ON INFINITELY DIVISIBLE LAWS AND A RENEWAL THEOREM FOR NON-NEGATIVE RANDOM VARIABLES.
Abstract
Proof is presented for the following theorem: Let (X sub n) be an infinite sequence of independent non-negative random variables such that, for some regularly varying non-decreasing function lambda(n), with exponent l/beta, o<beta<infinity, as n approaches limit of infinity, P (X sub l+...+X sub n/lambda (n) < or = x) approaches limit of K (x) at all continuity points of some d.f. K(x). Let A(x) be the function inverse to lambda(n), let R(x) be any other regularly varying function of exponent alpha > O. Then, if N(x) is the maximum k for which X sub l+...+X sub k < or = x, it is proved that, as x approaches limit of infinity, where E R(N(x)) approximately equals I(alpha beta) R(A(x)) and I(alpha beta) integral from O to infinity l/u to the alpha beta power d K(u) and this latter integral may diverge. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Feb 01, 1967
- Accession Number
- AD0648577
Entities
People
- Walter L. Smith
Organizations
- University of North Carolina at Chapel Hill