On probabilities that a random walk is negative

(1)

1.64033 Communicated 12 F e b r u a r y 1964 b y LENI~ART CARLESOI~

On the probabilities that a random walk is negative

B y BENGT ROSI~N

l. Introduction, notations and summary

Let X1, X~ . . . . be independent copies of a random variable X with distribution function F(x). The successive partial sums are denoted Sn = X I + X 2 + ... + Xn, n = 1, 2 . . . We define a~ = P(Sn < 0), n = 1, 2 . . . To every distribution function we get an associated sequence {an}[ r We list two immediate relations between the existence of moments of X and the asymptotic behavior of { a , } ~ .

A. The law of large numbers implies t h a t l i m ~ _ . ~ a n = 0 if E X > O and t h a t l i m ~ _ , ~ a = = l if E X < 0 .

B. F r o m the central limit theorem follows t h a t if E X ~ < co and E X = 0 then

lim a= = 89 (1.1)

The main aim of this paper is to answer the following question raised b y F. Spitzer in [3], p. 337. Does there exist a distribution F(x) for which the sequence { =}1 fails to have a (C, 1)-limit? a

I n Theorem 1 we show t h a t there is a distribution such t h a t E IXI2-~< co for every ~ > 0 , for which {a=}~ does not possess a (C, 1)-limit. I n Theorem 2 we discuss the limitability of {an}[ r for general limitation methods, and show t h a t for a n y regular linear limitation method there exists a distribution for which {an}F cannot be limited.

According to A, B and the result in Theorem 1, the condition E X ~ < co and E X = 0 is a weakest possible sufficient condition in terms of moments only for (1.1) to hold. In Theorem 3 we give a more general sufficient condition for (1.1). The essence of this theorem is t h a t (1.1) holds if F(x) does not deviate too much from a distribution which is symmetric around zero.

I wish to t h a n k Professor L. Carleson for having suggested the theme of this paper and for valuable guidance.

a oo

2. Existence of distributions for which { .}1 cannot be limited

Theorem 1. There exists a distribution F(x) with E IXI 2 ~ < cr /or every (~ > 0 /or which upper and lower (C, 1)-limits o~ {an}~ are respectively 1 and O.

(2)

B. ROS]~N,

On probabilities that a random walk is negative

Proo/.

W e show t h e e x i s t e n c e b y a n e x p l i c i t e x a m p e l . W e d e f i n e a d i s c r e t e d i s t r i b u t i o n w i t h m a s s p o i n t s {c:}~ r a n d c o r r e s p o n d i n g p r o b a b i l i t i e s

p ~ = P ( X = c ~ ) = [ ( e - 1 ) v ! ] -1,

v = l , 2 . . .

T h e e s s e n t i a l f e a t u r e of t h i s choice of t h e p r o b a b i l i t i e s is t h a t

p:+l/p:-+O

w h e n v--> + . W e f i r s t d e t e r m i n e t h e c's w i t h o d d indices. L e t

C 2 v _ 1 ~ ( _ _ l ~ v + l ~-( 89 ~] F 2 ~ - I , v = l , 2 , . . . ,

w h e r e ;t(2v - 1) = ~t(2v) = (log 2v) -+. (2.1)

The. e s s e n t i a l p r o p e r t y of +~(v) is t h a t i t t e n d s to 0, b u t n o t t o o f a s t , w h e n --> + . W e o b s e r v e t h a t c~ is a l t e r n a t i v e l y p o s i t i v e a n d n e g a t i v e w h e n v r u n s t h r o u g h o d d indices. F o r e v e n i n d i c e s we d e f i n e c2~ t h r o u g h t h e r e l a t i o n

pe~-lc2~=l+p2~c~=O,

v = l , 2 . . . . (2.2)

,'n89 l(2~) ~-1 (2.3)

w h i c h y i e l d s c2~ = ( - 1)" " r 2 , - 1 i02,.

T h e d i s t r i b u t i o n is n o w c o m p l e t e l y specified a n d we d e r i v e s o m e of its p r o p - erties. I t is e a s i l y c h e c k e d t h a t

~P~"~Pn

w h e n N--->co (2.4)

N

a n d t h a t [e2~-11 < [c2~[, v = l, 2 . . . (2.5)

Note.

