Weak Convergence to Brownian Meander and Brownian Excursion Author(s): Richard T. Durrett, Donald L. Iglehart, Douglas R. Miller Source: The Annals of Probability, Vol. 5, No. 1 (Feb., 1977), pp. 117-129 Published by: Institute of Mathematical Statistics Stable URL: http://www.jstor.org/stable/2242807 . Accessed: 15/04/2011 15:31 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=ims. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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The Annals of Probability 1977,Vol. 5, No. 1, 117-129

WEAK CONVERGENCE TO BROWNIAN MEANDER AND BROWNIAN EXCURSION' BY RICHARD

T. DURRETT,2 DONALD L. IGLEHART3 AND DOUGLAS R. MILLER

and University of Missouri StanfordUniversity We show that (i) Brownianmotionconditionedto be positiveis Brownianmeander;(ii) tied-downBrownianmeanderis Brownianexcursion; and (iii) Brownianbridgeconditionedto be positiveis Brownian of the suprema excursion.Using theseresultswe derivethedistribution of themeanderand excursion.

1. Introductionand summary. Brownian meander and Brownian excursion processeshave recentlyappeared as the limitprocessof a numberof conditional functionalcentrallimittheorems. These resultsmay be foundin Belkin (1972), Iglehart(1974, 1975), and Kaigh (1974, 1975, 1976). Our purpose in this paper is to investigaterelationshipsbetween Brownian meander,Brownian excursion,Brownian motionand Brownianbridge. In particular,we presentsome conditionedfamiliesof Brownian processeswhich have Brownian meanderor Brownian excursionas theirweak limit. (This is similar in spiritto weak convergenceto Brownian bridge of Brownian motion which is conditioned to be close to 0 at time 1; see Billingsley(1968), pages 83-86.) Employingthe continuous mapping theorem,Durrett and Iglehart (1977) use these resultsto determinethe distributionsof various functionalsof Brownian meanderand Brownian excursion. To describe our resultsin greaterdetail, we need to introducethe two processes mentionedabove. Brownian meander, W+ = {W+(t): 0 < t < 1}, can be described as follows: Let {W(t): t > 01 be standard Brownian motion, rz= sup {t e [0, 1]: W(t) = 01, and Al = 1 - -r. Then W+(t)

AlflW(rl + tA,)I,

0 < t< 1

In Belkin (1972), page 61, it is shownthat W+ is a continuous,nonhomogeneous b n,(x)dx, then Markov process. If n8(x)= (2irs)- exp( -x2/2s) and N8(a, b)

ReceivedMarch 16, 1976. 1This paper(preparedby D. R. Miller)combinespartsof two papers(references [5] and [13]). of theother. The researchin each of thesepaperswas performed independently 2 The work of this authorwas supportedby a National Science Foundation Graduate Fellowship. This authoris now at UCLA. 3 The workof thisauthorwas supported byNationalScienceFoundationGrantMPS74-23693 and Officeof Naval ResearchContractN00014-75-C-0521. AMS 1970subjectclassifications. Primary60J65;Secondary60B10. Key wordsandphrases. Brownianmotion,Brownianmeander,Brownianbridge,Brownian of supremum, weak convergence. excursion,distribution 117

118

RICHARD T. DURRETT, DONALD L. IGLEHART AND DOUGLAS R. MILLER

W+ has transitiondensitygiven by (1 .1l)

P{W+(t) e dy} = p+(, 0t, y) dy = 2t-ryexp(-y2/2t)N1_t(0, y) dy

for 0 < t < 1 andy > 0; for0< s < t < 1 and x,y > O (1.2)

where

P{W+(t) e dyI W+(s) = x} = p+(s, x, t,y) dy = g(t - s, x, y)[N1_t(0, y)/N1_8(0, x)] dy,

g(t,x,y) =

nt(y

-x)-

nt(y

+

x)

= P{W(t) e dy; W(s) > 0,0 < s < tI W(O) = x}/dy. The second equality followsfrom(11.10) of Billingsley(1968). - exp(-x2/2), x > 0, theRayleighdistribution. Note thatP{W+(1) < x} Throughoutthe paper we will use P{Z e dy) = f(y) dy to mean thatthedistribution of Z has the densityf with respectto Lebesgue measure dy. Brownian excursion, WO+= {WO+(t): 0 < t ? 1}, is also a continuous n6nhomogeneous Markov process. Let -2 = inf{t > 1: W(t) = 0} and set A2 = 1 Then 2-

0 < t< 1

WO+(t)= A2-iIW(rl+ tA2)1 ,

The transitiondensityis given by, (1.3)

