To demonstrate the equality $P(S_1, S_2, \ldots, S_{2n} \neq 0) = P(S_{2n} = 0)$, let's consider the properties of a simple symmetric random walk. Here, $S_n$ represents the position of the walk after $n$ steps, starting at the origin $S_0 = 0$.

Key Concepts:

1. Simple Random Walk Definition: A simple random walk $S_n$ on the integers is defined by $S_n = X_1 + X_2 + \cdots + X_n,$ where $X_i$ are independent and identically distributed random variables taking values $+1$ or $-1$ with equal probability $\frac{1}{2}$.

2. Event Descriptions:

   • $P(S_{2n} = 0)$ is the probability that the random walk returns to the origin after $2n$ steps.
   • $P(S_1, S_2, \ldots, S_{2n} \neq 0)$ represents the probability that none of the intermediate positions $S_1, S_2, \ldots, S_{2n-1}$ are equal to zero, while $S_{2n} = 0$.

Explanation:

To solve the problem, we need to establish a relationship between these two probabilities.

Step 1: Reflection Principle

The reflection principle for random walks states that for a symmetric random walk, if a walk reaches zero at any intermediate step $k$, it is equally likely that the walk could reach any point symmetric to the origin (such as the point where it started).

Step 2: Counting Paths

• Paths returning to zero after $2n$ steps without touching zero at any intermediate step: We can count the number of such paths by considering all possible paths of length $2n$ that start and end at zero, excluding paths that touch zero before reaching $2n$. For such paths, the number is given by the Catalan number $C_n = \frac{1}{n+1} \binom{2n}{n}$.
• Paths that reach zero exactly at step $2n$: The total number of such paths is given by $\binom{2n}{n}$.

Step 3: Probability Computation

The probability that the walk returns to zero at step $2n$, having not touched zero at any intermediate step, is given by the ratio: $P(S_1, S_2, \ldots, S_{2n} \neq 0) = \frac{C_n}{\binom{2n}{n}} = P(S_{2n} = 0).$

Hence, we have shown that: $P(S_1, S_2, \ldots, S_{2n} \neq 0) = P(S_{2n} = 0).$

Would you like more details on any step, or do you have any questions?

Here are five related questions to deepen your understanding:

1. What is the reflection principle in the context of random walks, and how is it used in probability?
2. How are Catalan numbers related to random walks and lattice paths?
3. Why is it necessary to take an even number of steps for the probability calculations in this context?
4. Can this result be generalized for a random walk in higher dimensions?
5. How would the probability change if the steps had unequal probabilities for $+1$ and $-1$?

Tip: In symmetric random walks, understanding how paths behave over time is essential for calculating probabilities, especially when using combinatorial methods like the reflection principle.