r ) W ( x ) W ( y ) W (z) dz dwdydx 0 · ¸ Z w E W (r ) W ( x ) W ( y ) W (z) dz dwdydx

y =0

w =r

y =0

¢ 1 ¡ − y xy − 2rw − 4xw + r2 + x2 dwdydx 2 w =r

0

Z x Z 1

1 4 r (r + 5) (1 − r ) , 12

such that ·

µZ

E W (r )

=

1

Z w

0

0

¶Z W (z) dzdw

r 0

W ( x ) dx

Z r 0

¸ W (y) dy

¢ 1 1 4¡ 13 6 r + r 4 (r + 5) (1 − r ) = r 30 − 24r + 7r2 . 72 12 72

The second type of term is · µZ r Z E W (r )

w

¶Z

x =r

y =0

0

·

w =0

0

W ( x ) dx

Z r

¸

W (y) dy ¸ Z 1 Z r Z Z w = E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z r Z r Z r Z w = E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z 1 Z r Z r Z w + E W (r ) W ( x ) W ( y ) W (z) dz dwdydx, 0 r

W (z) dzdw

1

0

37

0

where ·

Z 1 Z r Z r

¸

Z w

E W (r ) W ( x ) W ( y ) W (z) dz dwdydx 0 · ¸ Z w = E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x = r y =0 w =0 0 · ¸ Z w Z 1 Z r Z r E W (r ) W ( x ) W ( y ) W (z) dz dwdydx +

= +

x =r Z 1

y =0 Z r

w =0 Z y

x =r

y =0

w=y

Z 1 Z r Z y x =r

y =0

w =0

x =r

y =0

w=y

Z 1 Z r Z r

0

1 2 w (r + 2y) dwdydx 2 ¢ 7 5 1 ¡ 2 y 2w + 2rw − ry dwdydx = r (1 − r ) , 2 24

suggesting · E W (r )

µZ r Z 0

w 0

¶Z W (z) dzdw

1 0

W ( x ) dx

Z r 0

¸ W (y) dy

7 1 5 13 6 r + r 5 (1 − r ) = r (21 − 8r ) . 72 24 72

= We therefore obtain Cr1rr =

=

µ ¶ ¢ ¡ ¢ 1 4¡ 1 5 1 4 2 r (5 − 2r ) + µ1c τ r 30 − 24r + 7r + r (21 − 8r ) + O µ2c τ 2 6 72 36 ¡ ¢ ¡ ¢ 1 4 1 r (5 − 2r ) + µ1c τr4 10 + 6r − 3r2 + O µ2c τ 2 . 6 24

As for C11rr , we have · µZ E W (1)

1

Z w

¶Z

x =0

y =0

0

·

w =r

0

W ( x ) dx

¸

Z r

W (y) dy ¸ Z r Z r Z Z w = E W (1) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z r Z r Z r Z w = E W (1) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z r Z r Z 1 Z w + E W (1) W ( x ) W ( y ) W (z) dz dwdydx, 0 1

W (z) dzdw

r

0

0

where Z r

= 2 = 2 =

Z r Z 1

x =0 Z r x =0

Z r

x =0

y =0 w =r Z x Z 1

·

Z w

¸

E W (1) W ( x ) W ( y ) W (z) dz dwdydx 0 · ¸ Z w E W (1) W ( x ) W ( y ) W (z) dz dwdydx

y =0

w =r

y =0

¢ 1 ¡ − y x2 − 4xw + yx − w2 dwdydx 2 w =r

0

Z x Z 1

1 3 r (1 − r ) (5r + 1) (r + 2) , 36 38

suggesting ·

µZ

E W (1)

=

1 0

Z w 0

¶Z W (z) dzdw

r 0

W ( x ) dx

Z r 0

¸ W (y) dy

¢ 13 6 1 1 3¡ r + r3 (1 − r ) (5r + 1) (r + 2) = r 4 + 18r − 12r2 + 3r3 . 72 36 72

We also have ·

µZ r Z

w

¶Z

x =r

0 r

y =0

0

0

W ( x ) dx

¸

Z r

W (y) dy · ¸ Z 1 Z r Z Z w = E W (1) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z r Z r Z r Z w = E W (1) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z 1 Z r Z r Z w + E W (1) W ( x ) W ( y ) W (z) dz dwdydx, E W (1)

W (z) dzdw

1

w =0

0

0

with ·

Z 1 Z r Z r

¸

Z w

E W (1) W ( x ) W ( y ) W (z) dz dwdydx 0 · ¸ Z w = E W (1) W ( x ) W ( y ) W (z) dz dwdydx x =r y =0 w =0 0 · ¸ Z 1 Z r Z r Z w + E W (1) W ( x ) W ( y ) W (z) dz dwdydx x =r Z 1

y =0 Z r

w =0 Z y

x =r

y =0

w=y

Z 1 Z r Z y

=

x =r

y =0

w =0

x =r

y =0

w=y

Z 1 Z r Z r

+

0

1 2 w ( x + 2y) dwdydx 2 ¢ 1 ¡ 2 1 4 y 2w + 2xw − xy dwdydx = r (11r + 3) (1 − r ) , 2 48

which implies · E W (1)

=

µZ r Z 0

w 0

¶Z W (z) dzdw

1 0

W ( x ) dx

Z r 0

¸ W (y) dy

¢ 1 1 4¡ 13 6 r + r4 (11r + 3) (1 − r ) = r 9 + 24r − 7r2 . 72 48 144

By combining these last two results, C11rr =

=

µ ¶ ¢ ¢ ¢ 1 3¡ 1 ¡ 1 3¡ r 1 + 3r − r2 + µ1c τ r 4 + 18r − 12r2 + 3r3 + r4 9 + 24r − 7r2 6 72 72 ¡ ¢ ¡ ¢ ¡ ¢ 1 3 1 r 1 + 3r − r2 + µ1c τr3 4 + 27r + 12r2 − 4r3 + O µ2c τ 2 . 6 72

39

As for Cr11r , we have the terms · µZ 1 Z E W (r )

= = + + +

w

¶Z

x =r

y =0

0

0

w =r

W ( x ) dx

¸

Z r

W (y) dy · ¸ Z 1 Z r Z Z w E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z r Z r Z r Z w E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z 1 Z r Z r Z w E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x = r y =0 w =0 0 · ¸ Z w Z r Z r Z 1 E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =r 0 · ¸ Z 1 Z r Z 1 Z w E W (r ) W ( x ) W ( y ) W (z) dz dwdydx, 0 1