P{W9+(t)e dy} = po+(O,0, t,y) dy = 2y exp( -y2 /2t(1 (22rt3(1-ty1

-

t)) dy

for0 < t < 1; forO < s < t < 1 and x,y > 0

(1.4)

P{W.+(t) e dyI Wo+(s)= x} = po+(s,x, t,y) dy = g(t - s, xXY)

-s l-ti

y exp(-y2/2(I t)) x exp(-X2 /2(1 -s))dy

see It6-McKean (1965), page 76, for this result. Next we introducesome notation. For 0 < t < 1, let m(t) =inf{W(s):

O<

s?

t}

and

M(t) =sup{W(s):Os<

s?

t},

where W is standard Brownian motion. Let m = m(l) and M = M(1). For Brownian bridge, WO,Brownian meander, W+, and Brownian excursion, WO+, we use correspondingnotationfor the infimumand supremum;e.g. M+(t) = sup {W+(s): 0 < s
< mo+(t)= inf{WO+(s): 0 < s t?

.

Let C = C[O, 1] be the space of continuousfunctionson [O, 1], andjlet,' be the Borel sets of C when it is endowed with the topology generatedby the supremummetric, p. We shall need a concise definitionand notationforconditionedprocesses. Suppose Yis a randomfunctionof (C, k), i.e., Y is a measurable mappingfromsome probabilityspace (Q, J, P) into(C, W2). The random

119

BROWNIAN MEANDER AND EXCURSION

functioninducesa probabilitymeasure,Q = PY-', on (C, W). - Let A be a Borel subsetof C withQ(A) > 0. Then let (A, A n X, QA) be thetraceof (C, W', Q) on A A n v= {A n A: A e W} andQA(A) = Q(A)/Q(A) for A e A n W." Also let (Y'1(A), Y'1(A) n X, PY-1(A)) be-the trace of (Q, -- P) on Y-1(A). Then we definethe random function

YI A: (Y-1(A), Y-1(A) n

Py'(A)) I

(A, A n

',

QA)

as the restrictionof Y to Y-1(A). It follows that PY-1(A)((YI A)-'(.)) = QA(*). As discussedby Billingsley(1968), page 22, convergencein distributionof random functionsis equivalentto weak convergenceof theinduced probabilitymeasures. We now definethe followingconditioned random functionsof (C, W), for e > 0:

We= WI{m> -61,

WE = W+I{W+(1) -E1. WOE=WoI{MO The above processesare all Markov by virtueof thefollowinglemma whichwe state withoutproof. functionof C[O, 1]. Let A be a Borel LEMMA. Let Y be a Markovrandom (1.5) subsetof C withQ(A) > 0. Let 7r[Otl (iatt,1]) be theprojectionmap of C[O, 1] onto C[O, t] (C[t, 1]). If for all 0 < t < 1 thereexist sets At and Bt such thatA = 7rCO'tAtn fl7r']Bt, thenYI A is Markov. The main resultsof the paper are organizedas follows. In Section 2 we show that W" W+as e l 0. In Section 3 we presentan alternateproofof this result. In Section 4 we show that WE+ Wo+asksI 0. In Section 5, WOE Wo0'ase l . (Throughoutthispaper when consideringtightnessand convergence,"e l 0" and "'e > O." shall in realitycorrespondto a fixedsequence of numberstendingto 0; see footnote,Billingsley(1968), page 84.) In Section 6 the resultsof Sections 2 and 5 are used to obtain the distributionsof M+ and MO+. 2. Convergenceof conditionedBrQwnianmotionto Brownianmeander. This section is devoted exclusivelyto proving (2.1)

THEOREM.

WE

W+ase

1 0.

Let {W(t): t > 0} be standard Brownian motion definedon the probability triple(Q. A, P) witha-fieldsit = a{ W(s) : s < t} and shiftoperators{Ot: t > 01. The symbolPZ{A} means P{A I W(0) = x} and El the expectationwithrespectto PZ. When x = 0 the superscriptis omitted. To facilitatethe proofof (2.1) we breakit intothreelemmas. Definefore>0_ SC= inf{s > 0: W(s) = A,W(u) > 0 1 ZC(t) = W(s. + t). Thefirststepis fors< u < s+ 1}andfor0?< t (2.2) PROOF.

LEMMA.

For s > 0,

SE

For e > 0, let t --1

<

?? a.s.; SC and tk

-

so0and p(Z., ZO) -+0 a.s. as s j 0.

inf{t

_

t~k-i +

1: W(t)

=

_}.

