Arithmetics of aging distributions: Maximum -...

Acta Math. Hangar, 64 (1) (1994), 27-40.

A R I T H M E T I C S OF A G I N G D I S T R I B U T I O N S : M A X I M U M

T. F. M O R I 1 ( B u d a p e s t )

1. I n t r o d u c t i o n

1.1. O b j e c t i v e and p re l iminar ies . The present paper is the counterpart of [6], where the convolution structure of the reliability semigroups I F R , I F R A , N B U , N B U E , H N B U E and L are discussed. It is proved there that in each semigroup every distribution can be decomposed into the convolution product of at most countably many irreducible distributions and a degenerate one; degenerate distributions are the only anti-irreducible or infinitely divisible elements; and the set of irreducible distributions is dense. Uniqueness of the decomposition cannot be expected, since at least in the four upper classes no distribution is prime. This time our aim is to investigate these classes when the semigroup operation is the pointwise multiplication of distribution functions (which corresponds to taking the maximum of independent random variables) instead of convolution. In order to make the present work easily readable without having read its counterpart [6] we repeat here all the necessary definitions.

Arithmetical properties of probability distributions were first studied by Khinchin and L~vy, as early as in the thirties. Since then lots of relevant pa- pers have been published on the topic and interesting general theories have been developed. Most investigations have dealt with the convolution structure of probability distributions defined on more and more general struc- tures, but there exist results concerning semigroups of distributions with other operations such as the multiplication of distribution function (maximum of random variables). A recent monograph of Ruzsa and Sz~kely [8] provides an excellent synthesis of latest researches in the so called algebraic probability theory, which aims at proving arithmetical type results for a wide class of commutative semigroups. Their general theorems will be cited in this paper at every moment.

The arithmetic structure of the multiplicative semigroup of all real or nonnegative probability distributions is not so interesting, since every dis-

1 This paper was wri t ten while the author was visiting the Mathematical [nstituLe

of the Hungarian Academy of Sciences.

28 T F MdaI

F

F t > 0 .

F

t r ibut ion is infinitely divisible, therefore most ar i thmetic problems become meaningless. The si tuat ion is quite different in higher dimensions, where the pointwise mult ipl icat ion of dis tr ibut ion functions corresponds to the coordinatewise m a x i m u m of independent random vectors. Infinitely divisible elements in the mult iplicative semigroup of mult ivariate dis t r ibut ion functions were characterized by Balkema and Resnick [1]; Zempl6ni found condit ions for irreducibility and anti-irreducibility [9]-[11]. Another possi- bility to avoid triviality observed in one dimension is confining ourselves to certain subsemigroups, such as the aging classes of distributions. This is just what we wish to do.

Results to be communica ted below were discovered in 1987. Al though they were already reported in [5] and [8, Remark 6.3.9], no proofs have appeared so far. This may justify publishing the present paper.

1.2. A g i n g d i s t r i b u t i o n s . In order to introduce certain classes of aging dis t r ibut ions with finite expectat ion let us start with some definitions and nota t ions .

As it is usual in reliability theory, we shall only deal with the set D + of nonnegat ive probabili ty distributions. Distributions will be identified with their cumulat ive dis t r ibut ion functions defined right continuous. For an arbi t rary_distr ibut ion F E D + let us introduce the corresponding survival function F = 1 - F, expectation E ( F ) , variance V a r ( F ) , Laplace transform ~F(t) = f o e-tX dr(x) , t >= O, starting point aF and endpoint ~VF defined as a F = inf{t E It : r ( t ) > 0} and wv = sup{t E It: F(t) < 1}, resp.

The exponential dis tr ibut ion with expectat ion # > 0 will be denoted by s~, tha t is, E~(t) = 1 - e x p ( - t / # ) , t >= 0. The corresponding Laplace

1 t ransform is 1--4~, t > 0. In addit ion, we shall write 5, for the degenerate

dis t r ibut ion concentra ted onto # >__ 0. Clearly, 5 , ( t ) = 0 or 1 according as t < # or t >__ #, resp. The Laplace t ransform of 6, is e x p ( - # t ) , t >= 0.

