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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.

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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!

Consider a Markov chain {X_n, n >= 0} on the state space {0, 1, ...} with transition probabilities p_ij = (e^-i * i^j) / j!, j >= 0, i >= 0, and p_00 = 1. Show {X_n} is a martingale.

Learn how to prove that a Markov chain with Poisson-distributed transition probabilities is a martingale. This problem covers advanced probability concepts, including Markov chains and the martingale property. Suitable for university-level students.

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