Since

120

RICHARD T. DURRETT, DONALD L. IGLEHART AND DOUGLAS R. MILLER

P1{t'l < oo} = 1 forall y, an inductionargumentshows. P{t k+l

<

co} = E[Pw(t k+1)(tel < 00);

tE

<

P{tEk<

ca] =

oa} =

1

Now if W(t)hasno zeroin [t~k,t~l + 1], thenSC< tk so P{S < t I S > tE'} ? , PC{W(t) > 0 forall 0 < t < 1} > 0 and hence P{SE< oo} = 1. For 0 < < s, To see that S Iso, note that r= sa ,O so SW(o+er) - so inf{t-sO-I: t sO, > W(t) r forall 0 < < 1. SinceWhascontinuous pathsand Z,(t) = W(sE+ t), SEJ.soimpliesp(Z6,Zo)-?0. _

(2.3) LEMMA. For e > 0 therandomfunctionsZE - e and WEhave the same finite-dimensional distributions and hence induce the same probabilitymeasureson (C,'). PROOF. Let 0 < tj < *.. < tk < 1; letA1, A .., Ak be Borel sets of [0, oo), and F = {x e C: x(ti) e Ai; i = 1, * **, k}. Define the firstentrance time t, = inf{t > 0: W(t) = -} _ SE. Decompose {ZE e F} to obtain

P{Z e F} = P{Z, e F, SE= tj + P{Z, e F, SC> tj.

(2.4)

Since t. is a stoppingtime and W a strongMarkov process

P{ZEe F, SE= tj

e F, s. = tE,tE}I = E{Pw(tI){ We F, m > 0}} =

E{P{Z,

= Pe{We F, m >

whereF-

= {f-

0} = P{We F

(2.5)

rE

on {s. >

tEj.

+foo

ifthelastsetis empty.

Proceedingnow with the second termof (2.4) we

P{ZEe F, SE> tj

= -

P{Z e F,

rE

<

o}

E{E{116Z~F} IrTE};

. forsome On the set {zr < oo}, 11z.,j = - 0* the (2.5) and strongMarkov propertywe get

(2.6)

, m > -E},

: f e F}. If SE> t., then W(s) = 0 forsome s e (tC,tC+ 1].

Letzr= inf{s: s e (t6,t, + 1], W(s) = 0}, wherezr=

Clearly, SE> have

-

..

tj <

O}

measurable 5. Thus from

7* < oa} P{IZ e F, s. > tj} = E{Ew(r){}; = E0{10}P{zr< oo} - P{Z e F}[ - P{m >

-4]

.

Combining(2.4), (2.5) and (2.6) yields P{ZE e F} = P{We F-e, or

m > -}

+ P{Z e F}[1 -P{m > -}]

P{ZCeF} = P{WeF-sIm

> -,}.

(2.7) LEMMA. Thefinite-dimensional distributions of WEconvergeto those of W+ as jI 0.

121

BROWNIAN MEANDER AND EXCURSION

PROOF. We shall compute the transitionprobabilitiesof WE. From the reflectionprinciple P{m > -, = 2N1(O,e) - -(2/w)ias sj. 0. Using the Markov propertyfor W we obtain, for0 < t < 1, P{W(t) edy, m > -}

= P{W(t) edy, m(t)

-}PY{m(l

-

t) >

-}

= g(t, e,y + e) . 2N1-,(O,y + e) dy ~ (27Tt)-1[2_-(y/t)e -y2/2t] * 2N._t(0, y) dy

(2.8) as e l 0. Hence P{ Ws(t) e dy}

g(t, E,y + i) N1,(0,y

pj(O, 0, t,y) dy =

=

+ e)_ dy

N1(0,e

t-tye-'2/2t2N1-t(O,y) dy PP{W+(t) edy},

(2.9)

as

>

t0.

For 0 < s <

t

< 1 and x 0, >

by the Markov propertyof W

P{ We(t) e dy I We(s) = x}

e dx, m(s)

> -s}P{

-

e dy, m(t -

s) > -e} m X P"{m(l > -A}] t) > -e}]/[P{ W(s) e dx, s, x + e, y + e)2Nt(0, y + e) dx dy g(s, e, x + e)g(t g(s, A, x + e)2N1_8(0,x + A)dx

= [PI W(s)

W(t

s)

-

(2.10) -

g(t-s, = P{W+(t)

x,y)

N-t'

N18,(0,x)

