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A NNALES DE L ’I. H. P., SECTION B

L AJOS T AKÁCS

On the distribution of the supremum for stochastic processes

Annales de l’I. H. P., section B, tome 6, n

o

3 (1970), p. 237-247

<http://www.numdam.org/item?id=AIHPB_1970__6_3_237_0>

© Gauthier-Villars, 1970, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

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On the distribution of the supremum for stochastic processes

Lajos TAKÁCS (*)

Case Western Reserve University, Cleveland, Ohio.

Ann. Inst. Henri Poincaré, Vol. VI, 3, 1970, p. 237-247.

Section B : Calcul des Probabilités et Statistique.

SUMMARY. - In this paper mathematical methods are

given

for

finding

the distribution of the supremum for

compound

recurrent processes and for stochastic processes with

stationary independent

increments. New and

simple proofs

are

given

for a result of H. Cramer and for a result of

G. Baxter and M. D. Donsker. The paper also contains some extensions of these results.

I . INTRODUCTION

Our aim is to

give

mathematical methods for

finding

the distribution of = sup for

separable

stochastic (. -’B /?

0

2014 u oo

~.

J

Such methods were

given by

H. Cramer

[2]

in the case u

00 }

is a

compound

Poisson process and

by

G. Baxter and M. D. Donsker

[1]

in the case where

{ ~(M),

stochastic process with statio- nary

independent increments.

In this paper we shall

give

new and

simple proofs

for the results of H. Cramer

[2]

and G. Baxter and M. D. Donsker

[1].

Furthermore we shall

give

various extensions of these results. In

parti- cular,

we shall find the

joint

distribution of

17(t)

and

ç(t)

for stochastic pro-

cesses with

stationary independent

increments and for

compound

recurrent

processes.

(*) This research was supported by the National Science Foundation under Contract No. GP-7847.

(3)

238 LAJOS TAKACS

2. COMPOUND RECURRENT PROCESSES

First we shall consider the case

where { 03BE(u), 0 _ u ~ }

is a

separable compound

recurrent process which is defined in what follows. Let us

suppose

that Tn - (n

=

1, 2,

..., io =

0)

is a sequence of

mutually independent

and

identically

distributed

positive

random variables with distribution function

P ~ in - x ~ - F(x) and xn (n

=

1, 2, ... )

is a sequence of

mutually independent

random variables with distribution function

P ~ xn _ x ~ - H(x).

Furthermore let us suppose also that the

two

sequences {

are

independent.

Let us define

for

u >_

0 where c is a constant. We shall say that the process

{ ç(u), compound

recurrent process.

We shall write

for t >_ 0.

Let us introduce the

following

notation

for

Re(s) ~

0 and

for

Re(s)

= 0.

Denote

by Fn(x) (n

=

1, 2, ... )

the n-th iterated convolution of

F(x)

with itself and

by H"(x) (n

=

1, 2, ... )

the n-th iterated convolution of

H(x)

with itself. Let

Fo(x)

=

Ho(x)

= 1 for

x ~

0 and

Fo(x)

=

Ho(x)

= 0

for x 0.

Let us define also

for

n = 1, 2, ... and

(4)

ON THE DISTRIBUTION OF THE SUPREMUM FOR STOCHASTIC PROCESSES 239

for n =

1, 2,

... = 0. Let

for n =

0, 1, 2,

...

THEOREM 1.

If

c >_

0, Re(q)

>

0, Re(s

+

v) >_

0 and

Re(v) _ 0,

then

we have

Proof

- First let us suppose that i"

(n

=

0, 1, 2, ... )

and xn

(n

=

1, 2, ... )

are numerical

(non-random) quantities

for which

Let us define

~(t), 17(t)

for

0 ~

t oo and for

0 _ n

oo in

exactly

the same way as above. We shall show that if

Re(q)

>

0, Re(s + v) >__ 0, Re(v) _ 0,

then the

following

basic relation holds between the functions

?)? 17(t) (0 ~

t

oo )

and the sequences in,

’m

q"

(o __ n oo )

The

proof

of

(9)

is very

simple.

