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
o3 (1970), p. 237-247
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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, n° 3, 1970, p. 237-247.
Section B : Calcul des Probabilités et Statistique.
SUMMARY. - In this paper mathematical methods are
given
forfinding
the distribution of the supremum for
compound
recurrent processes and for stochastic processes withstationary independent
increments. New andsimple proofs
aregiven
for a result of H. Cramer and for a result ofG. Baxter and M. D. Donsker. The paper also contains some extensions of these results.
I . INTRODUCTION
Our aim is to
give
mathematical methods forfinding
the distribution of = sup forseparable
stochastic (. -’B /?0
2014 u oo~.
JSuch methods were
given by
H. Cramer[2]
in the case u00 }
is a
compound
Poisson process andby
G. Baxter and M. D. Donsker[1]
in the case where
{ ~(M),
stochastic process with statio- naryindependent increments.
In this paper we shallgive
new andsimple 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. Inparti- cular,
we shall find thejoint
distribution of17(t)
andç(t)
for stochastic pro-cesses with
stationary independent
increments and forcompound
recurrentprocesses.
(*) This research was supported by the National Science Foundation under Contract No. GP-7847.
238 LAJOS TAKACS
2. COMPOUND RECURRENT PROCESSES
First we shall consider the case
where { 03BE(u), 0 _ u ~ }
is aseparable compound
recurrent process which is defined in what follows. Let ussuppose
that Tn - (n
=1, 2,
..., io =0)
is a sequence ofmutually independent
andidentically
distributedpositive
random variables with distribution functionP ~ in - x ~ - F(x) and xn (n
=1, 2, ... )
is a sequence of
mutually independent
random variables with distribution functionP ~ xn _ x ~ - H(x).
Furthermore let us suppose also that thetwo
sequences {
areindependent.
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
notationfor
Re(s) ~
0 andfor
Re(s)
= 0.Denote
by Fn(x) (n
=1, 2, ... )
the n-th iterated convolution ofF(x)
with itself and
by H"(x) (n
=1, 2, ... )
the n-th iterated convolution ofH(x)
with itself. LetFo(x)
=Ho(x)
= 1 forx ~
0 andFo(x)
=Ho(x)
= 0for x 0.
Let us define also
for
n = 1, 2, ... and
ON THE DISTRIBUTION OF THE SUPREMUM FOR STOCHASTIC PROCESSES 239
for n =
1, 2,
... = 0. Letfor n =
0, 1, 2,
...THEOREM 1.
If
c >_0, Re(q)
>0, Re(s
+v) >_
0 andRe(v) _ 0,
thenwe have
Proof
- First let us suppose that i"(n
=0, 1, 2, ... )
and xn(n
=1, 2, ... )
are numerical
(non-random) quantities
for whichLet us define
~(t), 17(t)
for0 ~
t oo and for0 _ n
oo inexactly
the same way as above. We shall show that if
Re(q)
>0, Re(s + v) >__ 0, Re(v) _ 0,
then thefollowing
basic relation holds between the functions?)? 17(t) (0 ~
too )
and the sequences in,’m
q"(o __ n oo )
The
proof
of(9)
is verysimple.
Ifc ~ 0,
thenobviously 17(t)
= 17n and?) = ’n - c(t - in)
for in t Thusfor n =
0, 1, 2,
... If we add(10)
for n =0, 1, 2,
..., then weget (9).
Next suppose that
{ in ~
and are random variables as defined atthe
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 theexpectation
of(9),
then weobtain
that240 LAJOS TAKACS
where
Let us define a sequence of random variables
11: (n
=0, 1, 2, ... )
inthe
following way ~*0
= 0 andfor n =
0, 1, 2,
... where[x] + =
x forx ~
0 and[x] + =
0 for x 0.Then we can write also that
For
(12)
remainsunchanged
if wereplace ~
1,~2,
...,~n by ~n, ~n-1, ... , ~
1respectively
andby
thesesubstitutions ~n
becomesHere
Uo(s,
v,q)
= 1 andU"(s,
v,q)
for n =1, 2,
... can be obtainedrecursively.
Let us define an
operator
A in thefollowing
way.If (
is acomplex
random variable for which
E { ( ~ ~ ~
ooand q
is a real randomvariable,
thenE { ~e - S’’ ~
exists forRe(s)
= 0 and it determinesuniquely E ~ ~e S’’ ~ ~
for
Re(s) >_
0 wherer~ + -
max(0, r~).
