We are tasked with proving that the Markov chain $\{X_n, n \geq 0\}$ is a martingale, given the transition probabilities

$p_{ij} = \frac{e^{-i} i^j}{j!}, \quad j \geq 0, i \geq 0,$

and that $p_{00} = 1$.

Definition of a Martingale

A sequence $\{X_n\}$ is a martingale with respect to a filtration $\{\mathcal{F}_n\}$ if for all $n$, $\mathbb{E}[X_{n+1} | \mathcal{F}_n] = X_n,$ where $\mathcal{F}_n$ is the information available up to time $n$.

Approach to Solution

To show that $\{X_n\}$ is a martingale, we need to check the martingale condition: $\mathbb{E}[X_{n+1} | X_n = i] = X_n = i.$

Thus, we compute the expected value of $X_{n+1}$ given $X_n = i$: $\mathbb{E}[X_{n+1} | X_n = i] = \sum_{j=0}^{\infty} j \cdot p_{ij}.$

Given that $p_{ij} = \frac{e^{-i} i^j}{j!}$ (which corresponds to a Poisson distribution with parameter $i$), the expected value of $X_{n+1}$ given $X_n = i$ is the mean of this Poisson distribution, which is $i$.

Thus, we have: $\mathbb{E}[X_{n+1} | X_n = i] = i.$

This confirms that the Markov chain $\{X_n\}$ satisfies the martingale property.

Conclusion

Since the expected value of $X_{n+1}$ given $X_n = i$ is equal to $i$, the process $\{X_n\}$ is indeed a martingale.

Would you like more details on any part of this proof or have any questions?

Here are 5 related questions to deepen your understanding:

1. What are the general conditions for a process to be a martingale?
2. How does the Poisson distribution play a role in this Markov chain?
3. What are other examples of martingale processes in probability theory?
4. How would the analysis change if $p_{00} \neq 1$?
5. What is the significance of martingales in stochastic processes?

Tip: Understanding martingales is crucial in stochastic processes, as they are widely used in modeling fair games and financial markets!