W (z) dzdw

1

0

0

where ·

Z 1 Z r Z 1

¸

Z w

E W (r ) W ( x ) W ( y ) W (z) dz dwdydx 0 · ¸ Z w = E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =r y =0 w =r 0 · ¸ Z 1 Z r Z 1 Z w + E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =r Z 1

y =0 Z r

w =r Z x

x =r

y =0

w= x

0

Z 1 Z r Z x

¢ 1 ¡ = − y r2 − 4rw + yr − w2 dwdydx 2 x =r y =0 w =r Z 1 Z r Z 1 ¢ 1 ¡ + − y ry − 2xw − 4rw + r2 + x2 dwdydx 2 x =r y =0 w = x ¡ ¢ 1 2 = r 3 + 30r + 7r2 (1 − r )2 , 48 giving · E W (r )

= + =

µZ

1 0

Z w 0

¶Z W (z) dzdw

1 0

W ( x ) dx

Z r 0

¸ W (y) dy

1 1 13 6 r + r4 (11r + 3) (1 − r ) + r3 (1 − r ) (5r + 1) (r + 2) 72 48 36 ¢ 1 2¡ 2 r 3 + 30r + 7r2 (1 − r ) 48 ´ 1 2³ r 9 + 80r − 105r2 + 48r3 − 6r4 , 144

40

and · E W (r )

= = + +

µZ r Z

w

¶Z

x =r

y =r

0

0

W ( x ) dx

¸

Z 1

W (y) dy · ¸ Z 1 Z 1 Z Z w E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z r Z r Z r Z w E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z 1 Z r Z r Z w 2 E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =r y =0 w =0 0 · ¸ Z w Z 1 Z 1 Z r E W (r ) W ( x ) W ( y ) W (z) dz dwdydx 0 r

W (z) dzdw

1

w =0

0

0

where Z 1 Z 1 Z r

= 2 = 2

x =r y =r w =0 Z 1 Z x Z r x =r

y =r

·

y =r

¸

E W (r ) W ( x ) W ( y ) W (z) dz dwdydx 0 · ¸ Z w E W (r ) W ( x ) W ( y ) W (z) dz dwdydx

w =0

0

w =0

1 3 1 2 w (y + 2r ) dwdydx = r (8r + 1) (1 − r )2 2 18

Z 1 Z x Z r x =r

Z w

such that · E W (r )

=

µZ r Z 0

w 0

¶Z W (z) dzdw

1 0

W ( x ) dx

Z 1 0

¸ W (y) dy

¢ 7 1 1 3¡ 13 6 r + r5 (1 − r ) + r3 (8r + 1) (1 − r )2 = r 4 + 24r − 18r2 + 3r3 , 72 12 18 72

which yields Cr11r =

+ =

µ ´ ¢ 1 2¡ 1 2³ 2 3 r 1 + 6r − 5r + r + µ1c τ r 9 + 80r − 105r2 + 48r3 − 6r4 6 72 ¶ ¡ ¢ ¡ ¢ 1 3 r 4 + 24r − 18r2 + 3r3 + O µ2c τ 2 72 ³ ´ ¢ ¡ ¢ 1 2¡ 1 r 1 + 6r − 5r2 + r3 + µ1c τr2 3 + 28r − 27r2 + 10r3 − r4 + O µ2c τ 2 . 6 24

41

Consider C111r , where the first type of term is · µZ 1 Z w ¶Z 1 ¸ Z r E W (1) W (z) dzdw W ( x ) dx W (y) dy 0 0 0 0 · ¸ Z 1 Z r Z 1 Z w = E W (1) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z r Z r Z r Z w = E W (1) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z 1 Z r Z r Z w + E W (1) W ( x ) W ( y ) W (z) dz dwdydx x =r y =0 w =0 0 · ¸ Z w Z r Z r Z 1 E W (1) W ( x ) W ( y ) W (z) dz dwdydx + x =0 y =0 w =r 0 · ¸ Z 1 Z r Z 1 Z w + E W (1) W ( x ) W ( y ) W (z) dz dwdydx, x =r

y =0

w =r

0

which with ·

Z 1 Z r Z 1

¸

Z w

E W (1) W ( x ) W ( y ) W (z) dz dwdydx 0 · ¸ Z w = E W (1) W ( x ) W ( y ) W (z) dz dwdydx x =r y =0 w =r 0 · ¸ Z 1 Z r Z 1 Z w + E W (1) W ( x ) W ( y ) W (z) dz dwdydx x =r Z 1

y =0 Z r

w =r Z x

x =r

y =0

w= x

0

Z 1 Z r Z x

¢ 1 ¡ 2 = y 2w + 2xw − xy dwdydx x =r y =0 w =r 2 Z 1 Z r Z 1 ¢ 1 ¡ + − y x2 − 4xw + yx − w2 dwdydx 2 x =r y =0 w = x ¡ ¢ 1 2 = r 13 + 18r + 9r2 (1 − r )2 , 48 gives · E W (1)

= +

µZ

1 0

Z w 0

¶Z W (z) dzdw

1 0

W ( x ) dx

Z r 0

¸ W (y) dy

1 1 13 6 r + r4 (11r + 3) (1 − r ) + r3 (1 − r ) (5r + 1) (r + 2) 72 48 36 ¢ ¢ 1 2¡ 1 2¡ 2 r 13 + 18r + 9r2 (1 − r ) = r 39 − 16r + 3r2 . 48 144

42

The second type of term is · µZ r Z E W (1)

w

¶Z

= = + +

x =r

y =r

0

0

W ( x ) dx

¸

Z 1

W (y) dy · ¸ Z 1 Z 1 Z Z w E W (1) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z r Z r Z r Z w E W (1) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z 1 Z r Z r Z w 2 E W (1) W ( x ) W ( y ) W (z) dz dwdydx x =r y =0 w =0 0 · ¸ Z w Z 1 Z 1 Z r E W (1) W ( x ) W ( y ) W (z) dz dwdydx 0 r

