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The problem you're dealing with involves proving properties about a martingale sequence. Let’s break it down step by step:

1. **Martingale Properties**: You are given that \( X_1, X_2, \dots \) is a martingale. This implies two key properties:
   - \( E[X_n | X_{n-1}, \dots, X_1] = X_{n-1} \), meaning the expectation of \( X_n \), given the past, is equal to the previous value.
   - \( E[X_n] = 0 \), which indicates that the martingale has a mean of 0.

2. **First Step: Showing Variance Sum**:
   You need to show that:
   \[
   E[(X_{n+r} - X_n)^2] = \sum_{k=1}^r E[(X_{n+k} - X_{n+k-1})^2].
   \]
   This is essentially saying that the variance of the difference \( X_{n+r} - X_n \) can be broken down into a sum of incremental variances between consecutive terms.

   **Idea**: Since \( X_n \) is a martingale, the increments \( X_{n+k} - X_{n+k-1} \) form a martingale difference sequence. These increments are uncorrelated and have zero mean, which means that the variance of the sum of these increments is the sum of the variances of the individual increments. This is a well-known result for martingales.

3. **Second Step: Proving Almost Sure Convergence**:
   You are also asked to prove that \( X_n \) converges almost surely (with probability 1) under the assumption that:
   \[
   \sum_n E[(X_n - X_{n-1})^2] < \infty.
   \]
   
   **Approach**:
   - This condition is essentially saying that the total variance of the increments is finite.
   - The **Martingale Convergence Theorem** states that if \( X_n \) is a martingale and the sum of the variances of the increments is finite, then \( X_n \) converges almost surely.
   - Intuitively, the idea is that if the total variance of the increments is finite, then the sequence cannot fluctuate indefinitely and must settle down to some limiting value.

### Full Proof Outline:

1. **Variance Sum**:
   Start with the expression:
   \[
   E[(X_{n+r} - X_n)^2].
   \]
   Using the fact that \( X_n \) is a martingale, we expand the difference \( X_{n+r} - X_n \) as a sum of increments:
   \[
   X_{n+r} - X_n = \sum_{k=1}^r (X_{n+k} - X_{n+k-1}).
   \]
   Now, since the increments are uncorrelated, we can write the variance of the sum as the sum of the variances:
   \[
   E[(X_{n+r} - X_n)^2] = \sum_{k=1}^r E[(X_{n+k} - X_{n+k-1})^2].
   \]

2. **Almost Sure Convergence**:
   From the given condition \( \sum_n E[(X_n - X_{n-1})^2] < \infty \), apply the **Martingale Convergence Theorem**. This theorem tells us that if the total sum of the variances of the increments is finite, then the martingale sequence \( X_n \) converges almost surely.

Thus, under the assumption that \( \sum_n E[(X_n - X_{n-1})^2] < \infty \), you can conclude that \( X_n \) converges almost surely.

Would you like further details on any of the steps, or a more specific explanation of the martingale properties involved?

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**Relative Questions**:
1. How does the martingale property simplify calculations of conditional expectations?
2. What are the key assumptions needed for the Martingale Convergence Theorem?
3. Can you explain the concept of uncorrelated increments in the context of martingales?
4. How does the variance sum for martingales differ from other stochastic processes?
5. Why does finiteness of the total variance lead to almost sure convergence in martingales?

**Tip**: The Martingale Convergence Theorem is a powerful result. Whenever you're working with martingales, check whether the variance of the increments is finite—it’s a strong indicator of convergence.

Suppose that X1, X2,... is a martingale satisfying E[X1] = 0 and E[X1^2] < infinity. Show that E[(Xn+r - Xn)^2] = sum(k=1 to r) E[(Xn+k - Xn+k-1)^2] (the variance of the sum is the sum of the variances). Assume that sum E[(Xn - Xn-1)^2] < infinity and prove that Xn converges with probability 1.

Explore the proof that the variance of a martingale sum is the sum of the variances, and how this leads to almost sure convergence. This advanced probability problem involves martingales, variance calculations, and the Martingale Convergence Theorem.

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