An extension of a theorem of Cauchy with applications to probability theory

probability theory

Citation for published version (APA):

Vanderperre, E. J. (1975). An extension of a theorem of Cauchy with applications to probability theory. (EUT report. WSK, Dept. of Mathematics and Computing Science; Vol. 75-WSK-05). Eindhoven University of Technology.

Document status and date: Published: 01/01/1975

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ONDERAFDELING DER WISKUNDE DEPARTMENT OF MATHEMATICS

An extension of a theorem of Cauchy with applications to Probability Theory

by

E.J. Vanderperre

T.H.-Report 75-WSK-05 August 1975

We use the theorem to solve an integral equation of the Pollaczek type [7J and to study a Markov chain {w , n ~ O} with state space [O,oo[ closely

relat--n ed to Queueing Theory.

I. An extension of a theorem of Cauchy

1.1. Introduction

Let S be the complex w-plane, S± := {w I 1m w ~ O} and JR := {a I _00 < a < +oo}. A. Let few) be a function, holomorphic in S and continuous in S + JR

includ-ing the point at infinity i.e., few)

=

f(oo) + 0(1) as Iwl + 00 in S + JR. Then

(a) I 27fi f ( a ) - = do

{H(OO)

a -w -few) if w E S + i f w E S

The symbol C indicates the principal value (Cauchy value) of the integral. B. Let few) be a function, holomorphic in S+ and continuous in S+ + JR in-cluding the point at infinity. Then

(b)

f(a)~

a-w - 0 0

=

{feW) - H(oo) -!f(oo) if w E s+ ' if w E S

The formulae (a) and (b) are called "Cauchy formulae" for the regions Sand S+ respectively.

A simple proof of (a) and (b) is gLven Ln [6J.

Remark. The condition that few)

=

f(oo) + 0(1) as Iwl + 00 in S_ + JR (or in S+ + JR) is not necessarily satisfied when few) is a Fourier-Stieltjes trans-form (F.S.T.). Therefore, we shall prove the following extension related to Probability Theory.

I .2. The theorem

Let {~,B,P} be a probability space with ~ := J-oo,+oo[ the sample space, B the smallest a-algebra containing all open subsets (Borel sets) of ~ and P a pro-bability measure on B. Introduce a B-measurable mapping~: ~ +JR (a random variable) with probability distribution function (p.d.f.)

F(x) := P[x < x], x E JR. and define +00 E{ _{e -ia.:.} :=}

f

e-iaxdF(X) , _{a EJR..} -00

Denote by (A) de indicator of the event A, i.e.

if A occurs, i f not • Theorem 1. 1 2rri _00 = {"

~E

(,:, =

-!E(!

= 0) - E{e-iw

!(!

~

O)} if w E S -iwx 0) + E{ e -(!:;;; O)} i f w E S +

Proof. Since

(!

< 0) + (! 0) + (x > 0) = 1, we have Vw

I

w E S+ u S

(1. 1) I 21fi _00 + E(,.:. +00

J

E{e-ia!}~

=

j

a - w +00 1

~

da = 0)'2'" _{rr~ a - w} -00

provided that all principal values

We shall now prove that +00 1 2rri I +

Z-

_rr~ +00

I

E{ -iax

O)}~

+

'j

e

-(!

> a - w +00

f

He -iax -(x < O)}~ a-w _00 involved exist. -iwx W E S ( 1 .2) I 21fi

f

E{e-ia !(! > 0)"}--2.... d = {

-ne

-(x> O)}, a - w 0 -00

In order to show the statement, let w E S_, i.e., put w

=

Rei8, rr <

e

< 21f,

o <,R < 00.

Consider a closed curve

r

consisting of a segment [-T,T] of the real line and

of a semi-circle y lying in S_, with center at the origin and with radius

ie'

T

I

0 < R < T < 00. Finally, let w'

=

Te , rr <

e'

< 2rr. The contour integra-tion is clockwise.