W (z) dzdw

1

w =0

0

0

where Z 1 Z 1 Z r

= 2

x =r y =r w =0 Z 1 Z x Z r

= 2

x =r

y =r

·

y =r

¸

E W (1) W ( x ) W ( y ) W (z) dz dwdydx 0 · ¸ Z w E W (1) W ( x ) W ( y ) W (z) dz dwdydx

w =0

0

w =0

1 3 1 2 w ( x + 2y) dwdydx = r (5r + 4) (1 − r )2 , 2 18

Z 1 Z x Z r x =r

Z w

such that · E W (1)

= =

µZ r Z 0

w 0

¶Z W (z) dzdw

1 0

W ( x ) dx

Z 1 0

¸ W (y) dy

1 1 13 6 r + r4 (11r + 3) (1 − r ) + r3 (5r + 4) (1 − r )2 72 24 18 1 3 r (16 − 3r ) , 72

and so we obtain C111r =

=

µ ¶ ¢ 1 2 1 2¡ 1 3 2 r (4 − r ) + µ1c τ r 39 − 16r + 3r + r (16 − 3r ) 6 72 72 1 2 13 r (4 − r ) + µ1c τr2 . 6 24

43

Finally, we have Cr111 , which only contains one term, namely, · µZ 1 Z w ¶Z 1 ¸ Z 1 E W (r ) W (z) dzdw W ( x ) dx W (y) dy 0 0 0 0 · ¸ Z 1 Z 1 Z 1 Z w = E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z r Z r Z r Z w = E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =0 y =0 w =0 0 · ¸ Z 1 Z r Z r Z w + 2 E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =r y =0 w =0 0 · ¸ Z w Z r Z r Z 1 E W (r ) W ( x ) W ( y ) W (z) dz dwdydx + x =0 y =0 w =r 0 · ¸ Z 1 Z 1 Z r Z w + E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =r y =r w =0 0 · ¸ Z 1 Z r Z 1 Z w + 2 E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =r y =0 w =r 0 · ¸ Z 1 Z 1 Z 1 Z w + E W (r ) W ( x ) W ( y ) W (z) dz dwdydx, x =r

y =r

w =r

0

where · ¸ Z w E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =r y =r w =r 0 · ¸ Z 1 Z x Z y Z w = 2 E W (r ) W ( x ) W ( y ) W (z) dz dwdydx x =r y =r w =r 0 · ¸ Z 1 Z x Z w Z w + 2 E W (r ) W ( x ) W ( y ) W (z) dz dydwdx x =r w =r y =r 0 · ¸ Z 1 Z w Z x Z w + 2 E W (r ) W ( x ) W ( y ) W (z) dz dydxdw Z 1 Z 1 Z 1

w =r

= + + =

x =r

y =r

0

Z 1 Z x Z y

¢ 1 ¡ 2 2 r 2w + 2yw − yr dwdydx x =r y =r w =r 2 Z 1 Z x Z w ¢ 1 ¡ 2 − r y2 − 4yw + ry − w2 dydwdx 2 x =r w =r y =r Z 1 Z w Z x ¢ 1 ¡ 2 − r yr − 2xw − 4yw + x2 + y2 dydxdw 2 w =r x =r y =r ¢ 1 ¡ r 5 + 10r + 3r2 (1 − r )3 , 12

44

such that ·

µZ

E W (r )

1

Z w

0

0

¶Z W (z) dzdw

1 0

W ( x ) dx

Z 1 0

¸ W (y) dy

13 6 7 1 r + r 5 (1 − r ) + r 4 (r + 5) (1 − r ) 72 12 12 ¢ 1 3 1 ¡ r (8r + 1) (1 − r )2 + r2 3 + 30r + 7r2 (1 − r )2 18 24 ¢ ¢ 1 ¡ 1 ¡ r 5 + 10r + 3r2 (1 − r )3 = r 30 − 21r + 4r2 , 12 72

= + + implying

Cr111 =

¡ ¢ ¡ ¢ 1 1 r (2 − r ) + µ1c τr 30 − 21r + 4r2 + O µ2c τ 2 . 2 24

By putting everything together, " µZ ¶ # r

E U (r )

= + − + − = implying

0

3

Z (s) ds

µ ¶ ¡ ¢ 1 5 13 1 4 1 6 4 2 r + µ1c τr + 3r r (5 − 2r ) + µ1c τr 10 + 6r − 3r 2 24 6 24 µ ³ ´¶ ¡ ¢ 1 1 2 r 1 + 6r − 5r2 + r3 + µ1c τr2 3 + 28r − 27r2 + 10r3 − r4 3r2 6 24 µ ¶ µ ¶ ¡ ¢ 1 1 5 13 3 1 2 6 r r (2 − r ) + µ1c τr 30 − 21r + 4r −r r + µ1c τr 2 24 2 24 ¶ µ ¡ ¢ ¡ ¢ 1 2 3 2 3 2 1 3 r 1 + 3r − r + µ1c τr 4 + 27r + 12r − 4r 3r 6 72 µ ¶ µ ¶ ¡ ¢ 13 13 3 1 2 2 4 1 3r r (4 − r ) + µ1c τr + r + µ1c τ + O µ2c τ 2 6 24 2 24 ¡ ¢ ¡ ¢ 1 − µ1c τr4 8 − 24r + 7r2 (1 − r )2 + O µ2c τ 2 , 24 Z 1 0

" E U (r )