T h r o u g h o u t t h e p a p e r t h e s y m b o l ~ m e a n s t h a t t h e r a t i o of t h e q u a n - t i t y t o t h e r i g h t a n d t o t h e l e f t of ~ t e n d s t o 1.

N e x t we show t h a t

E [ X { 2 - o < c~

for e v e r y (~>0. L e t v be e v e n a n d (~>0.

A s ~t(n)-->0 v h e n n--> c~ t h e following i n e q u a l i t y h o l d s w h e n v is s u f f i c i e n t l y l a r g e

P~I ~: ll~-~ _- ~={ ~- ~<~)<2-~) ~ 1 = ~ < (v l) - ~ . F o r v o d d a n d s u f f i c i e n t l y l a r g e we h a v e

":i - r : 1 , : - 1 ~< - - ~ v [ ( v I)!]-+~.

\ P ~ l "

T h u s EIX[:- : < + . (2.6)

I n p a s s i n g we m a k e t h e following o b s e r v a t i o n s . W e a r e going t o s h o w t h a t f o r t h e d i s t r i b u t i o n we h a v e c o n s t r u c t e d i t h o l d s t h a t {an}~ r h a s n o t a (C, 1)- l i m i t a n d

a /ortiori

t h a t {an}~ h a s n o t a l i m i t . F r o m A a n d B in w 1, i t follows t h a t such a d i s t r i b u t i o n m u s t s a t i s f y

3 1 8

(3)

(i)

EX 2= ~ ,

(ii)

E X = 0

if t h e m e a n of X exists.

F o r t h e a b o v e d i s t r i b u t i o n (i) is easily verified a n d (ii) follows f r o m (2.2).

W e shall need t h e following e s t i m a t e later.

N

Ep~c~<HpNCUN, N

even, (2.7)

1

w h e r e H is a c o n s t a n t i n d e p e n d e n t of N . F r o m (2.2) if follows t h a t p2~[c2~]=

p2v-1]C2v-l[ and

(2.5) gives

p2~c2~ > P2~_ l c2~_1.

2 2 T h u s

N N/2 N/2

C2 1

~ 2pNc~ Y. p2~ 2,pN C;~ ~.

v = l v = l v = l

E s t i m a t e s w i t h Stirling's f o r m u l a yield

2 - 1 - 2 0

l i m P 2 ( v - 1 ) c 2 ( ~ - l ) p 2 v c2v = y.-~ or

c 2 - I c2~ ~< for

a n d t h u s p2(v-1) 2(v-1)P2v 89

V~Vo.

This implies

NI2 2 ~ 0 2 {

~l P2~C2~--~pNCN 1 + 89 + (89 +... + p~l c~2 ~ p2vc2~}

a n d n o w (2.7) follows as

p~C2N--> ~

w h e n N--> ~ . W e i n t r o d u c e t h e e v e n t s

A(n,

N ) : Sn a n d cN h a v e the s a m e sign.

B(n, N) : Xt, X~ .... , X,~

all a t t a i n t h e i r values a m o n g (ct, %, ..., CN),

Ck(n,

N ) : E x a c t l y k of X1, X 2 .. . . . Xn a t t a i n t h e value cN, k = 1, 2, . . . , n . F o r simplicity, we shall s o m e t i m e s suppress t h e indices n a n d N a n d we u n d e r s t a n d t h a t t h e y b o t h are t h e s a m e for A, B a n d C w h e n these e v e n t s occur simultaneously. T h e following inequalities are i m m e d i a t e

n

P(A) >1 P( [.J ABCk) = ~ P(ABCk)

k=0 k=O

>1 ~ P(B). P(Ck[B)" P(A I BC~),

k = K

w h e r e K is a n o n - n e g a t i v e integer

<~ n. P(A]BCk)

increases w i t h k for

k<~n

a n d we g e t

P(A(n, N)) >7 P(B). ~ P(C~ I B). P(A ]BCK).