The most frequently used classes of aging distributions are as follows.

F E I F R iff for every s > 0 the function t ~ F(t + s) /F(t) , t > 0 is decreasing.

F E I F I t A iff the function t ~ -F(t) 1/t, t > 0 is decreasing, or, equiva- lently, w(F,t) =: - } l o g F ( t ) is increasing in t, 0 < t < wF.

F E N B U iff F ( t + s) < F(t)F(s) for every nonnegative t and s.

E N B U E iff E ( F ) = # is finite and f o F(u)du <= #-F(t) for t __> 0.

E H N B U E iff E ( F ) = # is finite and f o - F ( u ) d u <= # e x p ( - t / # ) for

E L iff E ( F ) = # is finite and c2s(t ) < 1 = 1--4-~t, t > O. =

Acta Mathematica Hungarica 6~, 199~

ARITHMETICS OF AGING DISTRIBUTIONS: MAXIMUM 29

These classes form an increasing sequence in the order of definition: I F R C I F R A C N B U C N B U E C H N B U E C L. Properties of the first four classes are found in [2]. Classes H N B U E and L were first introduced and studied by Rolski [7] and Klefsj5 [4], resp. Both exponential and degenerate distributions belong to the smallest class IFR. In addition, exponential distributions lie on the common boundary of the above aging classes, since they satisfy each inequality-type definition with equality and show constant rate where monotonicity is required.

All these classes are closed subsets of D + with respect to the usual topology of convergence in distribution (i.e., pointwise convergence at the continuity points of the limit distribution function).

1.3. A r i t h m e t i c a l def ini t ions . For any pair of distributions F, G E E D + let F V G denote their pointwise product. This corresponds to the maximum of independent random variables with distributions F and G, resp. Operation V is clearly commutative , associative.and continuous with respect to the weak topology. This makes it possible to extend the operation to an infinite sequence of distributions as the weak limit of the finite sections. The main difference with respect to the case of convolution discussed in [6] is that the semigroup (D+,V) is not cancellative, which makes the basic arithmetical notions a little more complicated. Let us examine the above aging properties whether they are preserved under this operation.

As it is well-known, I F R is not closed in this sense; for example, E, V V ~ • I F R if # # ~. Classes I F R A and N B U are closed under a more general (multivariate) operation: the life distribution of a coherent system with independent components all belonging to I F R A (NBU) is I F R A ( N B U ) again [2, Theorems 4.2.6 and 6.5.1]. Particularly, the case of parallel systems shows that both classes are subsemigroups of D +, thus they can be subjects of our further investigations. Though N B U E is not preserved under the formation of coherent systems ([2] contains a counterexample consisting in a series system of two independent components), it can be shown that V does not lead out from N B U E . Maybe this is well-known; for the sake of completeness we nevertheless give a short proof below.

LEMMA 1. Let F and G belong to N B U E , then so does F V G.

PROOF. Suppose E(F) >___ E(G). Since F V G(t) = F(t) + F(t)G(t), we have

// // F V G(t)dt = (-F(t) + F(t)-G(t)) dt =

/ / - / / - / / = F(t) dt + F V G(x) E(t)G(t) d t+ F(x)G(x) F(t)G(t) dt.

Here the first two terms do not exceed E(F)F(x ) and (E(F v G)- - E(F)) F V G(x), resp. The third term can be estimated in the following

Acta Mathematica Hungarica 64, 1994

30 T.F. MORI

way:

jfX C, O - - ~(X) - -

F(x)G(x) F(t)G(t)dt < F(x) G(t)dt <__

Hence

< <= E ( r ) r ( x ) a ( x ) .

