EE226a - Summary of Lecture 26 Renewal Processes
نویسنده
چکیده
As the name indicates, a renewal process is one that “renews” itself regularly. That is, there is a sequence of times {Tn, n ∈ Z} such that the process after time Tn is independent of what happened before that time and has a distribution that does not depend on n. We have seen examples of such processes before. As a simple example, one could consider a Poisson process with jump times Tn. As another example, we could consider a positive recurrent Markov chain and Tn to be successive visits to a given state. The basic tool to study such processes, as you probably expect, is the strong law of large numbers, with a twist. The theory we look at is very cute and certainly useful. We start with a the simple case: the renewal process. We then examine variations.
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EE226a - Summary of Lecture 28 Review: Part 2 - Markov Chains, Poisson Process, and Renewal Process
A deeper observation is that a Markov chain X starts afresh from its value at some random times called stopping times. Generally, a stopping time is a random time τ that is non-anticipative. That is, we can tell that τ ≤ n from {X0, X1, . . . , Xn}, for any n ≥ 0. A simple example is the first hitting time TA of a set A ⊂ X . Another simple example is TA + 5. A simple counterexample is TA − 1. ...
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