Renewal process

A renewal process is a special stochastic process that is studied in renewal theory. It is a counting process whose interarrival times are independent, identically distributed, non-negative random variables .

Origin of the term

The term renewal has its origin in industrial applications of probability theory. System components (e.g. machines, tools, lighting fixtures) typically have lifetimes that have the character of non-negative random variables. If such components fail, they must be replaced ( renewed ) by components of the same type in order to ensure the functioning of the system.

Definitions

${\displaystyle X_{i},i=1,2,\dotsc }$be the inter-arrival times, e.g. B. the lifetimes of components. These random variables are assumed to be independently and identically distributed . In addition, they are almost certainly positive with an expected value . ${\displaystyle X_{i}}$ ${\displaystyle 0<E(X_{i})<\infty }$

${\displaystyle (X_{i})_{i\in \mathbb {N} }}$is called the renewal sequence.

Their joint distribution function will be using designated, that is, it is . If they have a probability density , this is denoted by. ${\displaystyle F}$ ${\displaystyle F(t)=P(X_{i}\leq t)}$ ${\displaystyle X_{i}}$ ${\displaystyle f}$

Next is the time of the -th renewal, that is ${\displaystyle S_{n}}$ ${\displaystyle n}$

${\displaystyle S_{n}=\sum _{i=1}^{n}X_{i},\qquad S_{0}\equiv 0.}$

Let the distribution of be denoted by, i.e. H. . ${\displaystyle S_{n}}$ ${\displaystyle F_{n}}$ ${\displaystyle F_{n}(t)=P(S_{n}\leq t)}$

The renewal process is now through ${\displaystyle \{N(t),t\geq 0\}}$

${\displaystyle N(t)=\sup\{n\in \mathbb {N} _{0}|\,S_{n}\leq t\}}$

defined stochastic process , that is , the number of renewals up to the point in time . ${\displaystyle N(t)}$ ${\displaystyle t}$

The equivalence of the description above and is expressed in the following fundamental relationship ${\displaystyle N(t)}$ ${\displaystyle S_{n}}$

${\displaystyle \{N(t)\geq n\}=\{S_{n}\leq t\}}$.

Both sets contain precisely those elements of the underlying probability space for which at least renewals have taken place up to the point in time . ${\displaystyle t}$ ${\displaystyle n}$

properties

${\displaystyle S_{n}}$is the sum of identically distributed, independent random variables, therefore the -fold convolution of the distribution and is calculated recursively as follows ${\displaystyle F_{n}}$ ${\displaystyle n}$ ${\displaystyle F}$

${\displaystyle F_{n}(t)=\int _{0}^{t}F_{n-1}(s)f(t-s)ds}$,

where the probability density is from above. ${\displaystyle f}$

It applies

${\displaystyle P(N(t)=n)=P(S_{n}\leq t)-P(S_{n+1}\leq t)=F_{n}(t)-F_{n+1}(t)}$

With the above notation we see that the following integral equation is fulfilled.

${\displaystyle {\begin{matrix}P(N(t)=n)&=&\int _{0}^{t}P(N(s)=n-1)f(t-s)ds\\\end{matrix}}}$

proof

We assume and replace and come in and receive ${\displaystyle P(N(t)=n)=F_{n}(t)-F_{n+1}(t)}$ ${\displaystyle F_{n}(t)=\int _{0}^{t}F_{n-1}(s)f(t-s)ds}$ ${\displaystyle F_{n-1}(t)=\int _{0}^{t}F_{n-2}(s)f(t-s)ds}$

${\displaystyle P(N(t)=n)=\int _{0}^{t}F_{n-1}(s)f(t-s)ds-\int _{0}^{t}F_{n-2}(s)f(t-s)ds}$

After summarizing the integrals and taking into account , the assertion follows. ${\displaystyle F_{n-1}(s)-F_{n-2}(s)=P(N(s)=n-1)}$

The integral equation just presented serves as the starting point for a theory of counting processes whose waiting times are not exponentially distributed . It is thus a basis for the generalization of the theory of Poisson processes .

The mean number of renewals in the time interval is called the renewal function and is denoted by. It applies ${\displaystyle (0,t)}$ ${\displaystyle m}$

${\displaystyle m(t)=E[N(t)]=\sum _{n=1}^{\infty }n\left[F_{n}(t)-F_{n+1}(t)\right]=\sum _{n=1}^{\infty }F_{n}(t)\,}$

Individual evidence

1. ↑ Geoffry R. Grimmett, David R. Stirzaker: Probability and Random Processes . Clarendon Press, Oxford 1982, ISBN 0-19-853185-0 .
2. ^ Rainer Winkelmann: Duration Dependence and Dispersion in Count Data . In: Journal of Business & Economic Statistics . 13 (4), 1995, pp. 467-474.
3. ^ Blake McShane, Moshe Adrian, Eric T. Bradlow, Peter S. Fader: Count Models Based On Weibull Interarrival Times . In: Journal of Business & Economic Statistics . 26 (3), 2008, pp. 369-378.