Limit Theorem of Markov Chain and Applications

First let us consider few simple examples, which will motivate us about the idea why limiting theorems for Markov chains are important and their implications with respect to renewal process.

Example 1.21

In order to obtain the pricing expressions for financial instruments, whose underlying asset may be described through a simple continuous-time random walk (CTRW) market model, one may use renewal equations pertinent to the renewal process to derive the expressions.

Example 1.22

Suppose one is interested to find the software reliability and the costs are both deterministic as well as probabilistic. Then using the concepts of renewal process one can estimate the different metrics like mean error free time, number of errors remaining in the software product, etc.

Example 1.23

Consider light bulbs are being replaced consecutively one after the another by a new one after the previous one fuses, then one may be interested to find the expected number of light bulbs replaced in some stipulated time, and for that one may use the concept of renewal process.

Having discussed these simple examples let us start with full earnestness the theorems necessary to under Markov chains