Hirohisa Hatori

Some Tauberian theorems for stochastic processes

(1) ]im t-az(f)= uniformly in λsΛ, where Ax is bounded on Λy then ( 2 ) Mm sfx(s)=Ax s-»+0 uniformly in λ

Some theorems in an extended renewal theory, II

Λ. Conversely if there exist constants K and f >0 such that for every XGΛ the functions Reax(t)+Kf and lmax(t)-}-Kt are non-decreasing in0^t<oo and if (2) holds uniformly in λzΛ with Ax bounded on Λ, then (1) holds uniformly in teΛ. The proof of this Lemma will not be given here, since it is similar in the main to the proof of well-known Tauberian theorem (see [1]). 3. We shall now prove the following THEOREM 1. Let {X{t)\t^} be a stochastic process such that ^X(t)dt exists for every finite T>0, and assume that there exist positive constants M and γ such that \/E{\X(t)\} ^Mffor every t>0. Then a necessary and sufficient condition that (3) U^Tζχm = is that