Some recent characterizations of the exponential and geometric distributions

Some recent characterizations of the exponential and

geometric distributions

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Steutel, F. W. (1990). Some recent characterizations of the exponential and geometric distributions. (Memorandum COSOR; Vol. 9015). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1990

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Memorandum COSOR 90-15

SOME RECENT CHARACfERIZATIONS OFTHE

EXPONENTIALANDGEOMETRIC DISTRIBUTIONS

F.W. Steutel

Eindhoven University ofTcclmology

Department of Mathematics and Computing Science P.O. Box 513

5600 MB Eindhoven The Netherlands

Eindhoven,Juni 1990 The Netherlands

AND GEOMETRIC DISTRIBUTIONS

F.W. Steutel

Eindhoven U iversity of Technology, Eindhoven, The Netherlands

1. Introduction, not tion and summary

In this paper we resent some recent characterizations of the exponential distribution and its discrete analogue, the geometric distribution.

The following definition also serves as a model for the notation that we shall use.

1.1. Definition: A r. notation

x

E

E(

X has an exponential distribution with parameter A,

if, equivalently,

(i) _FX(X) P (X~x) = 1

-

e-AX (x>OJ A> 0)

,

(ii) _fx(x) F' (x) = A e

-Ax

(x> 0)

,

x

(iii) _fx(x) E e-sx = A (Res >- A)

1.2. Definition: A r.v. N has a geometric distribution with parameter P,

notation:

NEG(p),

if, equivalently,

(i) P(N=n) (1-p) pn (n=O, 1 ,2, ... ; O<p<l) I

(ii)

1 - P

1 - pz

Remark: We shall write X E E, if X E E(A) for some unspecified A; in a similar way we use NEG.

E and G are much used in applications; to name only a few: reliability, queueing, statistical physics. Many characterizations are known (see e.g. Galambos and Kotz (1978) and Azlarov and Volodin (1986».

The best known characterizations of E(A) are:

1. "Lack of memory":

X E E iff P (X > x +a

I

X> a) P(X>x) .

x

E

E iff min (Xl , •.. ,X )d .!.X 1 • n = n

Property 1. is also characteristic for G:

1 : NEG iff P (N > n +k) IN> k) = P (N > n) (n E:IN) I

but for 2. there is no direct analogue.

1.3. Proposition. N E G(P) iff

N [X] with X E E(/I.) and Ie = -log P

here [x] denotes the 'integral part' of x, i.e., the largest integer not exceeding x.

We shall consider three characterizations of E, some related characteri-zations of G, and some generalicharacteri-zations. The three charactericharacteri-zations of E concern the following properties and their (partial) converses:

I If X

E

E, then [x] and {x}

=

X - [X] are independent.

I I Let X

1'X2' ... be i.i.d. and nonnegative, and Y1,Y2, .•. i.i.d. in E. Let N == min {nE:IN I X

n

<

yn}, and Z == Y1 + •.• + YN-1 + XN' Then

d

Z X, if Xl E E

III Let U, Y

1 and Y2 be independent r.v. 's, U uniformly distributed on

d

(0,1) and Y

l

=

Y2 ~ O. Then

We shall see that the three properties above lead to characterizations

of E and G.

2. Independence of [X] and {X}

Let X be a r.v. with if. F

X= F. Then it is easily verified that [X]

and {X} (see I above) are independent iff

2.1. P ({X} ~ y

I

[X] n) = -.;::;....:.;,::...~-...;:;...=-~-F(n+y) - F(n-O)

F(n+l-O) - F(n-O) is constant in n.

As is easily checked, 2.1 holds for X E E, but it does not chararacterize E, since 2.1 is true for all r.v.'s X of the form X = N + Z with Nand Z

The following theorem is proved, independently, in Kopocinsky (1988) and Steutel and Thiemann (1989).

2.2. Theorem: Let X be a non-constant r.v .• Then X E E iff

2.3. [CXJ and {cX} are independent for all c

>

0 .

We sketch a proof of this result by giving five lemmas without proof; for details we refer to [16J.

2.4. Lemma: I f 2.3 holds and i f 0 < a < b < 1, then from P (k ~X< k + 1)

>

0

and P(k+a<cX<k+b) = 0 it follows that

P(£. +a

<

cX

<£.

+b)

o

for all £.E:?Z.

This means that once there is a gap in the probability distribution

between to integers, this gap is repeated periodically.

2.5. Lemma: If 2.3 holds, then

P (X~0) 1 or P (X~0) 1 •

From here on we assume that p(X ~0) 1.

2.6. Lemma: For all y

>

0 one has

P(O~X<y)

»

0 and P(X

>

y)

>

0 •

2.7. Lemma: If 2.3 holds, then [xJ E G.

The final lemma has earlier been given in Bosch (1977).

2.8. Lemma: If [cxJ

E

G for all c

>

0, then X E E.