~o ~176 F V G(t) dt <

<= + r(x)C(x)) + ( E ( r v a) - E(r)) r v a ( x ) =

= E(F v a ) r v a(x).

REMARK 1. If a F ~ x <~ WF and aa < x < wa, in the above line strict inequality holds.

H N B U E is not closed with respect to V, as it can be seen by considering the mixture F = 0.98 51 + 0.02 55 E H N B U E , F V F = 0.9604 51 + + 0.0396 55 r H N B U E . So far I have been unable to decide if L is closed or to find any reference on the subject.

In the ar i thmetic of (D +, V) the unity is ~0, the point mass at 0. Now, let S be an arbi t rary subsemigroup of D + containing 50. For F and G belonging to S let us introduce the following arithmetical notions.

G is a divisor (or a factor) of F if there exists an H in S such that F = G V H. We use the notation G[F. An equality of the form F = G V H is called a decomposition of F. A pair of elements mutually dividing each- other are called associates. Since G[F implies F __< G pointwise, it follows tha t D + is associate-free.

G is an effective divisor of F if F = G V H with H E S, H ~ F. A decomposition F = G V H is effective, if neither G, nor H is equal to

r ~ F F

tion. F

is irreducible, if F ~ 50 and it has no divisor but the unity and itself. is effectively irreducible, if F ~ 50 and it has no effective decomposi-

is anti-irreducible, if it has no irreducible effective divisor. F is effectively anti-irreducible, if it is not effectively divisible by any

effectively irreducible element. F is idempotent, if F 2 = F. F is bald, if it has no idempotent divisor

but ~5o. F is infinitesimally divisible, if it can be decomposed into a product of

distributions all coming from an arbitrarily preassigned neighbourhood of 50. F is infinitely divisible, if for every positive integer n there exists an Fn E S such that F = Fn n (F~ is called the nth root of F) .

F is prime, if it is different from 50 and FIG V H implies FIG or FIH.



2. M a i n resul ts

2.1. D e c o m p o s i t i o n s . In this section a Khinchin-type decomposition theorem will be proved in our three aging classes. This can be done by applying the general theory of Hun semigroups developed in [8, Chapter 2]. We start with some simple lemmas.

LEMMA 2. F E D + is idempotent iff F = (~c, c > O, F is bald iff t~F = O.

PROOF. Obvious.

LEMMA 3. I F R A , N B U and N B U E are stable normable Hun semigroups (see Definitions 2.2.2, 2.10.6 and 2.15.2 in [8]).

PROOF. It is sufficient to show that D + itself possesses all these properties, since they are inheritable to closed subsemigroups. Since a distribution is always stochastically larger than its divisors, the set of divisors of a compact set is tight, hence relatively compact.

Thus D + is stable Hun. D + is normable, since for any non-degenerate F and aF < x < ~F, AF(G) = - log G(x) defines an F-norm on the set of divisors of F such that AF(F ) > 0.

THEOREM 1. In I F R A , N B U and N B U E every distribution can be decomposed

(a) into the product of at most countably many irreducible distributions and an anti-irreducible one,

(b) into the product of at most countably many effectively irreducible distributions and an effectively anti-irreducible one.

PROOF. This is a simple corollary of our Lemma 3 and Theorem 2.23.3 of [s].

REMARK 2. The above decomposition can sometimes be simplified due to the results of the next section. Firstly, by Theorem 2.24.16 of [8], every effectively anti-irreducible element is infinitely divisible, hence we obtain that in I F R A every distribution can be decomposed into a (finite or countably infinite) product of effectively irreducible elements and a degenerate distribution. Secondly, the (effectively) anti-irreducible factor can be omit- ted for bald distributions in each of our semigroups, since by Theorem 2.8.9 of [8] every bald anti-irreducible element is infinitesimally divisible.

2.2. In f in i t e ly divis ible d i s t r ibu t ions . In this section we first deal with infinitesimally divisible elements.

