To find the generating function for $T_a$ for $a > 1$, let's break this down step-by-step, using concepts from probability and generating functions.

Problem Breakdown

• $S_n$ is some stochastic process (possibly a random walk).
• $T_a = \min \{n \geq 1 : S_n = a\}$ is the first time the process reaches $a$, where $a \geq 1$.
• $G_1(t)$ is the generating function for $T_1$, i.e.,

$G_1(t) = \mathbb{E}(t^{T_1}) = \sum_{n=1}^{\infty} \mathbb{P}(T_1 = n) t^n.$

We are tasked with finding the generating function for $T_a$ for $a > 1$.

Step 1: Recurrence Relations for $T_a$

The idea is that the time $T_a$ (the first time the process reaches $a$) can be related to hitting 1 first and then hitting $a$. This suggests the following recurrence relationship:

$T_a = T_1 + T_{a-1} \circ T_1,$

where $T_1$ is the time to reach $1$, and $T_{a-1} \circ T_1$ represents the time to reach $a-1$ after reaching $1$. This is based on the Markov property, where once you hit 1, the process "resets" and starts anew.

Step 2: Generating Function Relation

We can now express the generating function $G_a(t)$ for $T_a$ in terms of $G_1(t)$. Using the fact that the generating function of a sum of independent random variables is the product of their generating functions, we get the recurrence:

$G_a(t) = G_1(t) G_{a-1}(t).$

This indicates that the generating function for $T_a$ is the product of generating functions for the previous steps.

Step 3: Solving the Recurrence

We can solve this recurrence relation for $G_a(t)$. Starting from $G_1(t)$, we have:

$G_2(t) = G_1(t) G_1(t) = G_1(t)^2,$ $G_3(t) = G_1(t) G_2(t) = G_1(t)^3,$ and in general:

$G_a(t) = G_1(t)^a.$

Thus, the generating function for $T_a$ is given by:

$G_a(t) = G_1(t)^a.$

Final Result

The generating function for $T_a$ for $a > 1$ is:

$G_a(t) = G_1(t)^a.$

This expresses $G_a(t)$ in terms of $G_1(t)$, the generating function for $T_1$.

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Would you like further clarifications on any step or detailed examples of generating functions? Here are some questions that might expand your understanding:

1. How does the generating function $G(t)$ help in understanding the distribution of $T_a$?
2. Can this approach be generalized for processes with different boundary conditions or more complicated states?
3. What is the interpretation of $G_1(t)$ in terms of probability distributions for $T_1$?
4. How would you apply this concept to a random walk problem with drift?
5. What is the relationship between recurrence relations and generating functions in this context?

Tip: Generating functions are powerful tools not only for finding sums but also for solving recurrence relations, as they transform complex recursive problems into algebraic ones.