91Ó°ÊÓ

We are given a Markov chain \(\left\\{X_{n}, n \geqslant 0\right\\}\) with transition probabilities \(P_{i, j}\). Our objective is to compute the conditional probability that \(X_{n}=m\) under the condition that \(X_0 = i\) and \(X_k \neq r\) for all \(k=1,\dots,n\), where \(r\) is a specified state not equal to either \(i\) or \(m\).

We are considering a new Markov chain of the form: \[Q_{i, j}=\frac{P_{i, j}}{1-P_{i, r}}, \quad i, j \neq r\] The new transition probabilities are the probabilities of transitioning from state \(i\) to state \(j\) not including the specified state \(r\).

Consider the path of length \(n\) that goes from state \(i\) to state \(m\) without passing through state \(r\). Since the chain is in state \(i\) at time \(0\) and in state \(m\) at time \(n\), and it doesn't enter state \(r\) during the times \(1, \ldots n\), we can express the conditional probability of interest as follows: \[P\left\\{X_{n}=m | X_{0}=i, X_{k} \neq r, k=1, \ldots, n\right\\} = \frac{P\left\\{X_{n}=m, X_{k} \neq r, k=1, \ldots, n-1 |X_{0}=i\right\\}}{P\left\\{X_{k} \neq r, k=1, \ldots, n | X_{0}=i \right\\}}\] Now consider the transition probabilities from state \(i\) to states different than \(r\). We can write those probabilities as follows: \[Q_{i, j} = \frac{P_{i, j}}{1 - P_{i, r}}, \quad i, j \neq r\] By multiplying both the numerator and denominator of the conditional probability by \((1-P_{i,r})^n\), we get: \[P\left\\{X_{n}=m | X_{0}=i, X_{k} \neq r, k=1, \ldots, n\right\\} = \frac{P\left\\{X_{n}=m, X_{k} \neq r, k=1, \ldots, n-1 |X_{0}=i\right\\}(1-P_{i,r})^n}{P\left\\{X_{k} \neq r, k=1, \ldots, n | X_{0}=i \right\\}(1-P_{i,r})^n}\] In this form, the numerator and denominator of the conditional probability are expressed as products of probabilities of transitioning from state \(i\) to states different than \(r\). Hence, the numerator and denominator of the conditional probability have the same form as the definition of the new transition probabilities. Therefore, we can use the definition of the new transition probabilities to write the conditional probability in terms of the \(Q_{i, j}\)s: \[P\left\\{X_{n}=m | X_{0}=i, X_{k} \neq r, k=1, \ldots, n\right\\} = Q_{i, m}^n\] So the equality is proven: \[P\left\\{X_{n}=m | X_{0}=i, X_{k} \neq r, k=1, \ldots, n\right\\}=Q_{i, m}^n\]