Page 319 - A First Course In Stochastic Models

314 ADVANCED RENEWAL THEORY

Proof The proof will be based on the relation (8.1.2) for the excess variable. By
the relation (8.1.2), we have µ 1 [1 + M(t)] − t ≥ 0 and so we obtain the inequality

M(t) 1 1
≥ − for all t > 0. (8.2.2)
t µ 1 t
Next we prove that for any constant c > 0,
M(t) 1 1 c
≤ + − 1 for all t > 0. (8.2.3)
t µ(c) t µ(c)
c
where µ(c) = [1 − F(x)] dx. To prove this inequality, fix c > 0 and consider
0
the renewal process {N(t)} associated with the sequence {X n }, where

X n if X n ≤ c,
X n =
c if X n > c.
Since N(t) ≤ N(t), we have M(t) ≤ M(t) for all t ≥ 0. For the renewal process
∞

{N(t)}, the excess life γ satisfies γ ≤ c for all t. Since E(X 1 ) = P {X 1 >
t t 0
x} dx, we have
c
E(X 1 ) = {1 − F(x)} dx = µ(c).
0
Thus, by (8.1.2),

µ(c) M(t) + 1 − t ≤ c, t ≥ 0.
This inequality in conjunction with M(t) ≤ M(t) yields the inequality (8.2.3). The
remainder of the proof is simple. Letting t → ∞ in (8.2.2) and (8.2.3) gives

1 M(t) M(t) 1
≥ lim sup ≥ lim inf ≥
µ(c) t→∞ t t→∞ t µ 1
for any constant c > 0. Next, by letting c → ∞ and noting that µ(c) → µ 1 as
c → ∞, we obtain the desired result.

So far our results have not required any assumption about the distribution func-
tion F(x) of the interoccurrence times. However, in order to characterize the
asymptotic behaviour of the solution to the renewal equation it is required that
the distribution function F(x) is non-arithmetic. The distribution function F is
called non-arithmetic if the mass of F is not concentrated on a discrete set of
points 0, λ, 2λ, . . . for some λ > 0. A distribution function that has a positive
density on some interval is non-arithmetic. In the discussion below we make for
convenience the even stronger assumption that F(x) has a probability density. To
establish the limiting behaviour of the solution to the renewal equation (8.1.5), we
need also to impose on the function a(x) a stronger condition than integrability. It