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.