Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) holding times that have finite expectation. A renewal-reward process additionally has a random sequence of rewards incurred at each holding time, which are IID but need not be independent of the holding times. A renewal process has asymptotic properties analogous to the strong law of large numbers and central limit theorem. The renewal function m ( t ) {\displaystyle m(t)} (expected number of arrivals) and reward function g ( t ) {\displaystyle g(t)} (expected reward value) are of key importance in renewal theory. The renewal function satisfies a recursive integral equation, the renewal equation. The key renewal equation gives the limiting value of the convolution of m ′ ( t ) {\displaystyle m'(t)} with a suitable non-negative function. The superposition of renewal processes can be studied as a special case of Markov renewal processes. Applications include calculating the best strategy for replacing worn-out machinery in a factory; comparing the long-term benefits of different insurance policies; and modelling the transmission of infectious disease, where "One of the most widely adopted means of inference of the reproduction number is via the renewal equation". The inspection paradox relates to the fact that observing a renewal interval at time t gives an interval with average value larger than that of an average renewal interval.
Renewal processes
Introduction The renewal process is a generalization of the Poisson process. In essence, the Poisson process is a continuous-time Markov process on the positive integers (usually starting at zero) which has independent exponentially distributed holding times at each integer i {\displaystyle i} before advancing to the next integer, i + 1 {\displaystyle i+1} . In a renewal process, the holding times need not have an exponential distribution; rather, the holding times may have any distribution on the positive numbers, so long as the holding times are independent and identically distributed (IID) and have finite mean.
Formal definition
Let ( S i ) i ≥ 1 {\displaystyle (S_{i})_{i\geq 1}} be a sequence of positive independent identically distributed random variables with finite expected value
0 < E [ S i ] < ∞ . {\displaystyle 0<\operatorname {E} [S_{i}]<\infty .}
We refer to the random variable S i {\displaystyle S_{i}} as the " i {\displaystyle i} -th holding time". Define for each n > 0 :
J n = ∑ i = 1 n S i , {\displaystyle J_{n}=\sum _{i=1}^{n}S_{i},}
each J n {\displaystyle J_{n}} is referred to as the " n {\displaystyle n} -th jump time" and the intervals [ J n , J n + 1 ] {\displaystyle [J_{n},J_{n+1}]} are called "renewal intervals". Then ( X t ) t ≥ 0 {\displaystyle (X_{t})_{t\geq 0}} is given by random variable
X t = ∑ n = 1 ∞ I { J n ≤ t } = sup { n : J n ≤ t } {\displaystyle X_{t}=\sum _{n=1}^{\infty }\operatorname {\mathbb {I} } _{\{J_{n}\leq t\}}=\sup \left\{\,n:J_{n}\leq t\,\right\}}
where I { J n ≤ t } {\displaystyle \operatorname {\mathbb {I} } _{\{J_{n}\leq t\}}} is the indicator function
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