II.—Asymptotic Renewal Theorems

Abstract

Synopsis A sequence of non-negative random variables { X i } is called a renewal process , and if the X i may only take values on some sequence it is termed a discrete renewal process . The greatest k such that X 1 + X 2 + … + X k ≤ x (> o) is a random variable N ( x ) and theorems concerning N (x) are renewal theorems . This paper is concerned with the proofs of a number of renewal theorems, the main emphasis being on processes which are not discrete. It is assumed throughout that the { X i } are independent and identically distributed. If H ( x ) = Ɛ { N (x) } and K ( x ) is the distribution function of any non-negative random variable with mean K > o, then it is shown that for the non-discrete process where Ɛ{ X i } need not be finite; a similar result is proved for the discrete process. This general renewal theorem leads to a number of new results concerning the non-discrete process, including a discussion of the stationary “age-distribution” of “renewals” and a discussion of the variance of N ( x ) . Lastly, conditions are established under which These new conditions are much weaker than those of previous theorems by Feller, Täcklind, and Cox and Smith.