2.6. Terminating Renewal Processes

2.6. Terminating Renewal Processes

Up to this point, the restrictions have been:

• the absence of a unit jump of the distribution function F(t) at zero, required for the existence of the renewal function;
• the existence of moments, required for the validity of the asymptotic expansion of this function.

Under these conditions the renewal process develops over time: in a finite time a finite number of renewals occur (there is no infinite accumulation), while over an infinite time an infinite number of renewals occur, that is, as t→∞, with probability one ξ(t)→∞.

Such a situation occurs when the distribution defining the renewal process is proper, that is, P{ξ<∞}=lim_t→∞F(t)=F(∞)=1 (a condition necessary for the existence of moments).

A different picture arises when the distribution F(t) is improper, F(∞)<1, P{ξ=∞}=1-F(∞)>0. Then, with positive probability 1-F(∞)>0, the renewal process may terminate at some step, that is, the time until the next renewal will be equal to infinity.

A renewal process whose distribution of the intervals between successive renewal epochs is improper is called a terminating renewal process.

Before formulating the theorem on the limiting behavior of a terminating renewal process, we prove a lemma on the limiting behavior of convolution integrals.

LEMMA 2.2. If the functions A(x) and B(x) are positive, nondecreasing and uniformly bounded for x>0, then

PROOF. By virtue of the conditions of the theorem, for any ε₁>0 there exists t₁(ε₁)>0 such that for t>t₁(ε₁)

.

For any ε₂>0 there exists t₂(ε₂)>0 such that for t>t₂(ε₂) and

.

Then for t>max[t₁(ε₁), t₂(ε₂)] we obtain the estimate

which proves the assertion of the lemma. *

THEOREM 2.2. For a terminating renewal process starting at time t=0, the following statements hold:

1. the process terminates with probability one, or, with probability one, over an infinite time only a finite number of renewals occur, that is

; (2.24)

1. the renewal function is bounded on the extended half-line [0, ∞] and

(2.25)

1. the termination epoch has a proper distribution

. (2.26)

PROOF. For an improper distribution, direct passage to the limit as t→∞ in the convolution integral gives F^(k)(∞)=[F(∞)]^k<1, k>0, and F⁽⁰⁾(∞)<1. This fact is easily proved by induction, using Lemma 2.2. The distribution functions satisfy the conditions of the lemma. Hence . If , then it follows from the assertion of the lemma that

Then from (2.3) we obtain

P{ξ(∞)=k}=lim_t→∞ P{ξ(t)=k}=lim_t→∞ [F^(k)(t)-F^(k+1)(t)]=(F(∞))^k[1-F(∞)] (2.27)

and hence (2.24) holds.

To prove (2.25) we make use of equality (2.4)

where the interchange of the order of summation and passage to the limit is legitimate, since the series (2.4) converges uniformly for 0<=t<∞.

Equality (2.27) shows that the number of terms ξ_m before the renewal process terminates has a geometric distribution. Then, by the total probability formula, we obtain

When computing the conditional probability , we note that the following equality of events holds

Therefore, by the independence of the random variables ξ_m+1 and t_m we have

Thus, all the statements of the theorem are proved. *

COROLLARY 2.1. If the nondecreasing function B(x) has a limit as x→∞, B(∞)=lim_x→∞B(x), and H(x) is the renewal function of a terminating renewal process, then

PROOF. The proof follows directly from equality (2.25) and the assertion of Lemma 2.2.*

See also

• Poisson's random measure
• random process
• random walks
• renewal process
• the Cramér–Lundberg model
• empirical measures
• Poisson random measure