(2.8)

k=K

(4)

B. ROSfiN,

On probabilities that a random walk is negative

We shall let N tend to infinity and we consider the following choices of n and K as functions of N.

n ( N ) = N

~a(N)p~l for 8 9 1,

K(N)= 89

Our aim is to show t h a t

P(A(n(N), N)) ---> 1

(2.9)

uniformly in a for 89 ~< ~ ~< 1, when N--> ~o through odd values. We do this b y showing t h a t all three factors to the right in (2.8) tend to 1 uniformly in for 89 ~< ~ ~< 1. We start by showing

lim

P(B(n(N),

N ) ) = 1 (2.10)

N - - ~

and the convergence is uniform for 89 ~ a ~< 1.

~ n ( N ) ~ (1 ^~ , n ( N )

P(B(n(N),N))

= ( ~ p ~ ) - ~+lp. ) exp ( - n ( N ) p N + I ) according to (2.4).

and (2.10) follows.

Thus

P(B(n(N), N)) ,,~

exp ( - N -~(m) Next we prove

n(N)

lim

Z P(Ck(n(N),N)[B)=

1

N.--~oo k f K(N)

(2.11)

and the convergence is uniform for ~ <~ ~ ~< 1. We introduce the truncated random variable X (m and the random variable

y(N).

Then and

N - 1

. : : 1 , 2 . . . N ( 2 1 2 )

otherwise.

"2 P ( C k ( n ( N ) , N ) I B ) = P . . . .

k = K(N)

where the Y~(m's are independent. The random variable ~

y(N)

has a binomial distribution with mean

n(N)pN (~Np~)-l.

Estimates with Tchebyeheff's inequality give t h a t the right hand side in (2.13) tends to 1 uniformly for 89 1. Thus (2.11) is proved. Finally we show

3 2 0

(5)

P(A(n(N),

N)[

BCK(m) -+ 1 (2.14)

uniformly for 89 ~< ~ ~ 1 when N--> oo through odd values.

The conditioned r a n d o m w r i a b l e

Sn I BCK(m

is identical in distribution with the r a n d o m variable

n ( N ) - K ( N )

K ( N ) . cN + ~

y ( N - 1 ) 1

where the X~ N-l) are independent copies of the r a n d o m variable

X (N-l)

defined in (2.12). Thus

As N is assumed to be odd it follows from (2.2) t h a t

E X (~-1)= O.

Tchebycheff's inequality now yields

N - 1

( n - K(N)) ~, p,c~

P ( A ] B C K(N)) >1 1 1

N - 1

K(N) c V p,

1

a n d in virtue of (2.7)

>1

1 - H " n ( N ) p N - 1 C ~ - 1

N--1

K(N) 5 P,

= 1 - R ( N ) .

B y inserting the choices of

n(N)

and

K ( N )

we get

R ( N ) , - , 4 H PN+a "PN-2 p~,~(N)

~ 2 )~(N-2) "~Vga(N)

PN'PN-1 ~'N-2

9 ~20,( N ) - )'( N - 2))

" 4 H N ~ ( N ) [ N ( N -

1)] -2~(N) ~,N-2

_20(N)-~(N-~)) ~ 1 a n d we get An estimate with Stirling's formula gives t h a t ~,~_~

R ( N ) ~ 4 H e x p - 4) Vl- g

Thus

P(AIBCK(m)->

1 uniformly for ~ ~<m~< 1 a n d (2.14) is proved. Now for- mulas (2.8), (2.10), (2.11), a n d (2.14) together i m p l y (2.9).

F o r odd values of N, cN is every second time positive a n d every second t i m e negative. Thus we get as an immediate consequence of (2.9) t h a t l i m n _ ~ an = 1 and li_mmn_,~an=0. We w a n t to sharpen this to the result t h a t also upper and lower (C, 1)-limits of {a~}~ are respectively 1 a n d 0. Choose e > 0 . F r o m (2.9) it follows t h a t if N is odd a n d sufficiently large a n d

CN

< 0, t h e n

an/> 1 - e for

n 1 (N) <~ n <~ n~(N),

where n 1 (N) =

N-a(N)p~l+l

a n d

n2(N ) = N- 89 1.