If

c ~ 0,

then

obviously 17(t)

= 17n and

?) = ’n - c(t - in)

for in t Thus

for n =

0, 1, 2,

... If we add

(10)

for n =

0, 1, 2,

..., then we

get (9).

Next suppose that

{ in ~

and are random variables as defined at

the

beginning

of this section. Then

(9)

holds for almost all realizations of

{ 03BE(t), ~(t);

0 t

~ }

and

{

i",

03BEn,

~n;

0 __

n If we form the

expectation

of

(9),

then we

obtain

that

(5)

240 LAJOS TAKACS

where

Let us define a sequence of random variables

11: (n

=

0, 1, 2, ... )

in

the

following way ~*0

= 0 and

for n =

0, 1, 2,

... where

[x] + =

x for

x ~

0 and

[x] + =

0 for x 0.

Then we can write also that

For

(12)

remains

unchanged

if we

replace ~

1,

~2,

...,

~n by ~n, ~n-1, ... , ~

1

respectively

and

by

these

substitutions ~n

becomes

Here

Uo(s,

v,

q)

= 1 and

U"(s,

v,

q)

for n =

1, 2,

... can be obtained

recursively.

Let us define an

operator

A in the

following

way.

If (

is a

complex

random variable for which

E { ( ~ ~ ~

oo

and q

is a real random

variable,

then

E { ~e - S’’ ~

exists for

Re(s)

= 0 and it determines

uniquely E ~ ~e S’’ ~ ~

for

Re(s) >_

0 where

r~ + -

max

(0, r~).

Let us write

for

Re(s) ~

0. Here A is a linear

operator.

We note that for

Re(s)

> 0

we can write that

where e y

oo ~.

Since

(15)

is con-

tinuous for

Re(s) >_- 0, (16)

determines

(15)

for

Re(s)

= 0

by continuity.

Since

11:+ 1

= +

~n+ 1]+

and

~n+ 1

=

~n + ~"+ 1

for

n = 0, 1, 2,

... we

can write that for

Re(q)

>

0, Re(s

+

v) ~ 0, Re(v)

0 and n =

0, 1, 2,

...

and here A

operates

on the variable s. The

generating

function of the

sequence {Un(s,

v,

q)}

can be

expressed

as follows:

(6)

ON THE DISTRIBUTION OF THE SUPREMUM FOR STOCHASTIC PROCESSES 241

and

(18)

is

convergent if

1. It is easy to check that

(18)

is indeed

the correct solution. Let us denote

by U(s,

v, q,

p)

the

right-hand

side

of

(18).

If we form the coefficient of

p"

for n =

0, 1, 2,

... in the power series

expansion

of

U(s,

v, q,

p)

and

apply

the

operator A,

then every coeffi- cient remains

unchanged.

If we form the coefficient of

p"

for n =

0,1, 2,

...

in the power series

expansion

of

[1

-

pt/J(s

+

v)~p(q -

cs -

cv)]U(s,

v, q,

p)

and

apply

the

operator A,

then we obtain 1 if n = 0 and 0

if n >-

1. Hence

we can conclude that the coefficient of

p"

in the

expansion

of

U(s,

v, q,

p)

satisfies the same initial condition and the same recurrence relation as

Un(s,

v,

q),

that is the coefficient of

p"

in the

expansion

of

U(s,

v, q,

p)

is

exactly q).

This proves

(18).

We note that

(18)

can also be

proved by using

the method of F. Pollaczek

[5]

or the method of F.

Spitzer [6].

Since

we

get (8) by (11)

and

(18).

This

completes

the

proof

of theorem 1.

If v = 0 in

(8),

then we

get

the

Laplace

transform of

E ~ e-S’’~i~ ~,

and

E ~ e-S’’~t~ ~

and P

f x ~

can be obtained

by inversion.

By (8)

we can write also that

if

c ~ 0, Re(q)

>

0, Re(s

+

v) >_

0 and

Re(v)

0.