LEMMA 4 [8, Remark 5.9.8]. Expectation is a continuous operator on L (hence particularly on N B U E ) .

T•EOREM 2. There is no infinitesimally divisible element in I F R A , N B U and N B U E except the unity 5o.

Acta Mathematica Hungarica 6J, 1994

32 T .F . MORI

PROOF. Suppose F is infinitesimally divisible in N B U E (it is clearly sufficient to deal with the largest class). Let us fix x > 0 and e > 0 arbitrarily. Then by the previous lemma there exists a decomposition of the form F = F1 V F2 V . . . V Fn, where Fdx ) >= 1 - c and E(Fi) _5 ~, i = 1 , . . . , n . Now let y >= x, then F(y) > 0 and

- l o g F ( y ) = E ( - l o g F ~ ( y ) ) _<_ Fi(Y) < _1 ~ ( y ) . i=1 i=1 Fi(y) = 1 e i=1

Integrat ing this and using the N B U E property we obtain that

- l o g F ( y dy < - E i(x) < e ( ) < 1 e i=1 1 s i=1 i'3:"

n

( )) < e - log b~(x = 1 e 1 e

- - - ( - l o g F ( x ) ) .

Since ~ can be arbitrarily small, it follows that F(x) = 1 for every x > 0.

TItEOREM 3 (Characterization of infinitely divisible distributions). (a) F e I F R A is infinitely divisible iff F = ~c, c >__ O. (b) F E N B U is infinitely divisible iff WE <= 2C~F. (c) F E N B U E is infinitely divisible iff

~ OO

(1) ( - logF(y)) dy _< . F ( - l o g r ( x ) ) , Vx > -F.

PROOF. (a) Degenerate distributions are obviously infinitely divisible. On the other hand, consider a non-degenerate F E I F R A that is infinitely divisible. Then there exist positive numbers x and y, 0 < x < y, such that 0 < F(x) <_< F(y) < 1. Infinite divisibility means that F 1/n e I F R A for every n = 1 ,2 , . . . , thus w(F l/n, x) -< w(F l/n, y), from which it follows that

i_ _ log(n(1 - F1/n(x)) < _1 _ log(n(J - F1/n(y)) x x l o g n = y y l o g n

! a contradiction. Lett ing n --. oc we obtain ~ __< y,

(b) Suppose F is infinitely divisible in N B U . By the N B U property of F 1/'~, for arbi trary x > aF, we have

1 - r l / n ( 2 x ) ~ (1- F1/~(x)) 2



Since n ( 1 - c 1/n) ~ - l o g c , multiplying the above inequality by n 2 then tending with n to infinity we arrive at a contradiction unless F(2x) = 1. Hence WF <= 2aF.

The opposi te implicat ion easily follows from the simple observation tha t wF <= 2aF is sufficient for any F r D + to have N B U proper ty (the inequali ty in the definition holds automatical ly if t or s is less than ag or t+~ >~F).

(c) Suppose F is infinitely divisible in N B U E . Since FUn C N B U E , f o r x > a F w e have

(2) fxc~ n (1 - F1/n(t)) dt <= nE(F1/n)( l - FUn(x) ) .

Here n( 1 - r l /n ( t ) ) <= - log F(t) uniformly in n, and - log F is integrable

over ( x , + o o ) , since - i o g r ( t ) ~ F ( t ) as t --~ ~ . Let n tend to infinity in (2). Then applying the monotone convergence theorem and using tha t E ( F 1/'~) ~ a F we arrive at (1).

For the opposi te it suffices to show that inequality (1) implies the N B U E property, because (1) is inheri table from F to F1/~.

Let 0 == x < aF , then clearly

/ ~F( t)dt <= E(F)= E(F)F(x).

If a F ~ x and F ( x ) E ( F ) ~ ~F, we have

/ ~-F(t) dt = E ( F ) - if0 x F ( t ) d t <= E ( F ) - aF <= E ( F ) F ( x ) .