Thus

(6)

B. rtos~r(, On probabilities that a random walk is negative

1 n~(N) 1 n,(N, [

= n , ( N )

N o w n I ( N ) / n z (N) ---> 0 w h e n N---> ~ . T h u s lim 1 ~ a , = l .

7 t - - ~ r n v = 1

nl(N)~

n~(N)] "

I n t h e same m a n n e r , it follows t h a t

n

lim _1 ~. a~ = 0 n-~cr n~=l a n d T h e o r e m 1 is proved.

Concluding remark. As t h e an's are probabilities, t h e y lie b e t w e e n 0 a n d 1.

I t is well k n o w n t h a t Abel a n d (C, 1)-limitability are e q u i v a l e n t for b o u n d e d sequences (see e.g. [1] T h e o r e m 92). T h u s for t h e distribution c o n s t r u c t e d a b o v e it holds t h a t (an}; r cannot, be Abel limited. I n fact, it is n o t h a r d to show directly t h a t {an}~ has u p p e r a n d lower Abel limits respectively 1 a n d 0.

T h e p a r t of T h e o r e m 1 which concerns non-limitability of {a~}~ holds for general linear limitation methods. W e consider a regular limitation m a t r i x [Tmn], m, n = 1,2 . . . i.e. we assume

(ii) lim 7~= = 0 f o r all n,

m - - ~

(iii) ~ 7ran---> 1 w h e n m---> ~ .

Theorem 2. .For every regular limitation m a w i x [~'mn] there exists a distribution F ( x ) /or which the sequence {an}~ satisfies

amt

lim ~ )~mn an = 1

r n - - ~ o o n

lim ~ ~'mn an = O.

(2.15)

Remark. W e do n o t k n o w a n y general relation b e t w e e n [Tmn] a n d t h e o r d e r of t h e m o m e n t s t h a t F ( x ) can possess w h e n (2.15) holds.

Proo/. T h e m a i n idea in t h e proof is t h e same as in t h e proof of T h e o r e m 1 a n d therefore we m a k e t h e proof s o m e w h a t brief. W e c o n s t r u c t a discrete dis.

t r i b u t i o n with points of mass {c~}~ ~ a n d corresponding probabilities {p~}~. T h e successive signs of Ca, C 2... are chosen + - + - + - . . . . L e t {~v}~ ~ be a sequence of positive n u m b e r s which t e n d t o 0. W e d e t e r m i n e {p,}~o, {c~}~o a sequence {m~}r of integers a n d a sequence {i,}~o of intervals of integers I~ = [i,, ~'~]

recursively. W e assume t h a t p,, c~, mr, a n d Iv are d e t e r m i n e d f o r v = 1, 2 . . . N - 1. W e consider PN as a f u n c t i o n of t h e p a r a m e t e r 2N given b y t h e relation 322

(7)

N - 1

a n d we shall d e t e r m i n e PN b y d e t e r m i n i n g ;tN, 89 ~ 2N< 1. Firs~3~,we choose cN so

N C - - ~ v

t h a t IcN[ >

ICN_i I

a n d sgn (ZI P, ,) - sgn (cN) w h e n 2N = 89 L e t y(N) (2N), V 1, 2 , . . . , be i n d e p e n d e n t r a n d o m variables w i t h d i s t r i b u t i o n

N - 1

T c h e b y c h e f f ' s inequality implies t h e existence of a n u m b e r

iN=i(eN)SUCh

^{t h a t}

iN > ?'N-1 a n d

e (~1X(~N) (2N) a n d cN h a v e t h e s a m e sign) ~>1 - e N (2.16) w h e n n >~ iN a n d 89 ~< 2~ < 1. N o w choose m N > mN 1 S O large t h a t ~ n = l i N

I~)mlvn [ <~- ~N

a n d jN large e n o u g h for

~n%J,,+llYm~vnl <'~.eN

to hold. These choices are clearly possible. Finally, we fix RN a n d t h u s PN b y t h e condition

p~

)'-

/> 1 - eN. (2.17)

T h e d i s t r i b u t i o n is n o w c o m p l e t e l y determined. L e t

X1,

X~ . . . . be i n d e p e n d e n t r a n d o m variables w i t h this distribution a n d

@ (n, N) = P{S,~

a n d cN h a v e t h e same sign). T h e n

~(n,N)=P(Sn

a n d cN h a v e t h e same sign

IM nlX I < N)P(MaxlX l<cN).

