Approximations for Geometric Sums and Perturbed Renewal Equations

Jose Blanchet
Statistics Department
Harvard University


Wednesday, January 19, 2005
4:45 - 5:45 PM
Terman Engineering Center, Room 453


Abstract:

Consider a sequence X(1), X(2),... of independent and identically distributed random variables (iid rv's), and a geometric rv M with parameter p, independent of the X(i)'s. The rv S = X(1) + ... + X(M) is called a geometric sum. In this talk, we shall present asymptotic expansions for the distribution of S as p --> 0. When EX > 0, the expansion is given in powers of p, where as if EX = 0 the expansion is given in powers of p^(1/2). In addition, expansions for defective perturbed renewal equations that are close to proper are also discussed. We apply the results to obtain corrected diffusion approximations for the M/G/1 model and a perturbed risk process model introduced by Dufresne and Gerber (1991). This is joint work with Peter Glynn.

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