Edy I W+(s)

dy

= x},

as e I 0. (These transitionprobabilitiesare derived in [13] from distribution dx's and dy's.) functionsratherthan using differentials, Convergenceof the transitiondensitiesimplyconvergenceof thefinite-dimensional densities,which in turnimplies convergence of f.d.d.'s, completingthe proof of (2.7). sets Lemmas 2.2 and 2.3 implythat Ws ZOas e i 0. The finite-dimensional are a determiningclass (Billingsley(1968), page 15); therefore,convergence of distributionsof Wsto W+ implies Ws W+ as e i 0; see the finite-dimensional also Billingsley(1968), page 35. 3. An alternativeproofof Theorem2.1. Verificationof weak convergenceof probabilitymeasureson functionspaces usually involves two steps: namely, (i) and (ii) tightness. In thissection distributions convergenceof finite-dimensional anotherproofof (sequential)tightnessof{ Ws,e > 0} is presented;whencombined with convergenceof f.d.d.'s (Lemma 2.7) this will provide an alternate proof of Theorem2.1. The proofis based on a characterizationof tightnessof random elementsof C[O, 1], namely THEOREM. Let {X, n = 1, 2, * } be a sequence of random elements of C[0, 1]. Define random elements Yn,8of C[s, 1] as Yaws-U= ,o] o X" If(i) for any

(3.1)

122

RICHARD T. DURRETT, DONALD L. IGLEHART AND DOUGLAS R. MILLER

s > 0, {Y,J,,n = 1, 2, lim8 1lim

P0

.} inducesa tightfamilyof measureson C[s, 1], and (ii) > f} = 0 for all e > 0, the family of measures in-

s IXn(u)l P{sup0<5<

duced on C[O, 1] by {X,~n = 1, 2,

*} is tight.

PROOF. The modulus of continuityof an element x of C is wx(3,a, b) = Ix(s) - x(t)I. By Theorem 8.2 of Billingsley(1968) it will suffice SUPaa,t5b;jatj<5 rn lim"_+O to show thatforeach positivee, lima Pn{x: w( 0, 1) > E} = 0. Clearly {wX(3,0, 1) _ E} C {wX(310, S) > A/2}U {w,(Q,s, 1) > A/2}. Thus it sufficesto show that, given a > 0, thereexist so > 0 and 3 > 0 such that (3.2)

P{x: w( lim"O00

0, so) > A/2}< 7/2

and lim,-._Pn{x: wx(3,so,1) ? A/2}< 7/2

(3.3)

Clearly wx(s,0, s)-< supo0u<,Ix(u)I; consequentlyby assumption lima o lim,_OO P,(x: wx(s,0

s) -

/2}

Let sobe a value such that limO00P,{x: w,(s, 0, s) > A/2}< 7/2 for s < so. Now pick a < so such that (3.3) is satisfied. This is possible, again using Billingsley (1968), Theorem 8.2, because by assumption {Yno , n - 1, 2, * } is tightfor so > 0. This completesthe proof. In Sections 4 and 5 we shall have occasion to use the followingvariations of Theorem 3.1: (3.4)

THEOREM.

stricted to C[O, (}=0,for

(3.5)

1 >0.

The random elements {X., n =

1,

.*} of C[O,

2,

s] form a tight family and lim8I0 lim,-_o P{suP18J
THEOREM.

1] when re-

IX,(u)I >

1, 2, .* } of C[O, 1] when re-

stricted to C[s, 1 - s] form a tightfamily and lim81I lim,_oo P{supo05U.,1Then {X., n = 1, 2, * }is tight. d} = 0, for e > 0.

au1

Xn(u)I

>

We now proceed withthe proofof tightnessof { WE, > O} usingTheorem3. 1. It will sufficeto prove two lemmas: (3.6)

LEMMA.

{WE(t),

( <

t _

1}, e > 0, induces a tightfamily of measures on

C[3, 1] for all 3 > 0.

PROOF. Let WE= until] o WEand IV+ = o W+. Given a > 0, thereexists a compact subset, K, of C[3, 1] such that P{ W+ e K} > 1- 7/2. Note from x + A, equations(1.2)and(2.10)thatforO < a < s < t < 1, ps(s,X t,y) p+(s = P{IW+CKI W+(3) = x} for ty + A). Thus, PI WE e KeI WE(3)= x -A K* is compact by the Ascoli-Arzela almostall x. DefineK* = (Uo
P{I

eK* } > =

P{ t

5-E

eK

P

-} K - (C I'eK

|

)=

x}p'

(:,

0, 3, x) dx

BROWNIAN

MEANDER

123

AND EXCURSION

-

PI A{+ e K I W+(3) = x}pe(0,0, ,x - ) dx = x}p+(0, 0O3, x) dx > P{I+ e KI f7V+(3) 1 -vj/2, -P{We+K}>

-

-

where the convergenceis justifiedby Scheffe'stheorem ([2], page 224). Thus, forE < E,. As mentionedin Section thereexistseasuch thatP{WE e K*} > 1 1, we are interestedin sequences of r.f.'s. For any fixedsequence of E's tending to 0, the correspondingr.f.'s, WET will be tight. (3.7)

LEMMA. For a > 0,

limI0

lim I0P{Isup0:8

a

IW(s)1s

I

?