I n v i r t u e of (2.16) a n d (2.17) we g e t

Q ( n , N ) ~ > ( 1 - e N ) ~ w h e n

n E I N

a n d t h u s an >~ (1-- eN) ~ w h e n

n E I N

a n d N is even, while an ~ l -- (1-- eN) 2 w h e n n E IN a n d N is odd. T h e t h e o r e m n o w follows.

3. A s u f f i c i e n t c o n d i t i o n f o r l i m an = 89

A c c o r d i n g t o B in w 1

E X ~ < co

a n d

E X = O

is a sufficient condition for lim an = 89 However, this condition is n o t necessary. This follows i m m e d i a t e l y f r o m t h e fact t h a t if

F(x)

is c o n t i n u o u s a n d s y m m e t r i c a r o u n d 0 t h e n an = 89 for all n a n d t h u s lim a = = 8 9 I n t h e n e x t t h e o r e m we show t h a t t h e a s s u m p t i o n s a b o u t s y m m e t r y a n d the existence of a finite second m o m e n t a n d zero m e a n c a n be c o m b i n e d t o get a m o r e general sufficient condition.

(8)

B. ROSI~N, On probabilities that a random walk is negative

Theorem 3. I / F ( x ) is non-degenerate and can be decomposed F ( x ) = H ( x ) + G(x), where H and G are o/ bounded variation and satis/y

(1) H ( x ) is symmetric around O, i.e.

H( - x) - H ( - ~ ) = H( oo ) - H ( x ) /or all x/> 0 which are continuity points of H(x).

(2) S-~ x2 IdG(x){ < oo ant ~_% xdG(x)

^{= 0 ,}

hen limn_,r162 an = 89

Proo/. The proof will be based on the following formula from [2], p. 331.

J a n - 89 <_n ( ~ I~(t)l" larg q~(t)[ d t + R ( n , O ) , (3.1) Jo t

where ~(t) is the characteristic function of F(x) and where R(n,&)--->O when n--> ~ for every 6 > 0.

As H ( x ) is symmetric around 0, we have

Im {~(t)}= f ; sin

xtdF(x)=f[:csinxtdG(x)

and from (2) it follows thai',

lira t -2 ~ sin xt dG(x) = O.

t - - ~ 0 j-r162

Thus ]arg q~(t) l <~ t ~ h(t), (3.2)

where h(t) --> 0 when t --> 0.

We shall also use the fact t h a t there are positive numbers 60 and C such t h a t

I~(t) l <

^{1 -} ^{Ct 2} ^for

I t1-<6o (3.3)

For a proof of (3.3) see e.g. Lemma 1 in [2].

B y inserting the estimates (3..2) and (3.3) into (3.1), we obtain, for 0 < 6 ~<6 0,

, f:

{ a n - 8 9 sup h(t) nte-"~ 6)

1 sup h(t) + R(n, 6).

Thus lim I a n - 89 (2~C) -1 sup h(t)

n - - ~ O~<t~<~

and by letting 6--> 0, we obtain the desired result.

324

(9)

R E F E R E N C E S

1. HAm)Y, G. H., D i v e r g e n t Series, Oxford, 1949.

2. Ros~.~, B., On the a s y m p t o t i c d i s t r i b u t i o n of sums i n d e p e n d e n t identically d i s t r i b u t e d r a n d o m variables. A r k i v M a t e m a t i k 4, 323--332 (1961).

3. SPZTZER, F., A c o m b i n a t o r i a l l e m m a a n d ist application to p r o b a b i l i t y theory. Trans. of t h e Amer. Math. Soc. 82, 323-339 (1956)

Tryckt den 9 juni 1964

Uppsala 1964. Ahnqvist & Wiksells Boktryckeri AB