From the above definition of

r,

we obtain for w E S +

2;i

J

-iw'x dw' He -(_x> O ) } _ -w' -w y But ex> -iw 'x E{ e -(~ > O)} =

f

0+

Whence, by Cauchy's theorem, we obtain for w E S

Now +T -iwx -E{e -(~ > O)} =~ I 27T~

_J

E{e-icr~(_x

>

O)}~

(J - W 00

12~i

J

f

y 0+ -T ex> + _{27T i}

f

J

y 0+ e-iw'XdF(X) dw'

1<

T w' -w -

2i

T 27T 00

f

f

7T 0+ 00 7T/2 ~ T - R

J J

e-Txsin~d~dF(X)

0+

o

The permutation of both integrals in the right-hand side of the first ine-quality is justified by Fubini's theorem.

It may be verified that for any 2 E JO,7T/2[

7T/2

f

e -Txsin'Pd ~ :::; sec Q, e -Txsincp co s cp 'P d + ( / 2 " ) 7T - IV e -TxsinQ, •

o

o

After Some algebra we obtain (Q, fixed),

12~i

f

f

y 0+ 00 e-iw'xdF(x) dw'

I

S wi -w 00 0+

00

T -I T

+ T _ R tan t{F(T cosec t+) - F(O+)} + ~ sec t

_f

*'

_{Xa (x)dF(x)}

0+

where a := JT -I cosec t,oo[ and

r I X (x)

=

J

l

a 0 if x E a if x ¢. a • since I X (x) :0;

Tx a I, we have by a well-known dominated convergence theorem

1

27fi

J J

y 0+ 00

Finally, we remark that

ifwES i f W E S+

In a similar way, we can prove that

as T -+ +00 •

I

27fi

t

E{e~cr~ (x' > O)}~

=

+ oo . , d { E{ e

.

,

~W~ (x' > O)}, W E S+ ' -00 - cr-w 0

Hence, for x' = -~, we have

(I. 3) By (1.1), 1 27fi - 0 0 (I .2) 27fi -00 +00 and +00

T

E{e-icr~(x

_....

{ -~wx dcr E{e -(~ < O)}, < O ) } - = cr-w

o

(I .3) we obtain

.

{-jE(X

=

0) - E{e -~wx -(~ > E{e-~cr~}~

=

--iwx cr-w 0) +~E(~ = + E{e -(~ < W E S W E S 0) }. if W E O)}, i f W E S S+

Corollary I. Let x be a nonnegative random variable (r.v.) with characteris-

-.

-~wx

tic function (c.f.), E{e -}, 1m W :0; O. Then

W E S W E S +

( I .4)

00

Corollary 2. Let <~n>n=O be ~ sequence

. {-1WX 00

1ng sequence of c.f.'s <E e -n}>n=O'

of nonnegative r.v.'s with correspond-1m W ~ O.

For W € Sand z € [0, I [ we have

+00

f

00 1

L

nE{ -i txn} dt 2'd z e

- - =

n=O t-w -00 +00

f

00 2ni

L

znE{eit~n}....!!!.... t-w -00 n=O 00 co I

I

2 n=O 00 _1

L

2 n=O znp[x -n znp[x -n 00 OJ -

L

znE{e-iw~n}, n=O oJ •

Proof. Let <Pn>n=O be a discrete proper probability distribution, then

00 •

'i' p E{ e-1W~n}, 1m w ~ O ' f f t ' d 1

L , 1S a c •• 0 a nonnega 1ve r.v. an we may app y

n=O n

(1.4). Since p is arbitrary, we may put p = (1 - z)zn, Vz

I

z € [0,1[.

n n

2. An integral equation of the Pollaczek type In this section, we use the results of section tion of the Pollaczek type [7J.

to solve an integral

equa-In order to introduce the equation, let {T , n € ~} be a family of indepen--n

dent, identically distributedr.v. '8 with p.d.f. A(T) concentrated on ~ and

-icrT

with c.f. a(cr) = E{e -}, cr €~. Define

+ ••• + T ,

-n n€JN.

{~n' n € IN} is a random walk generated by A(·). Let us introduce the

follow-ing functions for z € [0,1[

(2. 1 ) n;::1 n z E{e-iw~n(s ~ O)} n -n

- L

= e 1m w ;:: 0 ,

(2.2) 1m w :S 0 •

By (2.1) and (2.2), we obtain for Z E [O,I[ and a E R

(2.3) Define

l

x _-n : = max{ 0, _-s 1 ' _-s 2' ••• , _-n-s 1 } ,

~I : =

a .