Finally,

for the sake of

completeness

we shall mention the

following

result

which, however,

will not be used in this paper. If

c ~ 0, Re(q)

>

0,

Re

(s

+

v) >_

0 and

Re(v) __ 0,

then we have

(7)

242 LAJOS TAKACS

where

and A

operates

on the variable s.

3. COMPOUND POISSON PROCESSES

Let us suppose that in the

previous

section

that

is, cp(s)

=

À/(À

+

s)

for

Re(s) ~

0. Here À is a

positive

constant.

In this case

{03BE(u),

0 ~ u

~ }

is called a

compound

Poisson process.

Then

for x ~ 0,

for

u >_

0 and

for

Re(s)

= 0.

In this case we have the

following

result.

THEOREM 2.

- If Re(q)

>

0, Re(s

+

v) ~ 0,

and

Re(v) __ 0,

then we have

(8)

ON THE DISTRIBUTION OF THE SUPREMUM FOR STOCHASTIC PROCESSES 243

If

v =

0,

then

(27)

reduces to

Proof.

- First we shall prove

(27)

for

c ~

0. In this case

(27)

is a

parti-

cular case of

(8).

Now 1 -

~p(q - cv) = (q - cv)/(~,

+ q -

cv)

in

(8)

and

by using (24)

and

(25)

the

exponential

factor in

(8)

can be

expressed

as

follows:

Since

we

get (27) by (29).

For

c ~

0 we can

easily

deduce

(27)

from the

previous

case. Since the

finite dimensional distribution functions of the two processes

are

identical,

we can conclude that

r~(t) - ç(t)

and -

ç(t)

have

exactly

the same

joint

distribution as sup

[ - ~(u)]

and -

~(t).

Furthermore if

c ~ 0,

then for the

process ( - ç(u), 0 ~

u

oo ~

we can

apply (27)

if

we

replace

c

by -

c and

ç(u) by - ~(M).

Thus we obtain that for

c ~ 0, Re(q)

>

0, Re(s

+

v) >-

0 and

Re(v) --

0

If we

replace v by - (v

+

s)

in

(31),

then we obtain

(27)

for

c ~

0. This

completes

the

proof

of theorem 2.

(9)

244 LAJOS TAKACS

In the case

compound

Poisson process

the

Laplace

transform of

E {

has

previously

been found

by

H. Cra-

mer

[2].

Here we

provided

a new

proof

for Cramer’s result and we also found the

Laplace

transform of

E ~ e ~ S’’to - "~tt~ ~

which makes it

possible

to find the

joint

distribution of

17(t)

and

4. PROCESSES

WITH STATIONARY INDEPENDENT INCREMENTS

Let us suppose now

separable

stochastic

process with

stationary independent

increments for which

P~ ~(0) = 0 ~ =1.

Then for

Re(s)

= 0 and

0 ~

u oo we can write that

where is an

appropriate

function.

THEOREM 3.

2014 If { 03BE(u), 0 __

u

~ }

is a

separable

stochastic process with

stationary independent increments,

then

formulas (27)

and

(28)

hold

unchangeably.

Proof

- We can find a sequence of

separable compound

Poisson pro-

0 ~

u

~ }

such that the finite dimensional distribution functions

0 ~

u

oo ~

converge to the

corresponding

finite

dimensional distribution functions

of { ç(u), 0 ~

u

oo ~.

If

for

Re(s)

=

0,

then we can achieve the desired convergence

by choosing

ck,

Àk

and in such a way that

for

Re(s)

= 0.

If

17k(t)

= sup

~k(u),

then it follows that the

joint

distribution of

I1k(t)

and

~k(t) (k

=

1, 2, ... )

converges to the

joint

distribution of and

ç(t)

as k - oo. Since

(27)

holds for all

~k(t)

and

(k = 1, 2, ... ),

if follows

by

the

continuity

theorem for

Laplace-Stieltjes

transforms that

(27)

holds

also for

ç(t)

and

17(t).

This

completes

the

proof

of the theorem.

In a somewhat different form formula

(28)

was found

by

G. Baxter and M. D. Donsker

[l].

Formula

(27)

for stochastic processes is similar to

(10)

245

ON THE DISTRIBUTION OF THE SUPREMUM FOR STOCHASTIC PROCESSES

a result of F.

