Show that for all sample functions except set of probability

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Reference no: EM13909460

Let {Xi; i≥1} be the inter-renewal intervals of a renewal process and assume that E [Xi] = ∞. Let b > 0 be an arbitrary number and X? i be a truncated rv defined by X? i = Xi if Xi ≤ b and X? i = b otherwise.

(a) Show that for any constant M > 0, there is a b sufficiently large so that E[X? i] ≥ M.

(b) Let {N? (t); t≥0} be the renewal counting process with inter-renewal intervals {X? i; i ≥ 1} and show that for all t > 0, N? (t) ≥ N(t).

(c) Show that for all sample functions N(t, ω), except a set of probability 0, N(t, ω)/t 2/M for all sufficiently larget. Note: Since M is arbitrary, this means that lim N(t)/t = 0 WP1.

Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.

Reference no: EM13909460

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