On some results for Bernoulli excursions

journal of statistical planning Journal of Statistical Planning and Inference 54 (1996} 45 54

ELSEVIER

and inference

On some results for Bernoulli excursions E n d r e Cs'aki t Mathematical lnstilule of the Hungarian Academy
Abstract

We give a short survey of some distribution results for Bernoulli excursion. The re=tin emphasis is given to quantities expressible in terms of additive functionals, such as local times, area, etc. A?vlS classification: Primary 60J15; secondary 60F05, 60J55, 60C05

Keywords: Random walk; Excursion; Additive functional

O. Introduction

The simple symmetric random walk has been extensively studied in the literature. It is; well known that a number of statistical and combinatorial problems are closely connected with counting random walk paths or, equivalently, counting lattice paths. In this context Mohanty (1979) gives a nice survey of the results up to 1979. Since then, however, this subject became a vivid and very fruitful area of research, the literature on this subject is enormous. In our list of references, which is far from complete, we just mention some of these papers. Recently much attention has been paid to the so-called excursion theory, first studied in the case of continuous processes such as Brownian motion, etc. The idea to derive results for Brownian excursion via random walk excursions however looks quite natural and simple. So it is quite straightforward to study random walk or Bernoulli excursions. It seems that for Bernoulli excursions, the first results are due to Szekely (1977) and Kaigh (1978) who derived the distribution of the maximum of lhe excursion. Some more exact and limiting distribution results are given in Csitki and Mohanty (1981,1986). In a series of papers, Tak/tcs (1986,1991a, b,c, 1992a, b. ct

1 Research supported by Hungarian National Foundation Ibr Scientific Research. Grant No. 1905. 0378-375896/$15.00 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 3 7 5 8 ( 9 5 ) 0 0 1 5 5 - 7

E. Cshki/Journal of Statistical Planningand inference 54 (1996) 45 54

46

investigated the area and some related r a n d o m variables for Bernoulli excursion and pointed out a n u m b e r of relations to other problems. The first chapter of the b o o k by R6v6sz (1990) is devoted to the study of the simple symmetric r a n d o m walk, where a n u m b e r of results concern Bernoulli excursions. In the present p a p e r we are interested mainly in distributions and joint distributions of r a n d o m variables expressible in the form of additive functionals. O u r aim is to survey these and closely related results. But, of course, we do not claim completeness, there are much m o r e interesting and i m p o r t a n t results and papers we do not mention.

1. Exact distribution results so Let {X ~}i~1 be a sequence of symmetric P(Xi = 1) = P(Xi = - 1) = 1/2 and put

So=O,

S , = X1 + ... + X , ,

Bernoulli

n = 1,2,...

random

variables, i.e.

(1.1)

Then So, $1, ... is a simple symmetric r a n d o m walk. Define p = min{i: i > 0, Si = 0},

(1.2)

i.e the time of the first return to zero. The section (So, $1 . . . . , Sp) is called Bernoulli excursion. Let p = 2 n , then either S 0 = 0 , S i > 0 , i = 1 , 2 , . . . , 2 n - 1 , $2,=0 or S 0 = 0 , S i < 0 , i = 1 , 2 , . . . , 2 n - 1 , $2,=0. In certain papers the section (So, $1 . . . . . $2,) with the p r o p e r t y So = 0, S~ ~> 0, i = 1,2 . . . . . 2n - 1, $2, = 0 is also called Bernoulli excursion. We shall call it non-negative excursion, while the former will be called positive (or negative) excursion. N o t e that if (So . . . . ,$2,) is a nonnegative excursion of length 2n, then (0, So + 1, ..., $2, + 1,0) is a positive excursion of length 2n + 2. Certain characteristics of the r a n d o m walk can be expressed in terms of additive functionals of the form

A, = ~ f(S,)

(1.3)

i=1

with certain function f(z).

Examples:

(1) f (z) = 1. Then Ap = p, i.e. the length of the excursion.

(2) f(z) =

{10 i f z = x , otherwise,

(1.4)

E. Cshki /Journal of Statistical t'lanning and ln/erence 54 (1996) 45 54

47

where x is an integer. Then A,,=~(x,n)=

# { i : 1 <~ i ~ n, Si = x~j,

11.5)

i.e. the local time of the r a n d o m walk at x. (3)

./(z)={~

if

[zl=x,

(1.6)

otherwise, where x is a positive integer. Then

(1.7)

A , = ~ ( x , n ) = ~ ( x , n ) + ~( - x,,+).

