Vol. 30, No. 3, 1969

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ISSN: 0030-8730
Some renewal theorems concerning a sequence of correlated random variables

G. Sankaranarayanan and C. Suyambulingom

Vol. 30 (1969), No. 3, 785–803
Abstract

Consider a sequence {xn},n = 1,2, of random variables. Let Fn(x) be the distribution function of Sn = k=1nxk and Hn(x), the distribution function of Mn = max1knSk. Here we study the asymptotic behaviour of

∑∞
anGn (x),
n=1
(1.1)

where Gn(x) is to mean either Fn(x) or Hn(x) (so that if a property holds for both Fn(x) and Hn(x) it holds for Gn(x) and conversely) and {an} a suitable positive term sequence, when {xn} form

(i) a sequence of dependent random variables such that the correlation between xi and xj is ρ,ij,i,j = 1,2, , 0 < ρ < 1,E(xi) = μi,i = 1,2, and

    μ1-+-μ2 +-⋅⋅⋅+-μn
nli→m∞        nα        = μ,α > 1,0 < μ < ∞
(1.2)

and

(ii) a sequence of identically distributed random variables with E(xi) = μ,i = 1,2, such that the correlation between xi and xj is ρij = ρ|ij|,i,j = 1,2,,0 < ρ < 1.

Suitable examples are worked out to illustrate the general theory.

Mathematical Subject Classification
Primary: 60.70
Milestones
Received: 15 November 1968
Published: 1 September 1969
Authors
G. Sankaranarayanan
C. Suyambulingom