convergence of life distributions in a family which is larger than the HNBUE family. Also ... PROOF. The fact ~ C ;t4 follows immediately from the characterization.
Ann. Inst. Statist. Math. Vol. 46, No. 3, 587-592 (1994)
ON EQUIVALENCE OF WEAK CONVERGENCE AND MOMENT CONVERGENCE OF LIFE DISTRIBUTIONS Gwo DONG LIN
Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan, R.O.C. (Received June 23, 1993; revised February 10, 1994)
Abstract. We investigate the equivalence of weak convergence and moment convergence of life distributions in a family which is larger than the HNBUE family. Also, we point out that within the HNBUE family the exponential distribution can be characterized by one value of its Laplace-Stieltjes transform.
Key words and phrases: Weak convergence, moment convergence, HNBUE family, convex order, characterization of distribution.
1.
Introduction
Let F be a life distribution, namely F has the support contained in the interval [0, oc), and let its mean #F E (0, oc). Then F is said to be harmonic new better than used in expectation (HNBUE) if
~
['(x)dx < #F e x p ( - t / p F )
for all
t_>0,
where/~ = 1 - F (see, e.g., Klefsj5 (1982)). For convenience, the degenerate distribution at the point zero, denoted by F0, is also defined to be HNBUE. It is known that the HNBUE family is closed under weak convergence and that the weak convergence within this family is equivalent to the convergence of each moment sequence of positive order to the corresponding moment of limiting distribution (see Basu and Bhattacharjee (1984) and note that in order to guarantee the closedhess of weak convergence it is necessary to define F0 to be HNBUE). In this paper we first introduce a family 34 which is larger than the HNBUE family (Section 2), and then extend the above-mentioned results from the HNBUE family to the family 34 (Section 3). Finally, we point out that within the HNBUE family the exponential distribution can be characterized by one value of its Laplace-Stieltjes transform; a general result for the convex ordering family is also given (Section 4).
587
588
2.
GWO DONG LIN
The family .Avl
In this section we denote by F the general life distribution and by G the exponential distribution. Further, let X and Y be two random variables (r.v.s) having distributions F and G, respectively. For convenience, we also call F0 an exponential distribution with mean zero, and denote 1 / p = + o c if p = 0. The HNBUE family can be written as 7-t = [1... tt>_o
: #F = #V = #,
F ( x ) d x
.
Here we define three more families of life distributions: .A.tI = U { F tt>o
: pF = PG = # ' E X ~ 1};
A42 = U { F : PF = PO = p, Ee *x o A4 = A41 n A42. (See Stoyan ((1983), p. 34) for related life families.) It is easy to see t h a t M : ¢: M 2 and Ad2 ¢~ 3//1. We now prove t h a t A4 is larger t h a n 7-/. LEMMA 2.1.
~ C AJ and ?-t ~ A4.
PROOF. The fact ~ C ;t4 follows immediately from the characterization theorem (Lemma 2.2 below) for the HNBUE distributions. To see ?-t ~ AJ, define the distribution K by
K(x)=
0 0.3 1
if x < 0.3 if 0 . 3 _ _ x < 3 if3_< x.
Then K E A4 but K ¢ ~ (see also Klefsj5 ((1983), p. 617)). The proof is complete. LEMMA 2.2. Let F be a life distribution and G an exponential distribution with mean pG -- PF E [0, cxD). Further, let X and Y be two r.v.s having distributions F and G, respectively. Then F E ~ if and only if the inequality E h ( X ) < E h ( Y ) holds for all convex function h defined on [0, oo). PROOF.
See Stoyan ((1983), Theorem 1.3.1).
WEAK CONVERGENCE AND MOMENT CONVERGENCE
3.
589
Weak convergence within the family 3,t T h e main result is stated as follows. THEOREM 3.1. Let F~ E M and be obeyed by a r.v. X n , where n = 1, 2 , . . . . (i) I f F~ ~ F weakly for some life distribution F, then F E A4 and
(3.1)
lim E X ~ = E X ~ < oo
for all
r > O,
n ----+ C2~
where X is a r.v. obeying distribution F . (ii) Conversely, if the equality in (3.1) holds for each integer r > 0 and for some nonnegative r.v. X with finite mean, then F~ ~ F weakly, where F is the distribution of X and F E A4. To prove T h e o r e m 3.1 we need the following lemma. LEMMA 3.1. Let F E A/ll U A42 and be obeyed by a r.v. X . uniquely determined by the sequence of its m o m e n t s { E X n } ~ _ I .
Then F is
PROOF. It is clear t h a t the conclusion holds for the degenerate distribution F0. Suppose now t h a t F ~ F0. T h e n in case F E A41, we have for
EX n < EY ~ = n!/~
n = 1, 2 , . . . .
Note t h a t the Carleman criterion holds for F , namely ~-]~=~(EXn) -1/(2~) = oc, so the distribution F is uniquely d e t e r m i n e d by the sequence of its m o m e n t s { E X n } ~ = l . In case F E 2t42, we have E C x E e - X ~ ~ E e - X
E
(0, co)
as
n ~ oc,
so t h a t there exists a constant B E (0, cx~) such t h a t (3.2)
#n E [0, B)
for all
n = 1,2,....
For each fixed r > 0, take a constant s > max{r, 1}. T h e n since F• E M C M 1 it follows t h a t EX~ 1,
E X ~ = lim E X ~ 00
n---+ OO
lim F ( r + l ) p ; = F ( r + l ) p ~ . ft ---+~
Therefore F E -/~1- To prove the part F E M 2 , let us define the r.v. Z~ = exp(X~) for n = 1, 2 , . . . . T h e n for each fixed s < 1 / p F , take a positive constant t E (s, 1 / p F ) , and we have
EZ~ = Ee tX~
