Extensions to the Renewal TheoremAuthor(s): D. McDonaldSource: Advances in Applied Probability, Vol. 8, No. 2 (Jun., 1976), p. 245Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1425894 .

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University of Maryland, 9-13 June 1975 University of Maryland, 9-13 June 1975

II G. Markov and renewal processes

Extensions to the renewal theorem

D. McDONALD, Universite de Montreal

I propose to discuss the application of space-time harmonic analysis to the renewal theorem and regenerative processes.

The first part of these results (to appear in the Annales de l'Institut Henri

Poincare) gives: If Ti, T2 are i.i.d. with distribution F such that F(0)= 0, and such that some

convolution F*" has a part absolutely continuous w.r.t. Lebesgue measure and such that foxdF(x) = j < oo then

lim |I Y, - e | = 0

where Y, is the excess (waiting) time at t; e is the probability measure

e(A) = (1//I)fA (1 - F(s))ds for any Borel set A, and I[ || is the total variation. Corollaries are Feller's renewal theorem, the weak convergence of Y, to e when we do not have the convolution condition and Blackwell's theorem.

The second part treats regenerative processes without resorting to the Key Renewal theorem by dealing directly with the spent waiting time.

The third part extends these results to the non-stationary situation. The main result (in the discrete case for simplicity) is: If V, (t E I a positive integer) is a

non-stationary regenerative process; that is V, renews itself after cycles of length T1, T2, ? ? which are independent non-identical, strictly positive, integer-valued random variables with distributions F (x) > G (x)Vm, x such that foxdG(x) <

oo, and if the tail field of {S}, n=I (Sn = E=l Ti) is trivial (in the product space (I)') then

lim Pr V, E A }-lim mean time in state A during cycle m P,(m)j =0 t--*Oc - , -- =I mean length of cycle m '

where P,(m)= Pr{Sm_i t < S, . This yields asymptotic results about converg- ing regenerative processes and the non-stationary renewal theorem.

On the limiting distribution of generalized semi-Markov processes

R. SCHASSBERGER, University of Calgary

For the M/G/1 queue with finite capacity N, i.e. N customers at most can be held by the system, the limiting distribution of the queue length is known

II G. Markov and renewal processes

Extensions to the renewal theorem

D. McDONALD, Universite de Montreal

I propose to discuss the application of space-time harmonic analysis to the renewal theorem and regenerative processes.

The first part of these results (to appear in the Annales de l'Institut Henri

Poincare) gives: If Ti, T2 are i.i.d. with distribution F such that F(0)= 0, and such that some

convolution F*" has a part absolutely continuous w.r.t. Lebesgue measure and such that foxdF(x) = j < oo then

lim |I Y, - e | = 0

where Y, is the excess (waiting) time at t; e is the probability measure

e(A) = (1//I)fA (1 - F(s))ds for any Borel set A, and I[ || is the total variation. Corollaries are Feller's renewal theorem, the weak convergence of Y, to e when we do not have the convolution condition and Blackwell's theorem.

The second part treats regenerative processes without resorting to the Key Renewal theorem by dealing directly with the spent waiting time.

The third part extends these results to the non-stationary situation. The main result (in the discrete case for simplicity) is: If V, (t E I a positive integer) is a

non-stationary regenerative process; that is V, renews itself after cycles of length T1, T2, ? ? which are independent non-identical, strictly positive, integer-valued random variables with distributions F (x) > G (x)Vm, x such that foxdG(x) <

oo, and if the tail field of {S}, n=I (Sn = E=l Ti) is trivial (in the product space (I)') then

lim Pr V, E A }-lim mean time in state A during cycle m P,(m)j =0 t--*Oc - , -- =I mean length of cycle m '

where P,(m)= Pr{Sm_i t < S, . This yields asymptotic results about converg- ing regenerative processes and the non-stationary renewal theorem.

On the limiting distribution of generalized semi-Markov processes

R. SCHASSBERGER, University of Calgary

For the M/G/1 queue with finite capacity N, i.e. N customers at most can be held by the system, the limiting distribution of the queue length is known

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