Finally, if F ( x ) E ( F ) > aF, then

f f ( )) oo -F(t) dt <= - log F(t) dt <= aF - - log F(x <__

1 - F(x) < E ( F ) F ( x ) , =< -F =

thus F E N B U E .

2.3. I r r e d u c i b l e d i s t r i b u t i o n s . Similarly to the case of convolution s t ructure , we cannot give the full characterization of irreducible distribu- t ions, but there exists a simple sufficient condition for a dis tr ibut ion to be irreducible even in the largest class N B U E , which is still sufficiently general to imply tha t irreducible distr ibutions are dense in all classes under consid- erat ion. The next l emma corresponds to Lemma 4 in [6]; in fact, formally it is the same assertion but with a different proof.


34 T. F, MORI

LEMMA 5. Let F E N B U E , F ~ 50, and suppose that

limsup x-2F(x) = +oc. xi0

Then F is irreducible.

PROOF. By the N B U E property,

(3) F(x) <__ 1 1 F(t) dt - - - E(F) 1 f0~ E(F) -F(t)dt <= x /E(F) .

Hence the lemma.

REMARK 3. Lemma 2.11.8 of [8] concerns the existence of an upper bound for the number of terms in certain decompositions of a bald element in a Hun semigroup. Inequality (3) above provides the following estimation in N B U E . Let F = F1 V F2 V . . . V Fn, aF = 0, then

n _< liminf logF(x) - xl0 log x

THEOREM 4. The set of irreducible elements is dense in I F R A , N B U and N B U E .

PROOF. By Lemma 5 it suffices to show that every distribution in each class can be approached by distributions with the property F(x) ~ ex, x I O. This is just what has been done in Theorem 3 of [6].

REMARK 4. Effective irreducibility is weaker than irreducibility, for example, every irreducible distribution is necessarily bald, while there are lots of non-bald effective irreducible elements, as we shall show it in the next section (they all are anti-irreducible). Evidently, the set of effective irreducible elements is also dense in our semigroups. The set of irreducible elements, as well as the larger set of effective irreducible elements, are G~ in each class, by Theorem 2.19.2 of [8].

2.4. A n t i - i r r e d u c i b l e d i s t r ibu t ions . In this section we are going to show that, unlike the case oflconvolution structure, the set of anti-irreducible elements is rich enough to be dense in each semigroup. Two simple steps of the proof may be worth separating as lemmas, being of independent interest.

LEMMA 6. Let F,G E I F R A . Suppose that w ( F V G,x) is a positive constant on a nonnegative interval (a, b). Then min{~y,~G} 5 a.

PROOF. Suppose on the contrary that F(a)< 1, G(a)< 1. Let us denote w(F,a) = 1 / f , w(G,a) = 1/g, w(F V G,a) = 1/h. By the I F R A property F(x) => S(x) and V(x) >= g(z) for x > a. Hence in interval



(a,b) we have Sh(X) = F V a(x) => sf V sg(x), and for x = a equality holds. Here e] V r E I F R A , consequently ef V sg(x) >__ Sh(X), x > a. Thus

(4) (1 - e x p ( - f x ) ) (1 - exp ( -gx ) ) = 1 - e x p ( - h x )

for every x E (a,b). Both sides of this equality are analytic functions of x, hence (4) must hold for every real x. But this is impossible, since for negative x the sides of (4) differ in sign.

LEMMA 7. Let F E N B U . Suppose that F is constant and less than 1 on a nonnegative interval (a, b). Then alz >= b - a.

PROOF. Let 0 < x < b - a, then by the N B U property F(a) = F(a + x) <= v ( a ) r ( x ) , hence r ( x ) = o.

TtIEOREM 5. The set of anti-irreducible elements is dense in I F R A , N B U and N B U E .

