A NOTE ON BLACKWELL'S RENEWAL THEOREM

摘要

Our point of departure is a renewal process for which we define a number of crucial quantities. All of these concepts can be found in [1, 5, 7, 10]. Definition 1. Let {X_i, i ∈ N} be a sequence of independent identically distributed (i.i.d.) random variables with common distribution F of X, where X ≥ 0 but X not ≡ 0. The sequence {X_i,i ∈ N} is called a renewal process. Let S_0 = 0 and S_n = X_n + S_(n-1), n ≥ 1. The sequence {S_n,n ∈ N} constitutes the set of renewal (time) points. Let also t ≥ 0, and define N(t) = sup[n: S_n ≤ t}; then the processes {N(t),t ≥ 0} and N_0(t) := 1 + N(t) are called the renewal counting processes. The renewal functions are defined and denoted by U(t) =: EN(t) and U_0(t) =: I(t) + U(t). Finally, we write μ := EX ≤ ∞. We call the processes of the associated functions processes generated by F or by X. Note that N_0(t) is a stopping time for the sequence {X_i, i ∈ N}.