Let us writefor
Re(s) ~
0. Here A is a linearoperator.
We note that forRe(s)
> 0we can write that
where e y
oo ~.
Since(15)
is con-tinuous for
Re(s) >_- 0, (16)
determines(15)
forRe(s)
= 0by continuity.
Since
11:+ 1
= +~n+ 1]+
and~n+ 1
=~n + ~"+ 1
forn = 0, 1, 2,
... wecan 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. Thegenerating
function of thesequence {Un(s,
v,q)}
can beexpressed
as follows:ON THE DISTRIBUTION OF THE SUPREMUM FOR STOCHASTIC PROCESSES 241
and
(18)
isconvergent if
1. It is easy to check that(18)
is indeedthe correct solution. Let us denote
by U(s,
v, q,p)
theright-hand
sideof
(18).
If we form the coefficient ofp"
for n =0, 1, 2,
... in the power seriesexpansion
ofU(s,
v, q,p)
andapply
theoperator A,
then every coeffi- cient remainsunchanged.
If we form the coefficient ofp"
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
theoperator A,
then we obtain 1 if n = 0 and 0if n >-
1. Hencewe can conclude that the coefficient of
p"
in theexpansion
ofU(s,
v, q,p)
satisfies the same initial condition and the same recurrence relation as
Un(s,
v,q),
that is the coefficient ofp"
in theexpansion
ofU(s,
v, q,p)
is
exactly q).
This proves(18).
We note that(18)
can also beproved by using
the method of F. Pollaczek[5]
or the method of F.Spitzer [6].
Since
we
get (8) by (11)
and(18).
Thiscompletes
theproof
of theorem 1.If v = 0 in
(8),
then weget
theLaplace
transform ofE ~ e-S’’~i~ ~,
andE ~ e-S’’~t~ ~
and Pf x ~
can be obtainedby inversion.
By (8)
we can write also thatif
c ~ 0, Re(q)
>0, Re(s
+v) >_
0 andRe(v)
0.Finally,
for the sake ofcompleteness
we shall mention thefollowing
result
which, however,
will not be used in this paper. Ifc ~ 0, Re(q)
>0,
Re
(s
+v) >_
0 andRe(v) __ 0,
then we have242 LAJOS TAKACS
where
and A
operates
on the variable s.3. COMPOUND POISSON PROCESSES
Let us suppose that in the
previous
sectionthat
is, cp(s)
=À/(À
+s)
forRe(s) ~
0. Here À is apositive
constant.In this case
{03BE(u),
0 ~ u~ }
is called acompound
Poisson process.Then
for x ~ 0,
for
u >_
0 andfor
Re(s)
= 0.In this case we have the
following
result.THEOREM 2.
- If Re(q)
>0, Re(s
+v) ~ 0,
andRe(v) __ 0,
then we haveON THE DISTRIBUTION OF THE SUPREMUM FOR STOCHASTIC PROCESSES 243
If
v =0,
then(27)
reduces toProof.
- First we shall prove(27)
forc ~
0. In this case(27)
is aparti-
cular case of
(8).
Now 1 -~p(q - cv) = (q - cv)/(~,
+ q -cv)
in(8)
andby using (24)
and(25)
theexponential
factor in(8)
can beexpressed
asfollows:
Since
we
get (27) by (29).
For
c ~
0 we caneasily
deduce(27)
from theprevious
case. Since thefinite dimensional distribution functions of the two processes
are
identical,
we can conclude thatr~(t) - ç(t)
and -ç(t)
haveexactly
the same
joint
distribution as sup[ - ~(u)]
and -~(t).
Furthermore ifc ~ 0,
then for theprocess ( - ç(u), 0 ~
uoo ~
we canapply (27)
ifwe
replace
cby -
c andç(u) by - ~(M).
Thus we obtain that forc ~ 0, Re(q)
>0, Re(s
+v) >-
0 andRe(v) --
0If we
replace v by - (v
+s)
in(31),
then we obtain(27)
forc ~
0. Thiscompletes
theproof
of theorem 2.244 LAJOS TAKACS
In the case
compound
Poisson processthe
Laplace
transform ofE {
haspreviously
been foundby
H. Cra-mer
[2].