PROOF. The method of proof is common for I F R A and N B U E : we show that every distribution can be approximated by non-bald effective irreducible ones to an arbi t rary extent. Since the only effective divisor of an effectively irreducible element is itself, and a non-bald distribution can never be irreducible, non-bald effective irreducible elements are anti-irreducible at the same time. In addition, in the proof of Theorem 4 we have seen that every distribution in each semigroup can be approximated by bald ones with arbi t rary accuracy. Therefore we can be confined to approximating bald distributions.

Case I F R A . Let us define the approximating distributions F~, n = = 1 ,2 , . . . by

O, if nx <__ 1,

(F , 1 ~ __k+l < n x < k w ( F n , x ) = w \ k n ] ' if k = = k - 1 ' k = 2 ' 3 " " '

1) w x - , if 2 _ ~ n x .

This function is increasing in x, that is, Fn C I F R A . Clearly, aFn _2 > 1/n. Consider an arbi t rary IFRA-decomposi t ion of the form F~ = G V V H. In virtue of Lemma 6 we can suppose wa _-< 1/n, hence H = Fn, thus

the decomposition is not effective. Finally, Fn w) F, since w(F~,x) w(F,~,x) at every F-cont inui ty point x > 0. Case N B U E . Let R be a fixed distribution, s and 5 positive parameters

which can be thought of as small. For any distribution F E N B U E let us


36 T.F. MORI

define min{ R ( x ) , r ( c ) } ,

Fs(~) = F(~ - 6 + ~), Then # ( 6 ) = E(F~) i s continuous in 6, since

/0 . ( 6 ) = r ( t ) d t +

i f x < 6,

if x>_&

max{ R( t ) ,F ( r } dt.

In addition, if F (e ) < 1, we have

/ ~ y ( , ( e F ( E ) ) __< t )dt + eF(~) <__ E(F) .

0 0

. ( ~ / T ( c ) ) => F(t)dt + T(~)(~/F(~)) > E(F).

Hence . ( 6 ) = E ( F ) for some 6 = 6(E), cF(~) _< 6 _< e/F(~). Let u s choose (~ in this way. We first show that if both F and R belong to N B U E and E ( F ) __< E(R) , then F~ E N B U E . Indeed, for x __< (i we can write

]o /o x- xT6(t)dt = > R( t )d t > = E ( R ) R ( x ) > = E(F6)F6(x),

while for x > 6

/0 Z T~(t)dt = (t)dt <= E ( F ) F ( x - 6 + e) = E(F6)F~(x).

It is t ime now to specify R. Let 0 < c < EF(s), c < E ( F ) F ( r and

R(x) = c exp x - c if x > c. 1 - 1 E ' =

Then E ( R ) = E ( F ) , and ar~ = aR = c. We ~ant to show that no N B U E - decomposition of the form F6 = G V H can' be effective. Let c < x < 6, then

Z F6(t) d t = E ( F ) - c - 1 E(-F) exp ~ d t =

= ( B ( F ) - c) exp

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A R I T H M E T I C S O F A G I N G D I S T R I B U T I O N S : M A X I M U M 37

By Remark 1, ei ther wa =< c or WH ~ C; we can suppose the former. Conse- quently, H(x) = Fh(x) for x => c. If aH were less than c, H could not satisfy the N B U E inequali ty for x C (c, 5), since

-H(t) dt = -F6(t) dt = E(F~)TE(x) > E(H)H(x) .

Thus H = F6. Finally, let s tend to 0, then the corresponding 5 also tends to 0 (c ~ 5),

and the dis t r ibut ion F6 constructed above converges to F weakly. Case N B U . Now let F~(x)= F([nx]/n), then F~ e N B U , since

-Fn(t)F---~(8) = -F([nt]/n) F([ns]/n) >= -F( ([nt] + [ns])/n) >=

>= F([nt + ns]/n) = -fn(t + ~).