µZ

r 0

¶3 # Z (s) ds

dr =

45

¡ ¢ 7 µ1c τ + O µ2c τ 2 . 4320

¡ ¢ Consider the O µ2c τ 2 term. We have µ ¶ Z s Z t Z u Crstu = E W (r ) W (w) dw W ( x ) dx W (y) dy 0 0 0 · µZ s Z w ¶Z t ¸ Z u + µ1c τE W (r ) W (z) dzdw W ( x ) dx W (y) dy 0 0 0 0 · µ ¶ ¸ Z s Z tZ x Z u + µ1c τE W (r ) W (w) dw W (z) dzdx W (y) dy 0 0 0 0 · µZ u Z y ¶¸ Z s Z t + µ1c τE W (r ) W (w) dw W ( x ) dx W (z) dzdy 0 0 0 0 · µZ s Z w ¶ µZ t Z x ¶Z u ¸ + µ2c τ 2 E W (r ) W (z) dzdw W (v) dvdx W (y) dy 0 0 0 0 0 · µZ s Z w ¶Z t µZ u Z y ¶¸ 2 W (z) dzdw W ( x ) dx W (v) dvdy + µ2c τ E W (r ) 0 0 0 0 0 · µZ t Z x ¶ µZ u Z y ¶¸ Z s + µ2c τ 2 E W (r ) W (w) dw W (z) dzdx W (v) dvdy 0 0 0 0 0 ¡ ¢ 3 + O µ3c τ . Thus, to able to compute Crrrr we need · µZ r Z w ¶ µZ r Z x ¶Z r ¸ E W (r ) W (z) dzdw W (v) dvdx W (y) dy 0 0 0 0 0 · ¸ Z r Z r Z r Z w Z x = E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x =0 0 0 · ¸ Z w Z x Z r Z y Z x E W (r ) W ( y ) W (z) dz W (v) dv dwdxdy = 2 0 0 y =0 x =0 w =0 · ¸ Z r Z x Z y Z w Z x + 2 E W (r ) W ( y ) W (z) dz W (v) dv dwdydx x =0 y =0 w =0 0 0 · ¸ Z r Z x Z w Z w Z x + 2 E W (r ) W ( y ) W (z) dz W (v) dv dydwdx x =0

w =0

y =0

0

Z r Z y Z x

0

¢ 1 2¡ = 2 w 3yx − yw + 3x2 dwdxdy y =0 x =0 w =0 6 Z r Z x Z y ¡ ¢ 1 + 2 − w2 −12yx + 2yw + 3y2 − 3x2 dwdydx 12 x =0 y =0 w =0 Z r Z x Z w ¢ 1 ¡ 167 7 + 2 − y 2x3 − 12x2 w + 3yx2 − 6xw2 + 3yw2 dydwdx = r , 12 2520 x =0 w =0 y =0 which gives Crrrr =

¡ ¢ 1 5 13 167 r + µ1c τr6 + µ2c τ 2 r7 + O µ3c τ 3 . 2 24 840

This implies C1111 =

¡ ¢ 1 13 167 + µ1c τ + µ2c τ 2 + O µ3c τ 3 . 2 24 840

46

Similarly, since · µZ r Z E W (1) 0

w 0

W (z) dzdw

we obtain

¶ µZ r Z 0

x 0

¶Z W (v) dvdx

r 0

¸ W (y) dy =

167 7 r , 2520

¡ ¢ 1 5 13 167 r + µ1c τr6 + µ2c τ 2 r7 + O µ3c τ 3 . 2 24 840

C1rrr =

Cr1rr is comprised of two types of terms. The first one is · µZ r Z w ¶ µZ 1 Z x ¶Z r ¸ E W (r ) W (z) dzdw W (v) dvdx W (y) dy 0 0 0 0 0 · ¸ Z r Z r Z 1 Z w Z x = E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x =0 0 0 · ¸ Z r Z r Z r Z w Z x = E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x =0 0 0 · ¸ Z r Z r Z 1 Z w Z x + E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy, y =0

w =0

x =r

0

0

Z w

Z x

0

0

where ·

Z r Z r

Z 1

y =0 Z r

w =0 Z y

x =r Z 1

y =0

w=y

¸

E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy 0 0 · ¸ Z w Z x = E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x =r 0 0 · ¸ Z r Z r Z 1 Z w Z x + E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy

= +

Z r Z y y =0

w =0

Z r Z r y =0 3

x =r

Z 1

x =r

−

Z 1

w=y

x =r

1 w 12

¡ 2

¢ −6rx − 12yx + 2yw + 3r2 + 3y2 dxdwdy

1 y(−12r3 + 12r2 w + 3yr2 + 6rx2 − 6yrx + 6rw2 12

2

+ 4x − 3yw )dxdwdy ¡ ¢ 1 3 = r (1 − r ) 10 + 30r + 54r2 + 35r3 , 720 such that · E W (r )

= =

µZ r Z 0

w 0

¶ µZ W (z) dzdw

1 0

Z x 0

¶Z W (v) dvdx

¡ ¢ 1 3 167 7 r + r (1 − r ) 10 + 30r + 54r2 + 35r3 2520 720 ´ 1 3³ r 70 + 140r + 168r2 − 133r3 + 89r4 . 5040

47

r 0

¸ W (y) dy

The second type of term is · µZ r Z E W (r )

=

0 r

Z 1 Z r

Z

y =0 Z r

w =0 Z r

x =0 Z r

y =r

w =0

x =0

w 0

W (z) dzdw

¶ µZ r Z

·

Z w

0

x 0

¶Z W (v) dvdx

1 0

¸ W (y) dy

¸

Z x

E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy 0 0 · ¸ Z x Z w E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy = y =0 w =0 x =0 0 0 · ¸ Z 1 Z r Z r Z w Z x + E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy, 0

0

where Z 1 Z r

= 2 = 2

Z r

y =r w =0 Z 1 Z r y =r

x =0 Z x

x =0

Z 1 Z r y =r

x =0

·

Z w

¸

Z x

E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy 0 0 · ¸ Z w Z x E W (r ) W ( y ) W (z) dz W (v) dv dwdxdy

w =0

Z x

w =0

0

0

¢ 19 6 1 2¡ w 3rx − rw + 3x2 dwdxdy = r (1 − r ) , 6 180

and therefore, · E W (r )

=

µZ r Z 0

w 0

W (z) dzdw

¶ µZ r Z 0

x 0

¶Z W (v) dvdx

1 0

¸ W (y) dy

19 6 1 6 167 7 r + r (1 − r ) = r (266 − 99r ) . 2520 180 2520

Hence, Cr1rr =

+ + = +

¡ ¢ 1 4 1 r (5 − 2r ) + µ1c τr4 10 + 6r − 3r2 6 24 ¶ µ ³ ´ 1 6 1 2 3 2 3 4 µ2c τ r 70 + 140r + 168r − 133r + 89r + r (266 − 99r ) 2520 2520 ¡ ¢ O µ3c τ 3 ¡ ¢ 1 4 1 r (5 − 2r ) + µ1c τr4 10 + 6r − 3r2 6 24 ³ ´ ¡ ¢ 1 2 3 µ2c τ r 70 + 140r + 168r2 + 133r3 − 10r4 + O µ3c τ 3 . 2520