Here weprovided
a newproof
for Cramer’s result and we also found theLaplace
transform ofE ~ e ~ S’’to - "~tt~ ~
which makes itpossible
to find the
joint
distribution of17(t)
and4. PROCESSES
WITH STATIONARY INDEPENDENT INCREMENTS
Let us suppose now
separable
stochasticprocess with
stationary independent
increments for whichP~ ~(0) = 0 ~ =1.
Then for
Re(s)
= 0 and0 ~
u oo we can write thatwhere is an
appropriate
function.THEOREM 3.
2014 If { 03BE(u), 0 __
u~ }
is aseparable
stochastic process withstationary independent increments,
thenformulas (27)
and(28)
holdunchangeably.
Proof
- We can find a sequence ofseparable compound
Poisson pro-0 ~
u~ }
such that the finite dimensional distribution functions0 ~
uoo ~
converge to thecorresponding
finitedimensional distribution functions
of { ç(u), 0 ~
uoo ~.
Iffor
Re(s)
=0,
then we can achieve the desired convergenceby choosing
ck,
Àk
and in such a way thatfor
Re(s)
= 0.If
17k(t)
= sup~k(u),
then it follows that thejoint
distribution ofI1k(t)
and
~k(t) (k
=1, 2, ... )
converges to thejoint
distribution of andç(t)
as k - oo. Since
(27)
holds for all~k(t)
and(k = 1, 2, ... ),
if followsby
thecontinuity
theorem forLaplace-Stieltjes
transforms that(27)
holdsalso for
ç(t)
and17(t).
Thiscompletes
theproof
of the theorem.In a somewhat different form formula
(28)
was foundby
G. Baxter and M. D. Donsker[l].
Formula(27)
for stochastic processes is similar to245
ON THE DISTRIBUTION OF THE SUPREMUM FOR STOCHASTIC PROCESSES
a result of F.
Spitzer [6]
forpartial
sums ofmutually independent
and iden-tically
distributed random variables.Note. - If we take into consideration that for
Re(q)
> 0 andRe(s)
= 0w here the
principal
value of thelogarithm
isused,
thenby (27)
we obtainthat ’
for
Re(q)
>0, Re(s
+v) ~ 0, Re(v) _
0 and theoperator
A is definedby (16).
5 . EXAMPLES
Suppose that { ç(u), separable
stochastic process withstationary independent
increments and has a stable distribution withparameters
aand f3
where 0 a~
2 and - 1~ f3 ~
1. Thenoo
where for (l =1= 1,0a_2and -1/31
and for a = 1 and - 1
~3 --
1In this case the distribution of = sup
ç(u)
has been determinedo s ust
by
several authors in variousparticular
cases. We refer to the worksof 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 theLaplace-Stieltjes
trans-form of
P { r~(t) x }
andby
inversion we can obtainP { ~(t) x ~.
How-ever, if is
given by (38)
orby (39)
with/3 = 0,
then17(t)
has the samedistribution as and this observation makes
possible
somesimpli-
246 LAJOS TAKACS
fications.
Following
D. A.Darling [3]
we can also express the left-hand side of(28)
in another way. Letand suppose that
P { q(t) _ x } =
where 0o: ~
2.If q
= 1in
(28)
then the left-hand side can beexpressed
asfor
Re(s)
> 0 wherefor
Re(s)
> 0. We observe that for0 ~
x oo the functionI(x)
is a dis-tribution function of a
positive
random variable. If we write s =1/y
in
(41)
where 0 y oo, thencan be
interpreted
as the distribution function of theproduct
of two inde-pendent positive
random variableshaving
distribution functionsI(x)
andW(x) respectively.
On the other hand
G(x)
can be obtainedby using (28)
or(36).
In(36) if q
=1,
thenfor
Re(s)
> 0. If we denote theright-hand
side of(44) by L(s),
then wecan write that
for 0 x oo.
Finally
the unknownW(x)
can be obtained from(43) by using
Mellin-Stieltjes
transform. Since247 ON THE DISTRIBUTION OF THE SUPREMUM FOR STOCHASTIC PROCESSES
for 1 - a s
1,
we obtain from(43)
thatfor 1 - a s 1
and I s
a.By
inversion we can obtainW(x).
If,
inparticular,
a =1,
then(47)
reduces tofor 0 s 1 hence
for x >
0,
where the definition ofG(s)
is extendedby analytical
continua-tion to the
complex plane
cutalong
thenegative
real axis from theorigin
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).