Reference no: EM13909497
The purpose of this exercise is to show that for an arbitrary renewal process, N(t), the number of renewals in (0, t] is a (non-defective) rv.
(a) Let X1, X2, ... be a sequence of IID inter-renewal rv s. Let Sn = X1 + ··· + Xn be the corresponding renewal epochs for each n ≥ 1. Assume that each Xi has a finite expectation X > 0 and, for any given t > 0, use the WLLN to show thatb limn→∞ Pr{Sn ≤ t} = 0.
(b) Use (a) to show that limn→∞ Pr{N(t) ≥ n} = 0 for each t > 0 and explain why this means that N(t) is a rv, i.e., is not defective.
(c) Now suppose that the Xi do not have a finite mean. Consider truncating each Xi to X? i, where for any given b > 0, X? i = min(Xi, b). Let N? (t) be the renewal counting process for the inter-renewal intervals X? i. Show that N? (t) is non-defective for each t > 0. Show that N(t) ≤ N? (t) and thus that N(t) is non-defective. Note: Large inter-renewal intervals create small values of N(t), and thus E [X] =∞ has nothing to do with potentially large values of N(t), so the argument here was purely technical.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.
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