This last lemma completes the proof-sketch of Theorem 2.2.

Remark: In Balkema (1989) it is shown that the independence of [cxJ and {cX} for only two positive values c

1 and c2 with c1/c2 irrational, is

3. A truncated sum of minima

In Khalil e.a. (1989) the following theorem is proved.

3.1. Theorem: Let X

1,X2, ... be i.i.d.; nonnegative, integer-valued, let Y

1,Y2, ••• be independent and in G(a). Let N inf{n E IN ; X

<

Y }

n n

and, finally, let

Z N

L

n=1 min (X , Y ) • n n Then Z d iff

In this section we consider the continuous analogue of Theorem 3.1. This analogue is also considered in Dimitrov and Khalil (1988) under much more restrictive conditions. The result is probably even more

generally true than in the form it is presented here: one value of

A

should suffice.

3.2. Theorem: Let X

1,X2, ••. be i.i.d. and nonnegative, let Y1,Y2, •.. be

inde-pendent and in E(A). Let

N inf {n E IN ; X

<

Y }

n n

and, finally, let

3.3. Z Then: Y l + •.. + YN-1 + XN N ==

L

n=l min (X , Y ) • n n a. (for all A) , b. I f ZA d= Xl for A Xl E E.

3.4.

(s +A)

£X

(s + A)

s +A

£X

(s +A)

"'. _'"

].l/ (].l+s). and from 3.4, substitution of fx(S)

=

].l/ (].l+ s) yields f

z

(s)

which proves part a.

'" '" '" _3.4.

To prove part b., put f

z =

fx f in to obtain 3.5. (S+A) £(S+A) s + A

£

(s+ A) If we put 1 -

£

(s) h(s) -'" s f(s) then from 3.5 it follows that

h(s) h(S+A) h(s +kA) (all k E :IN) ,

i.e. h is periodic with period A. Now,

if

3.6. then '" lim s f(s) s-+<>o exists, 3.7. or, h(s) '" 1 - f(s) s

£

(s) 1 -

£

(s + kA) lim -k~ (s + kA)

£

(s + kA)

a ,

1 1 + as

i.e. X E E(l/a), even if ZA

~

X for just one value of A. If 3.6. does not hold, then from

and the continuity of h it follows that h is constant, and hence (see

Remark 1: According to an example by S¢ren Asmussen (private communication) lim s 1(s) need not exist in general, but maybe it does for 1 satisfying

s~

3.4.

Remark 2: In the discrete case (Theorem 3.1) there are no such

difficul-ties: one value of a suffices.

4. Characterizations by stochastic equalities

In Runnenburg and Steutel (1985) the following question is examined: What X satisfy the equation

4.1.

where V, Xl and X

2 are independent, X~ Xl ~ X2 , and V is uniformly

distributed on (0,1). Denoting the characteristic function of X by ~,

from 4.1 one obtains

~(t) = 1

J

o

2 ~ (ut) dx t-1 t

f

o

2 ~ (x)dx ,

and from this the differential equation:

1 2

t ~ (t) + ~(t)

=

~ (t) ; q>(0) 1 •

The solution of this equation is given by (cf. Runnenburg and Steutel

(1985) )

4.2. (t) _{= -l-+-a...,!...,t....:I=--+-~-·}1 _{b-t- ,}

with a ~ 0 and b real, which is indeed a characteristic function. If we

require X to be nonnegative, then 4.1 yields a characterization of the

exponential distribution (see Kotz and Steutel (1988»:

4.3. Theorem: If P(X ~ 0) = 1, P(X = 0)

<

1, then X satisfies 4.1 iff X E E.

More general forms of equation 4.1 have been studied in Artikis (1982)

4.4.

4.5.

4.6.

4.7.

in terms of random variables. A quite general form of 4.1 is the follo-wing:

d 1/a

x(s) u X(s + t) ,

where X(o) is a Levy process (process with stationary independent in-crements, and with X(O)

=

0).

In this section we consider discrete analogues of 4.4. The simplest one is the direct analogue of 4.1:

where a @ Y is defined (in distribution, for a E [0;1], and nonnegative,

integer-valued Y) by

which P_y denoting the p.g.f. of Y. The distribution of U @ Y is then defined by

1

f

P_y(l - u + uz)du •

o

It is easily verified that the following result by a.o. Al Zaid and Al-Osh (1990) is true (compare Theorem 4.3.) •

4.8. Theorem: Let X be nonnegative and integer-valued with P(X=O)

<

1. Then X satisfies 4.5 iff X E G.

The multiplication defined by 4.6 was introduced in Steutel and Van Barn (1979). A more general version of this multiplication, was given in Van Barn e.a. (1982), and will now be used to obtain generalizations of Theorem 4.8. We give a formal definition.

