Accession Number : AD0648577

Title :   ON INFINITELY DIVISIBLE LAWS AND A RENEWAL THEOREM FOR NON-NEGATIVE RANDOM VARIABLES.

Descriptive Note : Mimeo series,

Corporate Author : NORTH CAROLINA UNIV CHAPEL HILL DEPT OF STATISTICS

Personal Author(s) : Smith,Walter L.

Report Date : FEB 1967

Pagination or Media Count : 43

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)

Descriptors :   (*RANDOM VARIABLES, THEOREMS), PROBABILITY, FUNCTIONS(MATHEMATICS), DISTRIBUTION FUNCTIONS

Subject Categories : Statistics and Probability

Distribution Statement : APPROVED FOR PUBLIC RELEASE