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  "body": "\n## Definition\n\nLet $T_1, T_2, \\ldots$ be [[independent and identically distributed|00MRN0FARV]] random variables with values in $[0, \\infty)$ and $P(T_1 = 0) \u003c 1$.\nLet $S_0 = 0$ and $S_n = T_1 + \\cdots + T_n$ for $n \\ge 1$; these are the renewal times.\nThe *renewal process* with holding times $T_i$ is the process\n$$N(t) = \\sup \\set{n \\ge 0 : S_n \\le t}, \\qquad t \\ge 0.$$\nIts *renewal function* is $m(t) = \\mathbb{E}[N(t)]$.\nRenewal theory is the study of $N$, of $m$, and of processes that start afresh at the times $S_n$.\n\n## Remarks\n\nThe condition $P(T_1 = 0) \u003c 1$ makes $S_n \\to \\infty$ almost surely, so $N(t)$ is finite for each $t$.\nExample: if the $T_i$ are [[exponential|0E99F51GVB]] with rate $\\lambda$, then $N$ is the Poisson process of rate $\\lambda$, and $m(t) = \\lambda t$.\nNon-example: if $T_1, T_2, \\ldots$ are not identically distributed, for instance $T_i = 2^{-i}$, then $N$ is not a renewal process, and here $N(t) = \\infty$ for $t \\ge 1$.\nLet $\\mu = \\mathbb{E}[T_1] \\in (0, \\infty]$ be the [[expected value|0WQNRCNY3M]] of a holding time.\nThe elementary renewal theorem says $m(t)/t \\to 1/\\mu$ as $t \\to \\infty$, with $1/\\infty = 0$; the strong law gives $N(t)/t \\to 1/\\mu$ almost surely.\n",
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