4.9. Definition: Let X be nonnegative and integer-valued, and let a E [0,1]. Then a<i>F X is defined (in distribution) by

4.10. P_a @ (z) = Px(Ft(Z» F X

where F == {F

t ; t~O} is a composition semigroup of p.g.f.'s defined by

00

I

P(Zl(t) n=O n n)z , 4.11.

with Zk (.) a subcritical branching process satisfying P(Zk(0)

=

k) = 1. Equivalent to 4.10 one has

t d

e .,X

=

Z (t) X

for details we refer to Van Harn e.a. (1982).

Remark 1: In the special case given in 4.6 we have

-t -tz

e + e (t~O) ,

-t and, with a

=

e

Before we can formulate a generalization of Theorem 4.8 we need some notation from Van Harn e.a. (1982).

Let F be a semigroup of p.g.f.'s as in Definition 4. Then one defines

V(z) Ft(Z) - z lim UO t (the 'generator' of F) z

A

(z ) = exp ( -

J

V

/X) )dX

o

in the case of Remark 1 we have V(z) ~ A(Z)

=

1 - z.

We now formulate a theorem from a forthcoming paper (Van Harn and Steutel

4.12. Theorem: Let X(o) be a compound Poisson process (nonnegative, integer-valued Levy process), and let F be a semigroup as in Definition 4.9. Further let a

>

0, and let U be uniformly distributed on (0,1), indepen-dent of X(o). Then the equation

4.13.

4.14.

X(a)

~

u

1/

a

@ X(b)

F

is uniquely solved by X(o) with

_ { y }-r

P

X(l)(Z) - 1 + C Z (z) ,

(O<a<b) ,

with Y

=

a(b - a) / a and r is sufficiently small.

1 / (b - a), i f y ~ - log E Zl(1), i.e. i f a

The proof is essentially the same as the solution 4.2 of equation 4.1, except for some technical details.

Remark 1: Since X(o) is supposed to be a compound Poisson process and hence

infiniteZy divisibZe,

it is important to note that P

X(l) (z) in 4.14 is indeed infinitely divisible (compound

geometric).

Remark 2: The solution X(a) of 4.13 can be interpreted as follows:

i.e. X(a) has the same distribution as the (decreasing) branching process

Z(o) at an exponentially distribution time T

E

E(a), started with X(b)

a

individuals at time O.

Remark 3: In Van Harn and Steutel (1990) it is shown that equation 4.13 can also be solved for general nonnegative (i.e. non-integer) r.v.'s. Then the operation ~F has to be defined by branching processes with a more general ('continuous') state space. The formalism stays exactly the same. A link with the discrete case can be established by the means of Poisson mixtures. The simplest (trivial) subcritical branching pro-cess with continuous state space is given by

- t

and this leads us back to an equation of type 4.4:

References

1. Alamatsaz, H.M. (1985), A note on an article by Artikis, Acta Math. Hung. 45, 159-162.

2. Alzaid, A.A., and Al-Osch, M.A. (1990), Some results on discrete a-monotonicity, Statistica Neerlandica

i.

3. Artikis, T. (1982), A note on certain power mixtures, Acta Math. Hung. 39, 69-72.

4. Azlarov, T. and Volodin, N. (1986), Characterization problems associated with the exponential distribution, Springer, Berlin.

5. Balkema, A.A., On the independence of integral and fractional parts, preprint Univ. of Amsterdam.

6. Bosch, K. (1977), Eine Characterisierung der Exponentialverteilungen, Z. angew. Math. Mech. 57 (10),609-610.

7. Dimitrov, B. and Khalil, Z. (1990), On a new-characterizing property of the exponential distribution, J. Appl. Prob. 27, 221-226.

8. Galambos, J. and Kotz, S. (1978), Characterizations of probability distri-butions, Lecture notes in Mathematics, Vol. 67, Springer, Berlin.

9. Barn, K. van, Steutel, F.W. and vervaat, W., (1982), Self-decomposable discrete distributions and branching processes, Z.

10. Harn, K. van, Steutel, F.W. (1990), Characterization problems and branch-ing processes; preprint.

11. Khalil, Z., Dimitrov, B. and Dion, J.P. (1989), A characterization of the geometric distribution related to random sums; preprint.

12. Kopocinski, B., (1988), Some characterizations of the exponential distri-bl1tion function. Probl. and Math. Stat., Vol. 9, Fase 2, 105-111.

13. Kotz, S. and Steutel, F.W. (1988), Note on a characterization of

expo-nential distributions, stat. and Prob. Letters~, 201-203.

14. Runnenburg, J.Th. and Steutel, F.W. (1985), Solution to Problem 159,

Statistica Neerlandica~, 57-58.

15. Steutel, F.W. and Van Harn, K., (1979), Discrete analogues of self-decomposability and stability, Ann. Prob.

2,

893-899.

16. Steutel, F.W. and Thiemann, J.G.F. (1989), On the indepence of integer

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5600 MB Eindhoven - The Netherlands Secretariate: Dommelbuilding 0.03 Telephone: 040 - 47 3130

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