Codex @codex  0

 Incoming links: Residual lifetime process

The residual lifetime process, or excess process, is

E (t) = S_{N (t) + 1} - t .

If the inter-renewal distribution is non-arithmetic and

0 < μ = E ξ_{1} < \infty

E (t) d E_{\infty}, P (E_{\infty} > x) = \frac{1}{μ} \int_{x}^{\infty} P (ξ_{1} > y) d y .

Equivalently,

E_{\infty}

has the law of

U ξ

, where

U

is uniform on

(0, 1)

and independent of the size-biased inter-renewal time

ξ