Quantitative Exponential Bounds for the Renewal Theorem with Spread-Out Distributions

J.-B. Bardet, A. Christen, J. Fontbona

2017, v.23, Issue 1, 67-86


We establish exponential convergence estimates for
the renewal theorem in terms of a uniform component of the
inter-arrival distribution, of its Laplace transform which is
assumed finite on a positive interval, and of the Laplace transform
of some related random variable. Although our bounds are not sharp,
our approach provides tractable constructive estimates for the
renewal theorem which are computable (theoretically and numerically,
at least) for a general class of inter-arrival distributions. The
proof uses a coupling, and relies on Lyapunov\tire Doeblin type arguments
for some discrete time regenerative structure, which we associate with the
renewal processes.

Keywords: renewal theorem, spread-out inter-arrivals, convergence rate, Lyapunov, coupling


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