From the results of Fluctuation Theory, (see e.g. Cohen [IJ, pag. 151) we have for Z E [O,I[ and 1m w :S 0

n

- L

~ ECs > 0) n -n

(2.4) Z e ne ]

- Z

A similar expression can be g1ven with respect to y+. To simplify notation, we define

n a. (z)

:=!

I

~ E(s

=

0) • net n -n By (2.1), (2.2) we have lim y+(w,z) = e -20. (z)

,

1m w-++oo (2.5) Re w=O (2.6) lim y_(w,z) = 1

.

1m w-+-oo Re w=O

Hence by corollary 1 and 2, (2.4), (2.5) and (2.6) we obtain after simplifi-cation (2.7) 1 _ 2 21Ti da y (a,z)---- , w E S - a-w - 0 0 (2.8) = ..,,-,-21T1 - 0 0

Rewriting (2.3) in the form yields by (2.7) and (2.8), (2.9)

-

..,...,..

z 2'1T~ Define (2. ] 0) _00 do a(o)y_(o , z ) -o - w 1m w ::;

° .

By (2.9) and (2.10) we have that u_(w,z) ~s the solution of the integral

equation (2.11) -00 Finally, we define Since n z P_n := -

n

00 log(] - z) , n EN, z EO ]0,1[.

I

p an(o) _n

=

log[] - za(a)] _{log(1 - z)}

n=1

00

*

~s the c.f. of the p.d.f.

I

PnAn (t), t E R, we have for W E S

n=1 +00

(2.12)

- 0 0

do

log[1 - za(o)J---- = a(z) + log y_(w,z) •

o-w

By (2.10) and (2.12) we finally obtain for w EO S_, Z E [O,I[

sech a(z)e -00

da log[1 -

Remarks. Integral equations similar to (2.11) have been studied in [7J and [6J using different methods. Pollaczek assumed that a(w) is analytic in some strip \Im w\ < 6 for some 6 > 0, whereas Ke'ilson and Kooharian made the as-sumption that a(o) 0(1) as

\0\

+ 00 • (In this case sech a(z) = 1.)

By theorem 1 and a combination of some of the techniques presented in [IJ, [2J and [7J, we have shown that the above assumptions concerning a(o) are no longer necessary to solve equations similar to (2.11).

As a second application of theorem I, we study:

3. The transient behaviour of a Markov chain {w ; n ~ O} with state space [O,oo[

-n

-3.1. Introduction

Let {~n' n ~ O} be a Markov chain with state space [O,oo[ recursively

deter-mined by ~O a,;,s.

o ,

(3.1) _ o(n)} _{if w} 0 '-H -G ' -n = .

tax{o cr(n)

::!n+1 max{O,w + o(n) _ £.~n)}, if w

_#:

0 -n -F -n

where ~(n) _{~G , n =} 0 I 2 _{, , , ••• are POSl. l.ve independent r.v.'s with p.d.f.} "t" r[o(n) < tJ

=

G(t) and p[o(n) ~

-G -G

We also assume that o(n) o(n)

-G '-H '

" (n) (n)

OJ = (dl.tto £H '£F ).

£in) are mutually independent for each n.

Remark. The transient behaviour of the Markov chain {w } has been studied -n

by J. Loris-Teghem [4J, [5J using an algebraic method based on a theorem due to G. Baxter [5]. Our approach is based on a Wiener-Hopf integral equation of the second kind and its reduction to a functional equation onR in terms of F.S.T.'s.

3.2. A Wiener-Hopf integral equation for the Markov chain {w } -n For t > 0 and n = 0,1,2, ••• we define

(3.2) W (t)

:=

pew < tJ

n -n

and W (0+) := lim W (t).

For t E lR, we define AF(t) := pea (0) -F _ a (0) -G < tJ

,

~(t) := P[a(O) - cr (0) < tJ

.