}

=

1.

First note that it sufficesto prove

PROOF.

lima10lime10P{ MO()

_r =

Using the definitionof WE,the Markov propertyof W and generalizationof (11.10) of Billingsley(1968) gives P{sup0<8<6W(s) ?< a} = P{MO3) < 7;,m > -E}/P{m > -, -PE

=

P{-s

SE Ek=_ -n&(z

<

m(3) < M(3)

< r I W(3) =

P{m > -El

z}n,(z)P;{m(l

> -e dz

-3)

+ 2k(n + s) [n8(z + 2k(Q,+ s) + 2e)]N14-(0,z + e) dz/Nl(Os,).

Using dominatedconvergence,it is possible to take the limit as -> 0. Then Fubini's theoremjustifieschangingorderof summation. Then evaluationof the termcorrespondingto k = 0 shows thatthis termconvergesto 1 as a3-> 0. The remainingterms(k t 0) can be bounded by quantitieswhose sum converges(by monotoneconvergence)to 0 as a3-> 0. For details see [13]. Combining(3.1), (3.6) and (3.7) proves the tightnessof {We, s > 0}. 4. Convergenceof conditionedBrownianmeanderto Brownianexcursion. Our goal in thissectionis to prove that tied-downBrownian meander is Brownian excursion. We shall prove (4.1)

THEOREM.

WE+

W0+ as

s j 0.

We begin by showingthat the finitedimensionaldistributionsconverge. LEMMA. Thefinitedimensionaldistributions of WE+ convergeto thoseof (4.2) 0. s J. as Wo+ Because WE+ and W0+are Markov, it sufficesto show that the probabilitytransitiondensitiesconverge. For 0 < t < 1 and y > 0. PROOF.

PI+ (0, 0,t, y)dy

(4.3)

= P{W+(t) e dyIW+(1) < el -tN(-y = HY exp(-y2/2t)dy Nl-t(-y, -y + 1 - exp(- E2/2)

-

s, -y)

124

RICHARD T. DURRETT, DONALD L. IGLEHART AND DOUGLAS R. MILLER

Using L'Hopital's rule twice on the ratio above gives 0PE(0, 0, t,y) = 2y exp( Eim1

y2/2t(1 -

t))

Y= -P0+(0,0, t,y) For 0 < s < t < 1 and x,y > 0 P+

(4.4)

(s, x, t,y) dy

= P{W+(t) e dyI W+(s) = x, W+(1) < e} = g(t

-

-

s, x, y) dy Nl-t(-y, -y + e)-N1-(-y ,-xN,, -x,-x + e)

-y) - x)

6,

Divide numeratorand denominatorby 1 - exp(-_2/2) and use the same application of L'Hopital's rule as in (4.3) to see that lime I 0pE+(s, X, t' y) = pO(s, x, t,y)

which completesthe proof. Next we must show that the r.f.'s {WE+,6 > 0} are tight. We shall use Theorem 3.5. of C[O, 1 - s] as Given s > 0, definethe random element WEJ+ o ? WE. Then{ WE, theprojectionof WE onto C[O, 1 - s]; thatis WE = 7 1-8] 6 > 0} is tight.

(4.5)

LEMMA.

o W+. Definether.f. W* on C[O, 1- s] as follows; PROOF. Define W+ = U[O,,-,] W*(0) = 0, P{W*( - s) e dy} = po+(O,O 1- s, y) dy, and P{W* e A I W*(1 s) = y} = P{W+ e A I W+(1 - s) = y} forally, forany Borel setA ofC[O, 1 - s]. Given '2 > 0, thereexistsa compact subset, K, of C[O, 1 - s] such thatP{W* e K} > 1 - 72/2. s) = y}pE+(O0, 1 - s, Y) dy KI W+(1 -s) y}p,+(0,0, 1 -s, y) dy

P{JWEe K} = 5' P WE e K I WE( 1 =

P0P{ We

-

-0>SP{W+eK I +(l1-s)=y}p+(O,0,1-s,y)dy -'/2 =P W* e K} > where convergencefollows fromScheffe'stheorem([2], page 224). Given any fixedsequence of 6's tendingto 0, the correspondingr.f.'s We+ will be a tight family. (4.6)

= 1.

