This is a martingale problem related to a sequence of independent and identically distributed (i.i.d.) random variables. Let me walk you through how to show that the given process $M_n = S_n^2 - n$ is a martingale with respect to the filtration $\mathcal{F}_n$, where $S_n = X_1 + X_2 + \cdots + X_n$ and $\mathcal{F}_n$ is the filtration generated by the random variables $X_1, X_2, \dots, X_n$.

Step-by-Step Solution:

1. Given Conditions:

   • The random variables $X_1, X_2, \dots$ are i.i.d. with:
     • Mean: $\mathbb{E}[X_i] = 0$
     • Variance: $\mathrm{Var}(X_i) = 1$
   • $S_n = X_1 + X_2 + \dots + X_n$, so $S_n$ is the sum of the first $n$ variables.
   • $M_n = S_n^2 - n$ is the process we are asked to show is a martingale.

2. Definition of a Martingale: A stochastic process $\{M_n\}_{n \geq 0}$ is a martingale with respect to the filtration $\{\mathcal{F}_n\}$ if: $\mathbb{E}[M_{n+1} | \mathcal{F}_n] = M_n, \quad \forall n \geq 0$ That is, the conditional expectation of $M_{n+1}$ given $\mathcal{F}_n$ must equal $M_n$.

3. Computing $M_{n+1}$: We need to express $M_{n+1}$: $M_{n+1} = S_{n+1}^2 - (n+1)$ Since $S_{n+1} = S_n + X_{n+1}$, we expand $S_{n+1}^2$: $S_{n+1}^2 = (S_n + X_{n+1})^2 = S_n^2 + 2S_n X_{n+1} + X_{n+1}^2$ Therefore: $M_{n+1} = S_n^2 + 2S_n X_{n+1} + X_{n+1}^2 - (n+1)$ Simplifying: $M_{n+1} = (S_n^2 - n) + 2S_n X_{n+1} + (X_{n+1}^2 - 1)$

4. Taking the Conditional Expectation: Now we compute the conditional expectation $\mathbb{E}[M_{n+1} | \mathcal{F}_n]$:

   • Since $S_n$ is $\mathcal{F}_n$-measurable, we have $\mathbb{E}[S_n | \mathcal{F}_n] = S_n$.
   • Also, $X_{n+1}$ is independent of $\mathcal{F}_n$, and given $\mathbb{E}[X_{n+1}] = 0$, we have: [ \mathbb{E}[2S_n X_{n+1} | \mathcal{F}n] = 2S_n \mathbb{E}[X{n+1}] = 0 ]
   • Moreover, $\mathbb{E}[X_{n+1}^2] = 1$, so: [ \mathbb{E}[X_{n+1}^2 - 1 | \mathcal{F}n] = \mathbb{E}[X{n+1}^2] - 1 = 0 ]

   Thus: $\mathbb{E}[M_{n+1} | \mathcal{F}_n] = S_n^2 - n = M_n$

5. Conclusion: Since $\mathbb{E}[M_{n+1} | \mathcal{F}_n] = M_n$, the process $\{M_n\}_{n \geq 0}$ is indeed a martingale with respect to the filtration $\{\mathcal{F}_n\}$.

Would you like a deeper explanation on any of these steps? Let me know if you'd like to go further.

Related Questions:

1. What are the properties of a martingale in general?
2. How does the concept of a filtration relate to the definition of a martingale?
3. What are some typical applications of martingales in probability and finance?
4. How would the result change if the random variables $X_i$ had non-zero mean?
5. Can we use this martingale to apply optional stopping theorems, and under what conditions?

Tip: Remember that martingales are often used in stopping time problems, especially in gambling and finance, due to their "fair game" property, where the future expectation equals the current value.