The first term in C11rr is · µZ r Z E W (1)

=

0 1

Z r Z r

Z

y =0 Z r

w =0 Z r

x =0 Z r

y =0

w =0

x =r

w 0

·

¶ µZ W (z) dzdw Z w

1 0

Z x 0

¶Z W (v) dvdx Z x

r 0

¸ W (y) dy

¸

E W (1) W ( y ) W (z) dz W (v) dv dxdwdy 0 0 · ¸ Z w Z x = E W (1) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x =0 0 0 · ¸ Z r Z r Z 1 Z w Z x + E W (1) W ( y ) W (z) dz W (v) dv dxdwdy, 0

48

0

where

·

Z r Z r

Z 1

y =0 Z r

w =0 Z y

x =r Z 1

y =0

w=y

Z w

Z x

0

0

¸

E W (1) W ( y ) W (z) dz W (v) dv dxdwdy 0 0 · ¸ Z w Z x = E W (1) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x = r 0 0 · ¸ Z r Z r Z 1 Z w Z x + E W (1) W ( y ) W (z) dz W (v) dv dxdwdy Z r Z y

x =r

Z 1

¢ 1 2¡ w −12yx + 2yw + 3y2 − 3x2 dxdwdy 12 y =0 w =0 x = r Z r Z r Z 1 ¢ 1 ¡ + − y 2x3 − 12x2 w + 3yx2 − 6xw2 + 3yw2 dxdwdy 12 y =0 w = y x = r ¡ ¢ 1 3 = r (1 − r ) −5 + 25r + 67r2 + 42r3 , 720

=

suggesting

· E W (1)

= =

−

µZ r Z 0

0

¶ µZ W (z) dzdw

1

Z x

0

0

¶Z W (v) dvdx

r 0

¸ W (y) dy

¡ ¢ 1 3 167 7 r + r (1 − r ) −5 + 25r + 67r2 + 42r3 2520 720 ´ 1 3³ r −35 + 210r + 294r2 − 175r3 + 40r4 . 5040

The second term is · µZ r Z E W (1)

=

w

Z 1 Z r

Z

0 r

w 0

W (z) dzdw

·

¶ µZ r Z Z w

0

x 0

¶Z W (v) dvdx

1 0

¸ W (y) dy

¸

Z x

E W (1) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x =0 0 0 · ¸ Z r Z r Z r Z w Z x = E W (1) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x =0 0 0 · ¸ Z 1 Z r Z r Z w Z x + E W (1) W ( y ) W (z) dz W (v) dv dxdwdy, y =r

w =0

x =0

where Z 1 Z r

= 2 = 2 such that

Z r

y =r w =0 Z 1 Z r y =r

x =0

Z 1 Z r y =r

x =0

x =0 Z x

·

0

Z w

Z x

¸

E W (1) W ( y ) W (z) dz W (v) dv dxdwdy 0 0 · ¸ Z w Z x E W (1) W ( y ) W (z) dz W (v) dv dwdxdy

w =0

Z x

w =0

0

0

0

¢ 1 2¡ 1 5 w 3yx − yw + 3x2 dwdxdy = r (29r + 9) (1 − r ) 6 360

· µZ r Z E W (1)

=

0

w 0

W (z) dzdw

¶ µZ r Z 0

x 0

¶Z W (v) dvdx

1 0

¸ W (y) dy

¢ 167 7 1 5 1 5¡ r + r (29r + 9) (1 − r ) = r 63 + 140r − 36r2 , 2520 360 2520 49

giving ¢ ¡ ¢ 1 3¡ 1 r 1 + 3r − r2 + µ1c τr3 4 + 27r + 12r2 − 4r3 6 72 µ ³ ´ 1 µ2c τ 2 r3 −35 + 210r + 294r2 − 175r3 + 40r4 2520 ¶ ¢ ¡ ¢ 1 5¡ 2 r 63 + 140r − 36r + O µ3c τ 3 2520 ¢ ¡ ¢ 1 1 3¡ r 1 + 3r − r2 + µ1c τr3 4 + 27r + 12r2 − 4r3 6 72 ³ ´ ¡ ¢ 1 µ2c τ 2 r3 −35 + 210r + 357r2 − 35r3 + 4r4 + O µ3c τ 3 . 2520

C11rr =

+ + = +

As for Cr11r , we have · µZ E W (r )

=

Z r Z 1

Z

1

Z w

0 1

0

¶ µZ W (z) dzdw

·

Z w

1 0

Z x 0

¶Z W (v) dvdx

r 0

¸ W (y) dy

¸

Z x

E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x =0 0 0 · ¸ Z r Z r Z r Z w Z x = E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x =0 0 0 · ¸ Z r Z r Z 1 Z w Z x + 2 E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x = r 0 0 · ¸ Z r Z 1 Z 1 Z w Z x + E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy y =0

w =r

x =r

0

0

Z w

Z x

with Z r Z 1

= 2 = 2

y =0 Z r y =0

·

Z 1

w =r Z 1

x =r Z x

E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy 0 0 · ¸ Z w Z x E W (r ) W ( y ) W (z) dz W (v) dv dwdxdy

x =r

w =r

x =r

1 − y(3r2 x − 3r2 y + 3r2 w − 3x2 w 6 w =r

0

Z r Z 1 Z x y =0

¸

0

3

+ x + 3ryx + 3ryw − 12rxw)dwdxdy ¡ ¢ 1 2 = r (1 − r )2 2 + 39r + 16r2 + 3r3 , 120 giving · E W (r )

= + =

µZ

1 0

Z w 0

¶ µZ W (z) dzdw

1 0

Z x 0

¶Z W (v) dvdx

¡ ¢ 1 3 167 7 r + r (1 − r ) 10 + 30r + 54r2 + 35r3 2520 360 ¡ ¢ 1 2 r (1 − r )2 2 + 39r + 16r2 + 3r3 120 ´ 1 2³ r 42 + 805r − 1120r2 + 378r3 + 77r4 − 15r5 . 2520 50

r 0

¸ W (y) dy

Also, · E W (r )