LEMMA. For ry> 0, limr0limrn P{sup1_8<'<1IWe (U)I <_

PROOF. Since We+(t)> 0 it sufficesto consider P{sUp1_8.U.1

WeN(U)

<

2} =-0

P{sup1_8Su.1

W+(u)

?_ )|

W+(1) < e}p,+(O 0, 1

(4.7) -

S'h(x, s,

+)Pe(O,

, 1 - s,x)

W+(1 - S) = X, s, x) dx

-

dx

.

125

BROWNIAN MEANDER AND EXCURSION

By Lemma 4.2, lim IOpe+ = Po+ thusiflime1I h(x, s, e) = h(x, s) and Ih(x,s, 6)l ? 1 forall (x, s, e) then by Scheffe'stheorem([2], page 224)

1 1 x) dx . limeI0j Ih(x, s, -)p+(, 0, - s, x) dx = j ' h(x, s)po+(O,0, - s, In fact, using equations (11.10) and (11.11) of [2], (4.8)

h(x, s, e)

P{M(s) _ ,21W(O) = x, mr(s)> 0, W(s) ? e} N8(2k - x, 2k17- x + e) - N8(2k2- x - s,2k/2- x) k= N8(-x, -x + e) - N8(-x - e, -x) n8t(2k) - x)/n8t(-x) = h(x, s) k=

= -

as e j 0 by dominatedconvergenceand L'Hopital's rule. For detailsof the domination see [5]. Now consider t'h(x, s)po+(O,0, 1 - x, x) dx "h(x, s)p8(dx) themeasurewithatomat x = 0, as s I 0. ThusbyTheorem5.5 by(1.3) p,8=0 of Billingsley(1968), it sufficesto show that limx,8_0h(x, s) = 1 to prove -

(4.9)

h(x, s)po+(O,0, 1 - s, x) dx = 1 .

1im840

When (4.7), (4.8) and (4.9) are combined theyprove the lemma. Thus consider h(x, s) = -1

k?

n,'(2k7 - x)

+

o (2k2 + x)n,(2k7 + x)

n8 (-X)

(2k2 - x)n8(2ka - x)

-

xn8(-x)

Let

ak(s,

ak(s,

x) be the kth termin the above summation. Then x)

=

expL

(2k72)+ 4k7x] + expL

[

exp

(2k72)2-4kx]

(2k2) + 4k7x] _ exp[ (2k7)g -4k7x]

+ 2k722s2

x

For x ? 2/2,theratio in theabove expressionis less than8krljf'(c)jwheref(y) = the second termdominatesthe exp(-y/2s) and c = (2k2)2 - 4k7x; furthermore, firstso (for x < )7/2) IaK(s,x)l ? 2 exp ? (2 +

(2k2)2- 4k7x] + 16k

exp [- (2k2)-

8k2'2s-1)e-k272/28

Since forc > 0, (4.10)

e-k2/2c<

rC

and

T,0I

k2e-k2/20< crC

4k72

126

RICHARD T. DURRETT, DONALD L. IGLEHART AND DOUGLAS R. MILLER

it followsthat (for 0 < x < 7/2) 'T_1k=l 5,2r1(27rs)i--+ 0 I1@ x)a X)l <_5 (2T) lk(S,

as s j 0, completingthe proofof Lemma 4.6. When combined with Theorem 3.4, Lemmas 4.5 and 4.6 imply tightnessof {W., e > 01which togetherwith Lemma 4.2 impliesTheorem 4.1. 5. Convergenceof conditionedBrownianbridgeto Brownianexcursion. This section is devoted to proving (5.1)

THEOREM.

Woe

W0+ as e I 0.

The proofwill follow the same patternas the method of Sections 3 and 4. First we considerfinitedimensionaldistributions,then the assumptionsneeded to apply Theorem 3.5. distributions LEMMA. Thefinitedimensional of Woeconvergeto thoseof

(5.2)

WO+as e j 0.

PROOF. Because Woeand WO+are Markov it sufficesto show that the transition probabilitydensitiesconverge. Using argumentssimilarto thosepreviously used the transitionprobabilitiesare derived. Pe(O, 0, t y)

-

Poe(S X, t, Y) -

[1

- exp(-2e(y

[ l-exp (-2s(y

+ s)/t)][1 - exp(-2e(y 1 - exp(-2 20)

+ -)/(I -t))]

+ s)/(l -t))][ l-exp (-2(y +e)(x 1 - exp (- 2e(x + -)/(I - s))

x n,,(y-

n~

_(y)

y

+ s)/(t-s))]

x)

and 0 < s < t < 1; where u = (t-s)(l-t)/(1-s). for x,y >-e L'Hopital's rule gives lime0pOe = Po+ completingthe proof.