-R -G Whence for t E lR 00 (3.3) _~(t)

=

J

F(t + ,)dG(,) 0 00 (3.4) _~(t)

₌

J

_R(t + ,)dG(,) 0

By (3.1) we have for t > 0 and n 2

(n-I) (n-I)} (~n < t) = (max{O'~R - ~G < t)(~n_1 = 0) + (n-I) (n-l)} + (max{O'~n_l + ~F - ~G < t)(~n_1 > 0) • Since (w > 0)

=

-

(w = 0) we have -n-I - n - I ' or

(w < t)

=

(max{O cr(n-I) - a(n-I)} < t)(w_

n_1

=

0) + -n ' - R - G pew < tJ -n { (n-I) (n-I) - (max O'~F - ~G } < t)(~n-l = 0) + { (n-I) _ cr(n-I)} + (max O'~n-l + ~F -G < t) • (n-l) (n-I)} - P[max{O'£F - £G < _{tJ}Wn_1(0+)} (n-I) (n-I)} + P[max{O'~n_1 + £F - ~G < tJ •

P[max{O'~n_1

+

~~n-I)

_

~~n-I)}

< tJ

=

P[~n-I

+

~~n-I)

_~G (n-I) < tJ ,

+0:>

=

J

P[~n-I

< t -

TJdP[~~O)

<"T(O) ':.G < T ] •

Whence for t > 0 and n ~ I we obtain

+00

(3.5)

J

_{Wn_l(t -}

T)d~(t)

Since W(O+) = I and WO(t) = UO(t) where UO(t) is the Heaviside unit-step

function at t = 0, we have that equation (3.5) is also valid for n = I

(WI (t)

=

~(t) for t > 0).

Remark. If H

=

F, then (3.5) reduces to Lindley's integral equation for the

Markov chain {; , n ~ O} recursively determined by

-n

{

''''

a.~. 0 ~O

;

=

max{O.; -n+1 -n

3.3. Solution of the integral equation (3.5)

In order to apply a Wiener-Hopf technique to equation (3.5), we define for

+ + + +

t ~ 0, n

Em;

W (t)

:=

W (t) (note that W (0)

=

0 or W (0) f W (0+)

=

Wn(O+»,

n n n n n and for t ~ 0, n E

m;

W-(t) := O. n For t ~ 0, n E m we define +00

J

W n_1 (t -

T)d~(T)

• -00

Finally we define for t E ~

+00

0.6)

I

W:_I(t-T)d~(T)

₊

W~(t)

We will now reduce equation (3.6) into a functional equation on R in terms

of Fourier-Stieltjes transforms. Therefore, we define for Z E [O,I[

00 00

J

e-iwtdW+(t), w_(w,z) :=

I

z n 1m w ~ 0

,

n=O n 0 0 00

J

w+(w,z) :=

I

z n e-iwtdW-(t) , 1m w ~ 0

,

n=O _-00 n 00 W(z) :=

L

z~ (0+), n=O n +00 aF(w) :=

_f

e-iwtd~(t), 1m w = 0 , -00 +00 ~(w) :=

J

e-iwtd~(t),

1m w

=

0 • -00

Put a

=

Re w. For a E Rand z E [O,I[ we obtain by (3.6) and the above

defi-nitions (3.7) Note that

-iaa(n) iaa(n)

aF(a) = E{e -F }E{e -G }

. (n) . (n)

-l.cr~R l.cr~G

aR(a) = E{e }E{e } •

In order to simplify (3.7),

the property that !n

~ ~~n)

by (2.1), (2.2), (2.3)

00

we introduce a sequence of r.v.'s <!n>n=1 with

(n) -iaT

~G , n E N. Since E{e _n} = aF(a), we have

y+(a,z)

Substituting in (3.7) yields

(3.8)

w+(a,z) w_(a,z)

y+(a,z) - y_(a,z) =

aF(a) - ~(a)

Since is a F.S.T. of a function of bounded variation, we have

y+(a,z) that +T , 1

J

aF(a) - ~(a)

~:

2T

y + (a ,z) da -T

exists and is finite. Hence we may put +T l _1m ' _--2T I

J

aF(a)-~(a) _{( ) da} T~ y+ a,z -T M(z) • Moreover +T (c:r, z) w (c:r, z) l' 1

J

w (3.9) _{~m 2T} y Jc:r ,z) da

=

lim y_(a,z) W(z)

.

T~ 1m w-l-OO -T Re w=O Whence +T 1

J

w+ (a, z) 1 ' _~m

_2T

_y+(c:r,z) d _a

₌

T~ -T W ( z ) {I + zM ( z)} • Or equivalently (3.10) w+(a,z) lim

=

W(z){1 + zM(z)} • 1m W"Hoo y + ( a , z) Re w=O

By (3.8), (3.9), (3.10) and corollary (2), we obtain for W E S_,.z E [O,I[

(3.11) +00 w+(a,z) I

~

da -~W(z){ 1 + zM(z)} 21fi y+Ca,z) - = a -w _00 +00 w_(a,z) w_(a,z)

j

da

21fi y (a, z)

- -

a-w 4W(z)

-

y_(a,z)

.

