On the Renewal Theorem in Higher Dimensions

Renewal theory has been treated by many pure and applied mathematicians. Among the former we may mention Feller, Täcklind and Doob. The principal limit theorem (for one-dimensional, positive, lattice random variables) was however proved earlier by Kolmogorov in 1936 as the ergodic theorem for denumerable Markov chains. A partial result for the non-lattice case was first proved by Doob using the theory of Markov processes, and the complete result by Blackwell. The extension of the renewal theorem to random variables taking both positive and negative values was first given by Wolfowitz and the author [1], for the lattice case. A partial result for the non-lattice case, using a purely analytical approach, was obtained by Pollard and the author [3].² For the literature see [1]…