Use of

LEMMA. Givens > 0, definethe randomelementWoe6 of C[s, 1 - s] as (5.3) theprojectionof Woe ontoC[s, 1 - s]. Then{Je6, e > 01is tight. PROOF. The proofis similarto (3.6) and (4.5). Define the random element of C[s, 1 - s] as follows: The joint distributionof (W*(s), W*(1 - s)) has fV_* - s) =y} = P{W* eA JV*(s) = x, JW7*(1 density po+(O, 0, s, x)pO+(s, x, t, y). Pt W+ e A J'-+(s)= x, JI'+(1- s) = y} forall x, y > O and for all Borel sets A of C[s, 1- s]. Note that,as in (3.6), P{JW+e A W+(s) = x, W+(1 - s) Pt We

e

A-s

I we(s)

=

x -

e, J`(1

')

=y

-

e} for almost all y and all Borel

A. As before,given ' > 0, thereexistsa compactset K such thatPt W* e K) > 1 - 7/2. DefineK* = (U0<,<1(K - -))- as beforeand show that limeI0Pt Woe e K*} > PtW* e K} > 1 - /2, followingthe proofof Lemma 3.6. (5.4)

LEMMA. For r,> 0,

and

| Woe(t)|<,2}=1 im810 lime1? Ptsupo0-t8 lim810

limeI0 P{sup1-S
(t)? I WoV

-

1.

127

BROWNIAN MEANDER AND EXCURSION

PROOF. It sufficesto consider Woe(t) Wq}

P{supO?1t:

?

=

P{MW(S)

= P{mo(s)

<

72}

?< 7, MO >

-e}/P{mO

>

-r

- exp(-2e2). By the Markov By (1 1.40) of Billingsley(1968), P{mo> -} property,(11.10), (11.40) of Billingsley(1968) and the relationshipbetween W and W0

P{MA(s) ? 7, MO''> -6}

caP{M(s) _ ,, mr(s)> -A W(s) X P{m(1 - s) > - | W(1 - s)

-

an6 [o~k=.

=

x [1

[n8(z +

=

z}n8(1-8)(z)

dz

+ 2k(7 + e) + 2s)]

2k(7 + e))-n8(z

s))]n8(-8,)(z) dz]/n8(z)

exp(-2e(z + -)/(1-

-

z}

Using L'Hopital's rule twice and the dominatedconvergencetheoremgives limeIoP{Mo0(s) ? =0

7}

z ~k=- 2s-'(z

+ 2kv) exp(-2k7(z

+ k7)/s)z(1

dz.

-s)-1n8(1-8)(Z)

The termfor k = 0 approaches 1 as s I 0. The remainingtermsare bounded by <0 <

_k=-oo;k0

(s(1

-

2(s(1 - s))-'(z + 2k7)zn8(,_8)(z)dz

s))-1 z1>, Z2n8(1-8)(z) dz

0

-*

as s 0 by dominatedconvergence. The second statementof the lemmafollows fromthe firstbecause of symmetryof W0on [0, 1]. Theorem 5.1 followsfromLemmas 5.4, 5.3, and Theorem 3.5 plus Lemma 5.2. .

6. The supremaof Brownianmeanderand Brownianexcursion. In thissection the continuitytheorem(Theorem 5.1, Billingsley(1968)) is used in conjunction withTheorems2.1 and 4.1 to derivethedistributionof M+ and MO+,respectively. Chung (1975, 1976) derivesthe distributionof MO+using a different approach. Durrettand Iglehart(1977) derivethedistributionof M+ as a corollaryof a more general theorem. (6.1)

THEOREM. P{M+ < x} = 1 + 2 Ik

(-1)'

k exp(-k2x2/2),for x > 0.

PROOF. It suffices to evaluate

10J,{Me < x}. By definitionof WEand (10.17) and (11.12) of Billingsley(1968) we have PIMe+
= P-6 =

-

< m < M < X}/Pt-6 (-1)'N(k(x

+

6)-6,

< ml} k(x +

6)

+ 6)/N(-6,

6).

By L'Hopital's rule and dominated convergencethis quantity has the desired limit. (6.2)

THEOREM. P{Mo
1+2

--(1-4k2x2)exp(-2k2x2),forx>0.