= = + + +

µZ r Z 0 1

Z 1 Z r

Z

y =0 Z r

w =0 Z r

x =0 Z r

y =r

w =0

x =r

¶ µZ

w 0

W (z) dzdw

·

Z w

1 0

Z x 0

¶Z W (v) dvdx

1 0

¸ W (y) dy

¸

Z x

E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy 0 0 · ¸ Z w Z x E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x =0 0 0 · ¸ Z 1 Z r Z r Z w Z x E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy y =r w =0 x =0 0 0 · ¸ Z w Z x Z r Z r Z 1 E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x = r 0 0 · ¸ Z 1 Z r Z 1 Z w Z x E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy, 0

0

Z w

Z x

0

0

where ·

Z 1 Z r

Z 1

y =r Z 1

w =0 Z r

x =r Z y

y =r

w =0

¸

E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy 0 0 · ¸ Z w Z x = E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy y =r w =0 x = r 0 0 · ¸ Z 1 Z r Z 1 Z w Z x + E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy Z 1 Z r

x =y

Z y

¡ ¢ 1 = − w2 −12rx + 2rw + 3r2 − 3x2 dxdwdy 12 y =r w =0 x = r Z 1 Z r Z 1 ¡ ¢ 1 + − w2 −6yx − 12rx + 2rw + 3y2 + 3r2 dxdwdy 12 y =r w =0 x = y ¡ ¢ 1 3 = r 1 + 10r + 3r2 (1 − r )2 , 48 such that · E W (r )

= + =

µZ r Z 0

w 0

¶ µZ W (z) dzdw

1 0

Z x 0

¶Z W (v) dvdx

1 0

¸ W (y) dy

¡ ¢ 19 6 1 3 167 7 r + r (1 − r ) + r (1 − r ) 10 + 30r + 54r2 + 35r3 2520 180 720 ¢ 1 3¡ r 1 + 10r + 3r2 (1 − r )2 48 ´ 1 3³ r 175 + 980r − 1512r2 + 819r3 − 128r4 . 5040

51

It follows that Cr11r =

+ + = +

³ ´ ¢ 1 2¡ 1 r 1 + 6r − 5r2 + r3 + µ1c τr2 3 + 28r − 27r2 + 10r3 − r4 6 24 µ ³ ´ 1 µ2c τ 2 r2 42 + 805r − 1120r2 + 378r3 + 77r4 − 15r5 2520 ³ ´¶ ¡ ¢ 1 3 r 175 + 980r − 1512r2 + 819r3 − 128r4 + O µ3c τ 3 2520 ³ ´ ¢ 1 2¡ 1 r 1 + 6r − 5r2 + r3 + µ1c τr2 3 + 28r − 27r2 + 10r3 − r4 6 24 ³ ´ ¡ ¢ 1 2 2 µ2c τ r 42 + 980r − 140r2 − 1134r3 + 896r4 − 143r5 + O µ3c τ 3 . 2520

The first term in C111r is · µZ E W (1)

=

Z w

1 0

0

Z r Z 1

Z 1

y =0 Z r

w =0 Z r

x =0 Z r

y =0

w =r

x =r

¶ µZ W (z) dzdw

·

Z w

1 0

Z x 0

¶Z W (v) dvdx

r 0

¸ W (y) dy

¸

Z x

E W (1) W ( y ) W (z) dz W (v) dv dxdwdy 0 0 · ¸ Z w Z x = E W (1) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x =0 0 0 · ¸ Z r Z r Z 1 Z w Z x + 2 E W (1) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x =r 0 0 · ¸ Z r Z 1 Z 1 Z w Z x + E W (1) W ( y ) W (z) dz W (v) dv dxdwdy, 0

0

where Z r Z 1

= 2

y =0

w =r Z 1

x =r Z x

x =r

Z w

¸

E W (1) W ( y ) W (z) dz W (v) dv dxdwdy 0 0 · ¸ Z w Z x E W (1) W ( y ) W (z) dz W (v) dv dwdxdy

w =r

Z r Z 1 Z x

Z x

0

0

¢ 1 y 2x3 − 12x2 w + 3yx2 − 6xw2 + 3yw2 dwdxdy 12 y =0 x = r w =r ¡ ¢ 1 2 r (1 − r )2 36 + 67r + 58r2 + 19r3 360

= 2 =

y =0 Z r

·

Z 1

−

¡

which implies · E W (1)

= + =

µZ

1 0

Z w 0

¶ µZ W (z) dzdw

1 0

Z x 0

¶Z W (v) dvdx

¡ ¢ 167 7 1 3 r + r (1 − r ) −5 + 25r + 67r2 + 42r3 2520 360 ¡ ¢ 1 2 r (1 − r )2 36 + 67r + 58r2 + 19r3 360 ´ 1 2³ r 252 − 70r − 70r2 + 84r3 − 35r4 + 6r5 . 2520 52

r 0

¸ W (y) dy

The second term is · µZ r Z E W (1)

=

0 1

Z 1 Z r

Z

y =0 Z r

w =0 Z r

x =0 Z r

y =r

w =0

x =r

¶ µZ

w 0

W (z) dzdw

·

Z w

1 0

Z x 0

¶Z W (v) dvdx

1 0

¸ W (y) dy

¸

Z x

E W (1) W ( y ) W (z) dz W (v) dv dxdwdy 0 0 · ¸ Z w Z x E W (1) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x =0 0 0 · ¸ Z 1 Z r Z r Z w Z x E W (1) W ( y ) W (z) dz W (v) dv dxdwdy y =r w =0 x =0 0 0 · ¸ Z w Z x Z r Z r Z 1 E W (1) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x = r 0 0 · ¸ Z 1 Z r Z 1 Z w Z x E W (1) W ( y ) W (z) dz W (v) dv dxdwdy.

