On the Central Limit Theorems for the Renewal and Cumulative Processes

In our dissertation we derive the Central Limit Theorems (CLTs) with the remainder terms for the renewal and cumulative processes. A renewal process based on the sequence of random variables {X n } is defined in one of the following ways: 1) 2) N 2 (t) = number of n such that Sn $ t, 3) N 3 (t) = Sup{n: Sn $ t}, where Sn = Xl + ... + X n. If X n > 0 then all three definitions are identical. In this case the renewal process is denoted N(t). It is well-known that a CLT for the Sn's allows an expansion if EIXllk < 00 for some k 2: 3. The order of the remainder term in the CLT depends on k. \Ve prove a similar CLT for N(t) when the Xn's are i.i.d. and positive. Next we allow the Xn's to have different distributions and to take negative values. Vle prove a CLT with the remainder term of the order of O(~) for each Ni(t), i = 1,2,3, when the Xn's have the uniformly bounded third moments and satisfy other "natural" conditions.