128

RICHARD T. DURRETT, DONALD L. IGLEHART AND DOUGLAS R. MILLER

PROOF. As above it suffices to evaluate limeI?P{M08< x}. By definitionof W0., and (1 1.38) and (1 1.40) of Billingsley(1968), we have P{MoS <

x}

= Pt-e < MO< MO< x}/Pt-e - k= - [exp(-2k2(x + e)2) -

exp(-2(x

1 + E,-

+ k(x +

< mO}

))2)]/(1 -exp(-2s2))

[2 exp(-2(k(x + -))2)

-

exp(-2(k(x + ;)

exp(-2(k(x + e) + s)2)]/(1-exp(-2s))

-

.

Multiplyingnumeratorand denominatorby e2 and usingL'HoJpital'srule twice on each shows that the kth term in the above expressionconverges to 2(1 4k2x2)exp(- 2k2x2). Therefore,the sum convergesto the desiredlimitby dominated convergence;the numeratorof the kthtermin the above sum equals exp(-2k2(x + s)2)[2

-

exp(-2s2)[exp(4ks(x + s)) + exp(-4ks(x + s))]]

which is less in absolute value than exp(-2k2x2) max (12 - exp(-2 2)[2]1, 12- exp(-2,2)[2 - (4ks(x + S))2]1) < exp(-2k2x2)[2 - 2 exp(-2I2) + exp(-2s2)(4ks(x + s))2] < exp(-2k2x2)(4s2 + (4ks(x + s))2) Thus since 1 - exp(- 2s2) > 2e2- 2 4 it follows that the kth term of the sum in questionis less in absolute value than2 exp(- 2k2x2)(1 + (2k(x + 6))2)/(1 - e2) which in turnis less than 2(1 + 16k2x2) exp(-2k2x2) for e < x. By (4.10) the above servesas a dominatingseriesin applicationof the dominatedconvergence theorem. REFERENCES [1] BELKIN, B. (1972). An invarianceprincipleforconditionedrandomwalk attractedto a stablelaw. Z. Wahrscheinlichkeitstheorie und Verw.Gebiete21 45-64. [2] BILLINGSLEY,P. (1968). Convergence of Probability Measures. Wiley,New York. [3] CHUNG,K. L. (1975). Maxima in Brownianexcursions.Bull.Amer.Math.Soc. 81 742-745. [4] CHUNG, K. L. (1976). Excursionsin Brownianmotion. To appearin Ark.Mat., December 1976. [5] DURRETT, R. T. and IGLEHART, D. L. (1975). Functionalsof Brownianmeanderand Brownianexcursion.Dept.ofOperationsResearchTechnicalReportNo. 37,Stanford University.

[6] DURRETT, R. T. and IGLEHART, D. L. (1977). Functionals of Brownian meander and Brownian excursion. Ann.Probability 5 130-135. [7] IGLEHART,D. L. (1974). Functionalcentrallimittheoremsforrandomwalks conditioned to stay positive. Ann.Probability 2 608-619. [8] IGLEHART,D. L. (1975). Conditionedlimittheorems forrandomwalks. In Stochastic Processesand RelatedTopics(Vol. 1 of the Proc. of the Summer Research Instituteon Sta-

tistical InferenceforStochastic Processes, M. L. Puri, ed.) 167-194. Academic Press, New York. [9] IT6, K. and McKEAN, H. P. JR. (1965).Diffusion ProcessesandTheirSamplePaths. SpringerVerlag, Berlin. [10] KAIGH, W. D. (1974). A conditional local limit theoremand its application to random walk.

Bull.Amer.Math.Soc. 80 769-770.

129

BROWNIAN MEANDER AND EXCURSION

randomwalk. Ann. [11] KAIGH, W. D. (1975). A conditionallocal limittheoremforrecurrent 3 883-888. Probability bya latereturn [12] KAIGH, W. D. (1976). An invarianceprincipleforrandomwalkconditioned 4 115-121. to zero. Ann.Probability of thesupremaof Brownianmeander,Brownian [13] MILLER, D. R. (1975). The distributions TechnicalReport,Univ. excursionandsomeotherBrownianpaths. Dept.ofStatistics of Missouri. T. DURRETT DEPT. OF OPERATIONS RESEARCH STANFORD UNIVERSITY

RICHARD

STANFORD, CALIFORNIA

94305

L.

DONALD DEPT.

STANFORD

DOUGLAS R. MILLER DEPT. OF STATISTICS UNIVERSITY OF MISSOURI COLUMBIA, MISSOURI 65201

IGLEHART

OF OPERATIONS

STANFORD,

RESEARCH

UNIVERSITY CALIFORNIA

94305