Clearly, Fn .W> F as n -~ ~c. This t ime we prove the anti-irreducibility of Fn directly. Suppose Fn = G V H with some G, H E N B U . Then, together with Fn, bo th G and F are constant and positive on the intervals (k k_+A~ ~ n ' n ] '

k >= 1. IfwG > 1/n = aF,,, then ~G _-> 1/n by Lemma 7. thus G i s not irreducible. On the other hand, if wv <= 1/n, then H = F~, thus G is not an effective divisor of F~. Hence F~ is anti-irreducible.

REMARK 5. In the above proof none of the constructed anti-irreducible dis t r ibut ions is bald. This is not a ma t t e r of chance: there are no bald anti- irreducible elements at all (see Remark 2). One can natural ly ask if the (smaller) set of effectively anti-irreducible distributions is still dense. The answer is negative, not only in our semigroups, but also in every stable normable Hun semigroup, in which not every element is infinitely divisible. This follows from Theorem 2.24.16 and Statement 2.19.1 of [8]. The former claims tha t effective anti-irreducible elements are infinitely divisible; the la t ter , tha t the set of infinitely divisible elements is closed, thus it could be dense but in the trivial case when it exhausts the whole semigroup. Another_ question is whether effectively anti-irreducible distributions are dense among the infinitely divisible ones (this is obvious in I F R A but not decided in N B U and N B U E ) .

2.5. P r i m e d i s t r i b u t i o n s . There is no general result known about the existence or non-existence of primes in semigroups, should they be as special as stable, normable , metrizable Hun. For this reason it is always interest ing to answer this question in specific semigroups. Ruzsa and Sz~kely proved t h e non-existence of primes in the convolution semigroup of probability dis tr ibut ions on the Borel field of a locally compact Hausdorff group (to be more precise, they found a finite number of exceptions, all on small cyclic groups, see [8, Section 4,7]). Similar results are known for some other


38 T.F. M6RI

semigroups, such as the multiplicative s tructure of mult ivariate dis t r ibut ion functions (due to Zempldni, see [8, Corollary 6.3.7]), or the convolution s t ruc ture of reliability classes N B U , N B U E , H N B U E and L [6].

Surprisingly, in out semigroups there are primes, however trivial.

THEOREM 6. Degenerate distributions are the only primes in I F R A , N B U and N B U E .

PROOF. The prime proper ty of degenerate distributions is obvious, since ~fclF iff c __< O~F, and o~Gv H -~ m a x { a a , al l} . In the opposi te direction, the pr ime proper ty of a given non-degenerate F can be disproved, for example, by considering an appropr ia te decomposit ion of F V 5c with c < wE, in which none of the factors is divisible by F .

Case I F R A . Let F be an arbitrary non-degenerate I F R A distribution. We dist inguish two cases according as l imw(F ,x ) = l i m f ( x ) / x is positive

zlO xl0 or not .

Firstly, when this l imit is equal to 0, let G be defined as

w(F,c ) , x < c,

= w ( F , x ) , if x >= c,

where aF < c < W F . Then G E I F R A , G r F , F V 5~ = G V 5c, hence F divides G V 5c, but does not divide either G or 5c (for G is irreducible by L e m m a 5 and F(c) < 1).

Secondly, when this limit is positive, then lim w(F 2, x) = 0; thus we can xl0

repeat the above a rgumenta t ion with F 2 in place of F. Case N B U . If F C N B U is a non-degenerate infinitely divisible distri-

but ion, it cannot be prime, since F divides F = v / F V x/T, but does not divide v /F . By Theorem 3 we can suppose that 2 C~F < ~VF. Let c be an F-cont inui ty point , 2 aF < c < wF, then by the N B U inequality F(c) >= >= 1 - (1 - F(c/2)) 2 > F(c/2). Now define G as

0, if x < c/2,

G(x) = F(c/2) , if c/2 <= x < c,

F(x) if c __< x.

m

Then a E N B U , because for 0 g t _< s, t < c/2 we have G(t)G(s) = G(s) _> > G(t + s), while for c/2 < t < s we can write a( t )G(s) > F( t )F(s ) > >__ F( t + s) = G(t + s). Again, F V (5c = G V 5c, thus FIG V 5c. Clearly, 5c

c (c : l is not divisible by F; nor can F divide G, since q(~/2) = 1, but

F ( c / 2 ) F ( c - ) -- F(c/~)

-~ F(c ) < 1.