= + + +

0

0

Z w

Z x

0

0

Hence, since ·

Z 1 Z r

Z 1

y =r Z 1

w =0 Z r

x =r Z y

y =r

w =0

y =r

w =0

¸

E W (1) W ( y ) W (z) dz W (v) dv dxdwdy 0 0 · ¸ Z w Z x = E W (1) W ( y ) W (z) dz W (v) dv dxdwdy y = r w =0 x = r 0 0 · ¸ Z 1 Z r Z 1 Z w Z x + E W (1) W ( y ) W (z) dz W (v) dv dxdwdy

= + =

Z 1 Z r Z 1 Z r

x =y

Z y ¢ 1 2¡ w 3yx − yw + 3x2 dxdwdy x =r

Z 1

6

¢ 1 2¡ w −12yx + 2yw + 3y2 − 3x2 dxdwdy 12 y = r w =0 x = y ¡ ¢ 1 3 r 13 + 19r + 10r2 (1 − r )2 , 144

−

we obtain · E W (1)

= + =

µZ r Z 0

w 0

¶ µZ W (z) dzdw

1 0

Z x 0

¶Z W (v) dvdx

1 0

¸ W (y) dy

¡ ¢ 1 5 1 3 167 7 r + r (29r + 9) (1 − r ) + r (1 − r ) −5 + 25r + 67r2 + 42r3 2520 360 720 ¢ 1 3¡ 2 r 13 + 19r + 10r2 (1 − r ) 144 ´ 1 3³ r 420 − 35r − 105r2 + 70r3 − 16r4 , 5040

53

which in turn implies C111r =

+ + = +

1 2 13 r (4 − r ) + µ1c τr2 6 24 µ ³ ´ 1 µ2c τ 2 r2 252 − 70r − 70r2 + 84r3 − 35r4 + 6r5 2520 ³ ´¶ ¡ ¢ 1 3 2 3 4 r 420 − 35r − 105r + 70r − 16r + O µ3c τ 3 2520 13 1 2 r (4 − r ) + µ1c τr2 6 24 ³ ´ ¡ ¢ 1 2 2 µ2c τ r 252 + 350r − 105r2 − 21r3 + 35r4 − 10r5 + O µ3c τ 3 . 2520

Finally, we have Cr111 , where · µZ 1 Z w ¶ µZ 1 Z x ¶Z 1 ¸ E W (r ) W (z) dzdw W (v) dvdx W (y) dy 0 0 0 0 0 · ¸ Z 1 Z 1 Z 1 Z w Z x = E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x =0 0 0 · ¸ Z r Z r Z r Z w Z x = E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x =0 0 0 · ¸ Z 1 Z r Z r Z w Z x + E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy y = r w =0 x =0 0 0 · ¸ Z r Z r Z 1 Z w Z x + 2 E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy y =0 w =0 x = r 0 0 · ¸ Z 1 Z r Z 1 Z w Z x + 2 E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy y = r w =0 x =r 0 0 · ¸ Z w Z x Z r Z 1 Z 1 E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy + y =0 w =r x =r 0 0 · ¸ Z 1 Z 1 Z 1 Z w Z x + E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy, y =r

w =r

x =r

0

54

0

with Z 1 Z 1

·

Z 1

Z w

¸

Z x

E W (r ) W ( y ) W (z) dz W (v) dv dxdwdy 0 0 · ¸ Z w Z x = 2 E W (r ) W ( y ) W (z) dz W (v) dv dwdxdy y =r x =r w =r 0 0 · ¸ Z 1 Z x Z y Z w Z x + 2 E W (r ) W ( y ) W (z) dz W (v) dv dwdydx x =r y =r w =r 0 0 · ¸ Z w Z x Z 1 Z x Z w E W (r ) W ( y ) W (z) dz W (v) dv dydwdx + 2 y =r w =r Z 1 Z y

x =r

= 2 + 2

x =r Z x

w =r

y =r

Z 1 Z y Z x y =r

x =r

w =r

Z 1 Z x Z y x =r 2

y =r w =r 3

0

−

0

¢ 1 ¡ 3 r 2x − 12x2 w + 3rx2 − 6xw2 + 3rw2 dwdxdy 12

1 r (−12y3 + 12y2 w + 3ry2 + 6yx2 − 6ryx 12

+ 6yw + 4x − 3rw2 )dwdydx Z 1 Z x Z w 1 + 2 − r (3y2 x − 3y2 r + 3y2 w − 3x2 w 6 x =r w =r y =r + x3 + 3yrx + 3yrw − 12yxw)dydwdx ¢ 1 ¡ = r 25 + 42r + 21r2 + 12r3 (1 − r )3 , 120 which implies

·

µZ

E W (r )

= + + =

1 0

Z w 0

¶ µZ W (z) dzdw

1 0

Z x 0

¶Z W (v) dvdx

1 0

¸ W (y) dy

¡ ¢ 19 6 1 3 167 7 r + r (1 − r ) + r (1 − r ) 10 + 30r + 54r2 + 35r3 2520 180 360 ¢ ¡ ¢ 1 3¡ 1 2 r 1 + 10r + 3r2 (1 − r )2 + r (1 − r )2 2 + 39r + 16r2 + 3r3 24 120 ¢ 1 ¡ 2 3 r 25 + 42r + 21r + 12r (1 − r )3 120 ´ 1 ³ r 525 − 651r + 280r2 + 770r3 − 1617r4 + 1078r5 − 218r6 , 2520

giving ¡ ¢ 1 1 r (2 − r ) + µ1c τr 30 − 21r + 4r2 2 24 ³ ´ 1 2 + µ2c τ r 525 − 651r + 280r2 + 770r3 − 1617r4 + 1078r5 − 218r6 840 ¡ ¢ + O µ3c τ 3 .

Cr111 =

By adding everything together, " µZ ¶ # E U (r )

r

0

3

Z (s) ds

¡ ¢ 1 µ1c τr4 8 − 24r + 7r2 (1 − r )2 24 ¡ ¢ ¡ ¢ 1 µ2c τ 2 r4 193 + 170r − 114r2 (1 − r )4 + O µ3c τ 3 , 420

= − −

55

giving Z 1 0

or

Z 1 0

"

µZ

E U (r )

0

" E

r

µZ c3i U

(r )

¶3 #

r 0

dr =

Z (s) ds

¡ ¢ 97 7 µ1c τ − µ2c τ 2 + O µ3c τ 3 , 4320 103950

¶3 # dr =

Z (s) ds

¡ ¢ 7 97 µ4c τ − µ5c τ 2 + O µ6c τ 3 . 4320 103950

This completes the proof of (c). As for (d), since Z r 0

Z (s)ds =

Z rµ 0

W1 (s) −

Z 1 0

¶ W1 (t) dt ds =

Z r 0

W1 (s) ds − r

Z 1 0

W1 (s) ds

we have · 3

= = − + −

µZ

r

¶¸

E U (r ) Z (s)ds 0 · µZ r ¶¸ Z 1 E (W (r ) − rW (1))3 W1 (s) ds − r W1 (s) ds 0 0 · ¸ · ¸ Z r Z 1 3 3 E W (r ) W1 (s) ds − rE W (r ) W1 (s) ds 0 0 · ¸ · ¸ Z r Z 1 2 2 2 3rE W (r ) W (1) W1 (s) ds + 3r E W (r ) W (1) W1 (s) ds 0 0 · ¸ · ¸ Z r Z 1 2 2 2 3 3r E W (r ) W (1) W1 (s) ds − 3r E W (r ) W (1) W1 (s) ds 0 0 · ¸ · ¸ Z r Z 1 3 3 3 4 r E W (1) W1 (s) ds + r E W (1) W1 (s) ds 0