(4) 10 if a ~ z < ~ b , otherwise,

f(z) =

(1.8t

where - ~ ~< a ~< b ~< m, i.e. A, is the occupation time of the interval [a, h]. f(z) = z, i.e. A,, = ~7 1 S, is the area under the r a n d o m walk path. (6) f ( z ) = Izl, i.e A, = 2i=1 [S,[ is the absolute area under the r a n d o m walk path. There are, of course, other r a n d o m variables not expressible in the form (31. b'or example, the m a x i m u m tin = m a x IS,I 1 <~i<~n

(1.91

of the r a n d o m walk path is not an additive functional and cannot be expressed by (3). The joint distribution of p, Itp and ¢(x, p) is given by Cs/~ki and M o h a n t y (19861 in the form E(z~';~(x,p) = j , laj, < c)

= 2(1-w~ 2 \1 + w /

wx

( w ~J-1(1-w

(1 2 ~,~)2 \ i ~ w ' /

x 1

1 -- w 7~ ~

1 __.___W~ x- l)j1 -w"

~

/

1 ,

I 1 10)

where 1 -(1 W =

-

Z2) 1/2

1 Jr- (1 - - Z2) 1 / 2 '

Iz] ~< 1,

x=

1,2,...,

c=x+l

.... ,

j=l,2

....

(l.llj

48

E. Csitki/Journal of Statistical Planning and Inference 54 (1996) 45 54

F r o m this formula we can obtain the marginal distributions of p, ((x, p) and/~p. F o r example,

l(

L~ j-1

P(~(x,p)=j)=~ l - 2 x j

' j = 1 ' 2 ....

(1.12)

(see also R~v~sz, 1990). In general, the distribution of A e cannot be given explicitly. We can give however a recursion for

Cj(q) = E(qA,';p = 2j)

(1.13)

(see Cs/~ki, 1994).

E(qA2"; $2,, = O) ~- ~ Cj(q)E(q A .... J; S 2 , - 2j = 0).

(1.14)

j-1

Hence if we k n o w E(qA'-,;S2, = 0) for all n, then we can obtain C / q ) recursively. The joint m o m e n t generating function of p and Ap can be determined from (l.14) as 1

E(qA ,'z ,') = 1 -- S~n~_oE(qA2,,; S2n = 0)z 2""

(1.15)

By putting q = 1 into (1.14) we can obtain 1 cj(a) = P ( p = 2j) = 22-577T_ ~ Cj-1,

(1.16)

where

Ck=k+lkk/ is the Catalan number. Hence Cj(q) can be considered as a q-extension of the Catalan number. We can also consider a q-extension of the ballot number. Define p(k) = min{i: i > 0, Si = k}

(1.17)

B(~)(q) = E(qA/~,; p(k) = j).

(l.18)

1 ~(k) B~:~)(I) = p ( p(k) = j) = ~ ~ J ,

(1.19)

and

Then

E1 C~'iJki/Journal o[' Statistical Planning and Inference 54 f1996) 45- 54

where

B~ik) =

k(.;+ ,/

-/k

4')

~ 1.2())

. / \ 2~/ is the ballot number. We can obtain the following recursion:

E(qA";& = k) = L B![)(q)Ek(q '''' ,:S,, .i j

k).

(1.21t

k

where n and k have the same parity and Ek denotes expectation under the condition that the r a n d o m walk starts at k. Hence for the joint generating function we oblain

E(q.%,~,_,,,~,) = ~ = k E ( q ";S,, = k)z ~ }~,__oEk(q ";S, = k)z"

ll.22)

N o w consider the case when f I z ) = z. Then

E(q s~+

~s";S.=k)=~

(n+k)/2

(1.,~3)

q~

where

[a+b]

(q"+b-1)(q"+"

h

~=

l-l)...(q"+'-l)

~--1)(T ~

11.24)

1)...iq-1)

is the so-called q-binomial coefficient. Eq. (1.21) gives

l[n

lqk(n+l)/2(t,2k:,4

2" 07 + k)/2 qe

L

E((/~'~ 1 @

= ~=~

--St k' "

(k)

' ;P

' [ /'--.j ~ =/)2~L(n-.j)/2jq:

i~[/((H j)(11

j) 2 "4-

{1.25)

F r o m (1.15), (1.22) and (1.23) we obtain

E(q s~+ " +S,,z')= 1 -

1

22- o [~,;,],eq- o~(~)~

11.26)

and

E(qA, '' 'z' ~)

Y~n=k (n + k)/2 q:~ :~,

Z.=o

r-

n

1

b/_q~:q

kn-n2,4.

(1,27)

(9"

All the series converge for q = e j', t real and Izl < 1.

E. Csitki/ Journal of Statistical Planning and Inference 54 (1996) 45-54

50

F o r the non-negative excursion (So . . . .

, S2n )

2tt

put T2n = ~i= 1 Si and

f,(n + 2j) , C,

P(T2, = n + 2j) -

(1.28)

where f,(n + 27") is the n u m b e r of sequences (So . . . . ,S2n ) such that So = S2n = O, Si/> 0, i = 1, ... ,2n - 1 and T2, = n + 2j. Takfics (1991a) shows that the generating function O

(a,(z) = Z f,(n + 2j)z ~

(l.29)

j=O

satisfies ~)n(Z) = Z ~)i_ l(Z)~)n i(Z)Z i-1.

