Let's go through this problem step-by-step.

Problem Overview

The question is about a simple random walk $(S_n)$ with specific properties. Each $S_n$ represents the position of a random walker at step $n$, and it's defined by $S_n = X_1 + X_2 + \dots + X_n,$ where the $X_i$ are independent and identically distributed random variables with $P(X_i = 1) = \frac{1}{2}$ and $P(X_i = -1) = \frac{1}{2}$.

Two times are defined:

1. $T_0 = \min\{n > 0 : S_n = 0\}$: The first return to zero after starting at zero.
2. $T_1 = \min\{n > 0 : S_n = 1\}$: The first time the random walk reaches 1.

The generating functions for these stopping times are defined as $G_0(t)$ and $G_1(t)$, respectively.

Part (i)

We need to use the law of total probability to show that $P(T_0 = n) = P(T_1 = n - 1) \quad \text{for } n > 1.$

Approach: The idea here is to condition on the first step $X_1$. Consider:

• If $X_1 = 1$, the random walk has to return to zero, so the first return time $T_0$ at step $n$ is related to $T_1$ at step $n-1$.

This approach leads to an understanding of the recurrence between probabilities involving $T_0$ and $T_1$.

Part (ii)

We want to show that $G_0(t) = t G_1(t).$

Approach: Using the result from part (i), we can express $G_0(t)$ in terms of $G_1(t)$ by leveraging the relationship between $P(T_0 = n)$ and $P(T_1 = n-1)$.

Part (iii)

Determine the generating function for $T_a = \min \{ n > 1 : S_n = a \}$ for any $a > 1$ and provide an explanation.

Approach: Using similar arguments as above, the generating function $G_a(t)$ will involve understanding the recurrence relations for reaching a point $a$ for the first time. This part builds on the concepts of generating functions and recurrence.

Would you like a detailed solution for each part?

Follow-up Questions

1. Can you provide a more detailed explanation of part (i) and the law of total probability application?
2. How would you construct $G_1(t)$ in terms of known probabilities or recursive relations?
3. What are some general properties of generating functions in random walks?
4. How does the concept of recurrence apply in random walks and influence the results here?
5. Can you explain the reasoning behind why $G_0(t) = t G_1(t)$ in simpler terms?

Tip

When working with generating functions, remember that they effectively "encode" probability distributions, making complex recurrences and probability computations manageable.