Remainder Term Estimates of the Renewal Function

Uniform renewal theory with applications to expansions of random geometric sums

“…Since part (i) is a special case of part (ii), this suggests that the error term in part (ii) is not the best possible result (as a function of µ F ). The proof technique is based on a Fourier analysis argument due to Carlsson (1983). Our expression for U F (·) coincides with the one provided by Carlsson (1983) and, as we shall see in the proof, our argument requires a normalization of the U F s Fourier transform by the factor µ 4 F in order to be amenable to uniform estimates over the family F .…”

“…The proof technique is based on a Fourier analysis argument due to Carlsson (1983). Our expression for U F (·) coincides with the one provided by Carlsson (1983) and, as we shall see in the proof, our argument requires a normalization of the U F s Fourier transform by the factor µ 4 F in order to be amenable to uniform estimates over the family F . Such uniform estimates are the most important part of our contribution here.…”

“…Siegmund (1979) calculated the first three terms in the power series for RWs having exponential moments, 1. We obtain a uniform version of reminder term estimates in the renewal theorem owing to Stone (1965) and Carlsson (1983) over a family of distributions having increment means arbitrarily close to 0 (Theorem 1).…”

“…The proof of part (i) is adapted from the work of Stone (1965), except at the end where we link the behavior of V F (x, h, a) to that of U F (x) as x ∞. For this latter part, we use the results of Carlsson (1983). Both Stone (1965) and Carlsson (1983) relied on the use of Fourier analysis estimates, but Carlsson applied Fourier transforms to distributions because he worked directly with U F (·) rather than with an expression such as V F (·) (which involves increments of U F (·) convolved with a smooth kernel).…”

“…For this latter part, we use the results of Carlsson (1983). Both Stone (1965) and Carlsson (1983) relied on the use of Fourier analysis estimates, but Carlsson applied Fourier transforms to distributions because he worked directly with U F (·) rather than with an expression such as V F (·) (which involves increments of U F (·) convolved with a smooth kernel). Basically, analyzing the increments of U F (·) removes the need for dealing with the Fourier transform of an indicator function which gives rise to a delta function.…”