On a Limit Theorem in Renewal Theory

If oo < m < 0 then the limits given above should be interchanged. In the case where PI{X <0) = 0 the above theorem is deducible from results of Kolmogorov [1] and was proved very expeditiously by Erdts, Feller, and Pollard [2]. In the case m 5 i o? , the theorem can be proved by a different method developed by one of the authors [3] for the "continuous" analogue. The case m = i oo presents special difficulties and has to be treated separately. We shall proceed with the proof for 0 < m _ oo, t = 1; the general case follows trivially from this. Our proofs for both cases use an idea in [2], and our proof for the case m = Xo makes use of ideas in [1] and [4]. Let