Acta Mathemat i ca Hungarlca 64, 1994


Case N B U E . IfwF = c < ~ but F ~ 5c, then 5~/2 VFi s non-degenerate and infinitely divisible (even in NBU) . It is easy to see that F does not divide V/~2 V F (because F(x) < ~ if aF < x < WF), thus F is not prime.

Suppose now W F = OO. Since - l o g F ~ F as x ~ oo, we have

( - l o g F ( x ) ) -1 ( - l o g F ( t ) ) d t ~ F ( x ) - 1 -F(t)dt <= E(F) .

Hence there exists a positive constant c for which

l o g F ( t ) ) dt _< _> c.

Consequently F Y ~c is infinitely divisible, in virtue of Theorem 3. Again, F does not divide v/-fi V 5c, thus F is not prime.

A c k n o w l e d g e m e n t . Research supported by the Hungarian National Foundation for Scientific Research, Grant N ~ 1405, 1905.

References

[1] A. A. BaJkema and S. I. Resnick, Max-infinite divisibility, J. Appl. Prob., 14 (1977), 309-319.

[2] R. E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing. Holt, Rinehart & Winston (New York, 1975).

[3] H. W. Block and T. H. Savits, The IFRA closure problem, Ann,. Probab., 4 (1976), 1030-1033.

[4] B. Klefsj6, A useful ageing property based on the Laplace transform, J. Appl. Probab., 20 (1983), 615-626.

[5] T. F. M6ri, Max-arithmetic of aging distributions, in: Fifth European Young Statis- ticans Meeting (/~rhus, 1987), J. L. Jensen et al (eds.), Dept. Theor. Statist. Inst. Math. /~rhus Univ. (1987), pp. 98-100.

[6] T. F. M6ri, Arithmetics of aging distributions: convolution, to appear. [7] T. Rolski, Mean residual life, Bull. Internat. Statist. Inst., 46 (1975), 226-270. [S] I. Z. Ruzsa and G. J. Szdkely, Algebraic Probability Theory. Wiley (New York, 1988). [9] A. Zempl~ni, On the arithmetical properties of the multiplicative structure of p r o b a -

bility distribution functions, in: Statistics with Applications (Proc. of the 5th Pannonian Symposium on Math. Statist., Visegrs 1985), W. Grossmann et al (eds.) Akad~mi~i Kiad6 and Reidel (Budapest - Dordrecht, 1988), Vol. A, pp. 221-233.


40 T. MORI: ARITHMETICS OF AGING DISTRIBUTIONS: MAXIMUM

[10] A. Zempl~ni, The description of the class I0 in the multiplicative structure of distribution functions, in: Mathematical Statistics and Probability Theory (Proc. of the 6th Pannonian Symposium on Math. Statist., Bad Tatzmannsdorf, 1986), M. L. Puri et al (eds.), Reidel (Dordrecht, 1987); Vol. A, pp.291-303.

[11] A. Zempl~ni, On the max-divisibility of two-dimensional normal random variables, in: Probability Measures on Groups IX (Proceedings, Oberwolfach 1988), H. Heyer (ed.), Lecture Notes in Math. 1379, Springer (Berlin, 1989), pp. 419-424.

(Received March 18, 1991; revised July 5, 1991)

DEPT. OF PROBABILITY THEORY AND STATISTICS E()TV(~S LORAND UNIVERSITY H-1088 BUDAPEST HUNGARY

Acia Maihematica Hungarica 64, 1994

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