Spitzer [6]

for

partial

sums of

mutually independent

and iden-

tically

distributed random variables.

Note. - If we take into consideration that for

Re(q)

> 0 and

Re(s)

= 0

w here the

principal

value of the

logarithm

is

used,

then

by (27)

we obtain

that

for

Re(q)

>

0, Re(s

+

v) ~ 0, Re(v) _

0 and the

operator

A is defined

by (16).

5 . EXAMPLES

Suppose that { ç(u), separable

stochastic process with

stationary independent

increments and has a stable distribution with

parameters

a

and f3

where 0 a

~

2 and - 1

~ f3 ~

1. Then

oo

where for (l =1= 1,0a_2and -1/31

and for a = 1 and - 1

~3 --

1

In this case the distribution of = sup

ç(u)

has been determined

o s ust

by

several authors in various

particular

cases. We refer to the works

of D. A.

Darling [3],

G. Baxter and M. D. Donsker

[1]

and C. C.

Heyde [4].

We can

always

use formulas

(28)

or

(36)

to find the

Laplace-Stieltjes

trans-

form of

P { r~(t) x }

and

by

inversion we can obtain

P { ~(t) x ~.

How-

ever, if is

given by (38)

or

by (39)

with

/3 = 0,

then

17(t)

has the same

distribution as and this observation makes

possible

some

simpli-

(11)

246 LAJOS TAKACS

fications.

Following

D. A.

Darling [3]

we can also express the left-hand side of

(28)

in another way. Let

and suppose that

P { q(t) _ x } =

where 0

o: ~

2.

If q

= 1

in

(28)

then the left-hand side can be

expressed

as

for

Re(s)

> 0 where

for

Re(s)

> 0. We observe that for

0 ~

x oo the function

I(x)

is a dis-

tribution function of a

positive

random variable. If we write s =

1/y

in

(41)

where 0 y oo, then

can be

interpreted

as the distribution function of the

product

of two inde-

pendent positive

random variables

having

distribution functions

I(x)

and

W(x) respectively.

On the other hand

G(x)

can be obtained

by using (28)

or

(36).

In

(36) if q

=

1,

then

for

Re(s)

> 0. If we denote the

right-hand

side of

(44) by L(s),

then we

can write that

for 0 x oo.

Finally

the unknown

W(x)

can be obtained from

(43) by using

Mellin-

Stieltjes

transform. Since

(12)

247 ON THE DISTRIBUTION OF THE SUPREMUM FOR STOCHASTIC PROCESSES

for 1 - a s

1,

we obtain from

(43)

that

for 1 - a s 1

and I s

a.

By

inversion we can obtain

W(x).

If,

in

particular,

a =

1,

then

(47)

reduces to

for 0 s 1 hence

for x >

0,

where the definition of

G(s)

is extended

by analytical

continua-

tion to the

complex plane

cut

along

the

negative

real axis from the

origin

to

infinity.

REFERENCES

[1] G. BAXTER and M. D. DONSKER, On the distribution of the supremum functional for processes with stationay independent increments. Trans. Amer. Math. Soc., 85,

1957, 73-87.

[2] H. CRAMER, On some questions connected with mathematical risk. University of Cali- fornia Publications in Statistics, 2, 1954, 99-124.

[3] D. A. DARLING, The maximum of sums of stable random variables. Trans. Amer.

Math. Soc., 83, 1956, 164-169.

[4] C. C. HEYDE, On the maximum of sums of random variables and the supremum func- tional for stable processes. Journ. Appl. Prob., 6, 1969, 419-429.

[5] F. POLLACZEK, Fonctions caractéristiques de certaines répartitions définies au moyen de la notion d’ordre. Application à la théorie des attentes. Comptes Rendus Acad.

Sciences (Paris), 234, 1952, 2334-2336.

[6] F. SPITZER, A combinatorial lemma and its application to probability theory. Trans.

Amer. Math. Soc., 82, 1956, 323-339.

(Manuscrit reçu le 8 décembre 1969).

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