(3.12)

By (3.11), (4.11) and (3.8) we have (3.13)

₌

_{y_(w,z)W(z){1 +} 2 z M(z) + 2~i z _00 Finally, we have Whence (3.15) lim w_(w,z)

=

(1 - z)-I • w+O WES y _ (0, z) [J +

1-

M (z) + 1 im w-+O WES z 2~i

By (3.13) and (3.15) we have the following:

Theorem 2. +00 aFCr) - aH(T) z z

f

+

i'

M(z)

+r

y +

h,

z) ~~ (3.16) w_(w,z)

₌

; (w, z) -00 +00 I +

f

M ( z) + 1 im z

f

a F ( T) - aH ( T ) 2~i y+CT,Z) W+O -00 WES where n -iws

- I

-

z E{ ( I - e -n)(s > O)} n -n ;_(w,z)

₌

e n~1 - z

Corollary 3. I f aF(a), aH(a) are 0 (I) as

I

_a

I

+ 00, then

+00 aFCT) - ~(T) Z

1

dT

+r

~~ _{y +}

_h,

_{z) T -w} (3.17) w_(w,z) ;_(w,z) -00 +00 + lim z

i

~(T) - ~CT) dT

w+O 2~i _-00 y+(T,Z) T -w

WES

-dT T -w

dT T-W

Proof. Since lim aF(o) = lim ~(O) = 0, we have by (3.7),

lol~ lol~

lim w+(o,z)

lol~ lol~ lim w_ (0, z) W(z) •

Hence M(z) =

o.

Remark I. If aF(o) and ~(o) are Holder continuous at infinity, then (3.17)

can be obtained by the methods of the classical Hilbert problem [8J, [IOJ.

Remark 2. Finally, we remark that J. Loris-Tegh~'s result [5J can be

obtain-ed from (3.13). In order to show this we

w_(w,z) w_(ie,z) +00 I

f

=~ ~ 21T 1. w_ (w, z) w_(ie,z) _{_00} w_(ie,z)

Since lim = I, we have

e+O w_(ie,z)

References

= w _ (w , z)[ I + lim

r

e+O 1T1. _{_00}

remark that for any e > 0,

zW(z){aF(,) - ~(l)} _I

U

- zaF«)}w_(T ,z) { T - W T - I.e I }dT.

zW(z){a_F (l) - ~ (T)} 1 I

U -

zaF(,)}w_("Z) {-:r=w-'1'_ie}d't'

[IJ J.W. Cohen,"The single server queue". North-Holland Publ. Comp., Amster-dam, 1969.

[2J W. Feller, "An introduction to probability theory and its applications

II". John Wiley & Sons, Inc., New York, 1966.

[3J J. Keilson and A. Kooherian, "On the general time dependent queue with

a single server". Ann. Math. Stat. 31, 1960.

[4J J. Loris-Teghem, "On the waiting time distribution l.n a generalized

GI/G/I queueing system". J. Appl. Probe 8, 1971.

[5J J. Loris-Teghem, "Dne application d'un theoreme de G. Baxter

a

la

deter-mination de la distribution des temps d'attente dans un modele

[6J N.T. Muskhelishvili, "Some basic problems of the mathematical theory of elasticity". P. Noordhoff Ltd. Groningen, 1962.

[7] F. Pollaczek, "Problemes stochastiques poses par Ie phenomene de for-mation d'une queue d'attente

a

un guichet et par des phenomenes apparentes". Gauthiers-Villars. Fas. CXXXVT. Paris, 1957.

[8J B.W. Roos, "Analytic functions and distributions in physics and engl.-neering". John Wiley & Sons, Inc. New York, 1969.

[9J L. Takacs, "On a method of Pollaczek". Stoch. Proc. and their applica-tions. Vol. I, 1973.

[IOJ E.J. Vanderperre, "A nonhomogeneous Hilbert problem with applications

to queueing theory". Memorandum CaSaR 73-08. Eindhoven, University of Technology. Eindhoven, 1973.