Acta Mathematica Academiae Scientiarum Hungaricae 31. (1978)
1978 / 1-2. szám - Sankaranarayanan, G. - Balakrishnan, V.: A renewal theorem for a sequence of dependent random variables
Acta Mathematica Academiae Scientiarum Hungaricae Tomus 31 (1—2) (1978), pp. 1—8. (LI) and (1.2) A RENEWAL THEOREM FOR A SEQUENCE OF DEPENDENT RANDOM VARIABLES By G. SANKARANARAYANAN and V. BALAKRISHNAN (Annamalai Nagar) Let {.v,} be a sequence of dependent random variables with E(x,)=/q, i=l, 2,.... Let n S„ = У X:, = max S, and N„ = max •=1 l^i^n 1^Шп Let Fn(x), Hn(x) and Kn(x) be the distribution functions of S„, M„ and Nn, respectively. Assume that H i+••• + (in a о hm ——£-=— ------— = ц, 0 < /í < a > О with the requirement that Var — as л —°°. Under a very general assumption on the asymptotic behaviour of (1.3) ßm(n) = E as л —we have proved that Here we have taken {a„} to be a positive sequence such that (1.5) an ~ nxL(n), n -*■ where L(n) is a non-negative slowly varying function and A is chosen such that 2, an is divergent. It has been indicated that (1.4) is true when F„(x) is replaced П = 1 by H„(x) or Kn(x). It has also been shown that many of the known results found in [1, 9] were special cases of this general theorem. 1. Introduction. Several authors [6, 8] have studied the asymptotic behaviour CO oo of 2anFn(x) and 2 anHn(x) when {*„} form a sequence of independent and n=1 n=l identically distributed random variables and {я„} a positive coefficient sequence (1.4) Var(S„) = Óin2* &), n — 2a =» <5 > 0, 2 (xi~Pi) i—1 (*//*)<*+1)/i£(*“) Жа"РЛх)-----itt—’ Act a Mathematica Academiae Scientiarum Hungaricae 31,1978