Blackwell renewal theorem

Consider a piece of equipment that has a finite but random life-time. Suppose one starts with a new one and, after that fails, replaces it with a second new one and, after that one fails, replaces it with a third new one and so on indefinitely. Such a process is called a renewal process (cf. also Renewal theory) and objects of interest are the behaviour for large (time) :

i) of the average number of units replaced in the interval , where is fixed;

ii) of the probability of renewal at time ;

iii) of the age, the remaining life and the total life of the (current) unit in operation at time . The mathematical theory of such processes is called renewal theory and Blackwell's renewal theorem plays a central role in it. See [a2], [a4], [a5].

Mathematical framework.

Let be a sequence of independent and identically distributed random variables that are non-negative (cf. also Random variable). The value of is to be thought of as the life-length of the th unit. Let, for ,

(a1)

Thus, is the time of the th renewal, is the number of units used up to time excluding the one in operation, is the age of the unit in place at time with being its remaining life-time and its total life-time. Let

(a2)

where denotes the expected value, or mathematical expectation, of the random variable .

Blackwell's renewal theorem says that for fixed , converges as to . A precise statement is given below. This was proved by D. Blackwell; see [a2]. This result has several variants and consequences and applications in applied probability theory; see [a1].

Let be the distribution function of the random variable (cf. also Distribution function); it is assumed that . The function is said to be arithmetic if there exist and such that

i.e. is a non-negative integer-valued random variable. The largest for which this holds is called the span; see [a5]. is said to be non-arithmetic if it is not arithmetic.

Blackwell's renewal theorem.

Arithmetic case.

Let be arithmetic with and span . Let, for ,

(a3)

i.e. the probability that there is a renewal at time . Then

(a4)

Non-arithmetic case.

Let be non-arithmetic. Then for any ,

(a5)

The arithmetic case was proved by P. Erdös, W. Feller and H. Pollard; see [a4]. The non-arithmetic case was proved by Blackwell; see [a5]. More recently, proofs of these using the coupling method have become available (see [a3]).

Key renewal theorem.

There is an equivalent of this result, known as the key renewal theorem. It is as follows. Since , its expected value , where is the fold convolution of , defined by

\[ F^k(t) \equiv \int_{(0,t]} F^{(k-1)}(t-u) \, \rd F(u) \, , \quad k \ge 1 \, . \] A discrete renewal equation, with probability distribution and forcing sequence is an equation for a sequence that satisfies

(a6)

where it is assumed that , and .

A renewal equation with distribution and forcing function is an equation for a function where both and are functions that are Borel measurable and bounded on finite intervals and satisfy

(a7)

It can be shown that (a6) and (a7) have unique solutions given by, respectively,

(a8)

and

(a9)

where is as in (a3) and is as in (a2).

A function is called directly Riemann integrable if for every ,

and as and and approach the same limit, where and are, respectively, the supremum and infimum of . This common limit is usually denoted by ; see [a5]. A very useful equivalent of the Blackwell renewal theorem is the key renewal theorem.

Arithmetic case.

Let satisfy and . Then the unique solution to (a6) is given by (a8) and

Non-arithmetic case.

Let be non-arithmetic and let be directly Riemann integrable. Then the unique solution to (a7) is given by (a9) and

Application.

A random or stochastic process in discrete or continuous time is called regenerative if there exist a sequence of random times such that , , are stochastically independent, identically distributed and independent of . Let be a measurable function on the state space of and . Then satisfies the renewal equation (a7) with and . So, if the distribution of is non-arithmetic, then from the key renewal theorem one can conclude that

This, in turn, can be used to prove the convergence to an equilibrium distribution for many regenerative processes, including positive recurrent irreducible Markov chains; see [a1].

References

[a1]	S. Asmussen, "Applied probability and queues", Wiley (1987)
[a2]	D. Blackwell, "A renewal theorem" Duke Math. J., 15 (1948) pp. 145–150
[a3]	T. Lindvall, "Lectures on the coupling method", II, Wiley (1992) (Edition: Second)
[a4]	W. Feller, "An introduction to probability theory and its applications", 1, Wiley (1968) (Edition: Third)
[a5]	W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1970) (Edition: Second)