(1.30)

i=1

He shows m o r e o v e r that

F(z,w) = w ~ O,(z)(zw)"

(1.31)

n=O satisfies

F(z, w) = w + F(z, w)F(z, zw)

(1.32)

from which the continued fraction expansion W

F(z, w) =

ZW Z2W

1 1

....

can be obtained.

2. L i m i t t h e o r e m s

There are various types of limit theorems concerning Bernoulli excursions. O n e m a y consider e.g. limit distributions under the condition p = 2n as n --* oo. In this context Szdkely (1977) and Kaigh (1978) show that lim P(#2, < z x / ~ / P n~°°

= 2n) =

~

(1 -- 4k2zZ)e -zk2z~.

(2.1)

k=-~

Some more related limit distributions are given in Csfiki and M o h a n t y (1981, 1986). In the case when f ( z ) = [zl, the limiting distribution of At, under p = 2n was investigated by Takfics (1991a, 1992a, c). Based on the formula (1.30) he determined

E. Csltki / Journal o f Statistical Planning and Inference 54 (1996) 45 54

51

the moments of the limiting distribution from which he obtained

/ A2.

..... lira e ~

) ,/'6 -~

<~ x/p = 2n = x - k_~le-[~V~ '3 U{~,5, 1 4 vk).

(22)

where U is the confluent hypergeometric function, vk = 2a3/(27x 2) and - a k are/he zeros of the Airy function. These limiting distributions give also the corresponding distribution results for Brownian excursion. Another type of limit theorems was investigated by Sinai (1992) and Csiki (1994}, where the limiting distribution of N

A,,~. = ~ {A,,,- A,,, ~)

{2.3)

i-1

was determined for f(x) = x, where 0 = P0 < Pl < " < PN < "' are the consecutive returns to zero of the random walk. Here the summands are i.i.d, random variables. Based on the results of Korolyuk and Borovskikh (1981) it was shown in Cs',iki (1994) that Nlim~E exp i t ~ - - v ~ 2

=exp

--G(t, vi "

{2.4}

where 1

'

Some more limit theorems can be obtained from (1.26). Put k = k~ ~ [yx/~], q -- exp{ --2/r/3/2} and ~(2) = ~(2;y) = ~im E ( exp { - 2

s~ + 'i~3~+ s"";"t //p{y,',T}= ' n)

.

(2.6)

Then using that

l

14

(n + k)/2 q2

qk{n+l)/2 {,2-k:,.'~

2" (n + k)/2

2 + 0

, (2.7)

one can see that ~(2) satisfies the integral equation exp

- ~ - 4 24

-x/~1

- -~- 2y(1-- z} + ~4 (1-- z)3 d z. f~ ~(2zS/2) z3/z(ly_ z) T75exp{ yZ_T?. (2.8)

52

E. Cshki/Journal of Statistical Planning and hference 54 (1996) 45 54

3. Relations to random trees and other problems A tree is a connected undirected graph which has no cycles, loops or multiple edges. Consider a rooted tree, i.e. one vertex, the root is distinguished from the other vertices. It has been shown (see Takfics, 1991a) that there is a bijection between the set of positive Bernoulli excursions (So, $1, ..., $2,) of length 2n and the set of rooted trees with n unlabeled vertices. By this correspondence: the total number of such trees is C,_ 1 = ( l / n ) \

1J tl2n -- 1 is the maximal height (the longest distance of a vertex from the root), Z(V 2n ; ± i : l k S i -- n) is the total height of the tree. Hence the results mentioned in Sections 1 and 2 can be translated into the corresponding results for random trees. There are other related problems closely connected to Bernoulli excursions. We mention briefly some of them. Tournament problems are discussed in Winston and Kleitman (1983) and Takfics (1986). Takfics (1991a, c, 1992a, c) also discusses related problems for empirical distributions, branching processes, etc. There is a vast literature treating q-numbers (q-binomial coefficients, q-Catalan numbers, q-ballot numbers, etc.). We may mention the book by Andrews (1976) and the papers by Carlitz (1972), Gessel (1980), Handa and Mohanty (1980), Fiirlinger and Hofbauer (1985), Niederhausen (1986), Krattenthaler (1989a, b), Krattenthaler and Mohanty (1993), etc. Other references can be found in these papers but this list is not exhaustive. Bernoulli excursion is also known for combinatorialists under the name of Dyck path. See Delest and Fedou (1993). Some counting problems for lattice path on the plane is also connected to Bernoulli excursions or Bernoulli random walk in general. We may mention DeTemple and Robertson (1984), Csfiki et al. (1990), Breckenridge et al. (1991), Guy et al. (1992), and Bercucci and Verri (1992). n

1

Acknowledgements This paper was presented at the Third International Conference on Lattice Path Combinatorics and Applications in Delhi, 12-14 January 1994. The author is indebted to the organizers for the invitation.

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