0

2

2

= Drrrr − rDrrr1 − 3rD1rrr + 3r D1rr1 + 3r D11rr − 3r3 D11r1 − r3 D111r + r4 D1111 , where ·

¸

Z u

Drstu = E W (r ) W (s) W (t) W1 (v) dv 0 · Z uµ Z = E W (r ) W ( s ) W ( t ) W ( v ) + ci τ 0

v 0

¶¸

=

W (w) dw

Z u 0

grstv dv,

with ·

µ

grstv = E W (r ) W (s) W (t) W (v) + ci τ

Z v

·

0

¶¸ W (w) dw

= E [W (r ) W (s) W (t) W (v)] + ci τE W (r ) W (s) W (t)

Z v 0

¸ W (w) dw .

Consider Drrrr . By using the same technique as in the proof of (a)–(c), for r ≥ v, 3 grrrv = 3vr + µ1c τv2 r, 2 56

suggesting Drrrr =

Z r 0

grrrv dv =

Z rµ 0

¶ 3 1 2 3vr + µ1c τv r dv = r3 (3 + µ1c τr ) , 2 2

which in turn implies D1111 =

1 (3 + µ1c τ ) . 2

In order to obtain Drrr1 we use the fact that for r < v, 3 grrrv = 3r2 − µ1c τr2 (r − 2v) , 2 which, together with the above results, yilds Drrr1 =

=

¶ Z 1µ 1 3 3 2 2 grrrv dv + grrrv dv = r (3 + ar ) + 3r − µ1c τr (r − 2v) dv 2 2 0 r r 1 3 3 3 1 r (3 + µ1c τr ) − r2 (r − 1) (2 + µ1c τ ) = r2 (2 − r ) + µ1c τr2 (3 − 3r + r2 ). 2 2 2 2

Z r

Z 1

Next, consider D1rrr . We have D1rrr =

Z r 0

g1rrv dv,

where, for r ≥ v, 3 g1rrv = 3vr + µ1c τv2 r, 2 such that D1rrr =

Z r 0

g1rrv dv =

Z rµ 0

¶ 3 1 2 3vr + µ1c τv r dv = r3 (3 + µ1c τr ) . 2 2

For r < v ≤ 1, we have 1 g1rrv = r (v + 2r ) − µ1c τr (2r2 − 4vr − v2 ), 2 suggesting D1rr1 =

= = =

Z r

g1rrv dv +

Z 1

g1rrv dv ¶ Z 1µ 1 3 1 2 2 r (3 + µ1c τr ) + r (v + 2r ) − µ1c τr (2r − 4vr − v ) dv 2 2 r ¡ ¡ ¢¢ 1 3 1 r (3 + µ1c τr ) − r (r − 1) 3 + 15r + µ1c τ 1 + 7r + r2 2 6 ¢ ¡ ¢ 1 ¡ 1 r 1 + 4r − 2r2 + µ1c τr 1 + 6r − 6r2 + 2r3 . 2 6 0

r

57

D11rr can be written as D11rr =

Z r 0

g11rv dv,

where, for r ≥ v, 1 g11rv = v (1 + 2r ) + µ1c τv2 (1 + 2r ) , 2 and therefore D11rr =

Z r 0

g11rv dv =

Z rµ 0

¶ 1 1 2 v (1 + 2r ) + µ1c τv (1 + 2r ) dv = r2 (2r + 1) (3 + µ1c τr ) . 2 6

For r < v ≤ 1, we have ¡ ¢ 1 g11rv = r (1 + 2v) + µ1c τr 2v2 + 2v − r , 2 which we can use to obtain D11r1 =

Z r

= = =

0

g11rv dv +

Z 1 r

g11rv dv

¶ Z 1µ ¡ 2 ¢ 1 2 1 r (2r + 1) (3 + µ1c τr ) + r (1 + 2v) + µ1c τr 2v + 2v − r dv 6 2 r ¡ ¡ ¢¢ 1 2 1 r (2r + 1) (3 + µ1c τr ) − r (r − 1) 12 + 6r + µ1c τ 5 + 2r + 2r2 6 6 ¡ ¢ 1 1 r (4 − r ) + µ1c τr 5 − 3r + r2 . 2 6

It remains to consider D111r , which can be written as D111r =

Z r 0

g111v dv,

where, for v ≤ 1, 3 g111v = 3v + µ1c τv2 , 2 giving D111r =

Z rµ 0

¶ 3 1 2 3v + µ1c τv dv = r2 (3 + µ1c τr ) . 2 2

58

By adding all the results, · µZ r ¶¸ 3 E U (r ) Z (s)ds = 0

− + + − −

µ ¶ ¡ ¢ 1 3 3 2 1 2 2 r (3 + µ1c τr ) − r r (2 − r ) + µ1c τr 3 − 3r + r 2 2 2 1 3r r3 (3 + µ1c τr ) 2µ ¶ ¢ 1 ¡ ¢ 1 ¡ 3r2 r 1 + 4r − 2r2 + µ1c τr 1 + 6r − 6r2 + 2r3 2 6 1 3r2 r2 (2r + 1) (3 + µ1c τr ) 6µ ¶ ¡ ¢ 1 3 1 2 r (4 − r ) + µ1c τr 5 − 3r + r 3r 2 6 1 1 r3 r2 (3 + µ1c τr ) + r4 (3 + µ1c τ ) = −µ1c τr3 (1 − r )3 , 2 2

such that ·Z E

1 0

U (r )

3

µZ

r 0

¶

¸

Z (s)ds dr = −µ1c τ

Z 1 0

r3 (1 − r )3 dr = −

1 µ1c τ. 140

¥

This establishes (d).

59