Blackwell Renewal Kit

The renewal rate of Blackwell is a mathematical theorem from probability theory , more precisely from the renewal theory , a branch of the theory of stochastic processes , and provides information about the expected number of renewals within a time interval. It goes back to David Blackwell and dates from 1948.

Background and definitions

The theory of renewal is based on a sequence of independent, identically distributed random variables with values ​​in which are interpreted as periods of time, so-called renewal cycles . could be the functional duration of a -th machine part, at the end of which this machine part has to be replaced again. One is then interested in the random variables ${\displaystyle (T_{n})_{n\in \mathbb {N} }}$ ${\displaystyle T_{n}}$ ${\displaystyle [0,\infty )}$ ${\displaystyle T_{n}}$ ${\displaystyle n}$

${\displaystyle N(t)=\max\{n\in \mathbb {N} |\,T_{1}+\ldots +T_{n}\leq t\}}$,

that means for the number of renewals up to the point in time or for ${\displaystyle t}$

${\displaystyle m(t)=E(N(t))}$,

that is, for the expected value of this number. The stochastic process is called a renewal process and its mean value function . ${\displaystyle (N(t))_{t\in [0,\infty )}}$ ${\displaystyle m}$

Blackwell's renewal theorem to be discussed here makes a statement about the behavior of for , that is, about the expected number of renewals in a time interval of length for the beginning times of these intervals which tend towards infinity. ${\displaystyle m(t+\delta )-m(t)}$ ${\displaystyle t\to \infty }$ ${\displaystyle \delta }$

A special technical feature must be observed when formulating the sentence. It is called an arithmetic renewal process if the values ​​of the renewal cycles are only integer multiples of a fixed time , and the largest real number with this property is called the span of the arithmetic renewal process. Otherwise one speaks of a non-arithmetic renewal process. ${\displaystyle \lambda >0}$ ${\displaystyle \lambda }$

Formulation of the sentence

There is a renewal process , the common expected value of the associated renewal cycles. The following applies to the mean value function ${\displaystyle (N(t))_{t\in [0,\infty )}}$ ${\displaystyle \mu =E(T_{n})}$ ${\displaystyle m}$

• in the non-arithmetic case for all ${\displaystyle \delta >0}$

${\displaystyle \lim _{t\to \infty }(m(t+\delta )-m(t))={\frac {\delta }{\mu }}}$

• in the arithmetic case with range ${\displaystyle \lambda >0}$

${\displaystyle \lim _{t\to \infty }(m(t+\lambda )-m(t))={\frac {\lambda }{\mu }}}$

Remarks

The sentence is intuitively clear immediately. If a renewal cycle lasts on average , then renewals are expected on average in an interval of length . If one sets in the non-arithmetic case , then the right-hand side becomes 1 and one receives the likewise very plausible statement that a renewal is to be expected on average in a time interval of average cycle duration. The remarkable thing about Blackwell's renewal theorem is therefore not the value, but the existence of the limit value. ${\displaystyle \mu }$ ${\displaystyle \delta }$ ${\displaystyle \textstyle {\frac {\delta }{\mu }}}$ ${\displaystyle \delta =\mu }$

The restriction in the arithmetic case is necessary because you always have to . Since then between and with certainty no renewal takes place, is , again and again for increasing , that is, the limit inferior of this difference for is 0. Therefore, the limit value statement made in the non-arithmetic case cannot apply here. The limit value statement in the arithmetic case can still be ${\displaystyle \delta <\lambda }$ ${\displaystyle [t,t+\delta ]\subset \mathbb {R} \setminus \lambda \cdot \mathbb {Z} }$ ${\displaystyle t}$ ${\displaystyle t+\delta }$ ${\displaystyle m(t+\delta )-m(t)=0}$ ${\displaystyle t}$ ${\displaystyle t\to \infty }$

${\displaystyle \lim _{t\to \infty }(m(t+k\lambda )-m(t))={\frac {k\lambda }{\mu }}}$ for all ${\displaystyle k\in \mathbb {N} }$

to be generalized. This follows very easily by transitioning to the telescope sum , because then you get a -fold sum of limit values ​​with Limes . ${\displaystyle k}$ ${\displaystyle \textstyle {\frac {\lambda }{\mu }}}$

If you bring this to the left in the non-arithmetic case , you get ${\displaystyle \delta }$

${\displaystyle \lim _{t\to \infty }{\frac {m(t+\delta )-m(t)}{\delta }}={\frac {1}{\mu }}}$for everyone . ${\displaystyle \delta >0}$

Since the right side no longer depends on, the limes of the left side exists for . It is a mistake to deduce the differentiability of for large from this, because the limit value formations and cannot be interchanged. ${\displaystyle \delta }$ ${\displaystyle \delta \to 0}$ ${\displaystyle m}$ ${\displaystyle t}$ ${\displaystyle \delta \to 0}$ ${\displaystyle t\to \infty }$

Individual evidence

1. ^ D. Blackwell: A renewal theorem. In: Duke Math. Journal. Volume 15 (1948), pp. 145-150.
2. ↑ R. Serfőző: Basics of Applied Stochastic Processes. Springer Verlag, 2009, ISBN 978-3-540-89332-5 , Chapter 2.6: Blackwell's Theorem. Theorem 33.
3. ^ RG Gallager: Stochastic Processes, Theory for Applications. Cambridge University Press, 2013, ISBN 978-1-107-03975-9 , Theorem 5.6.3, without proof but with explanations.