To find the generating function for the random time before a simple random walk reaches height $a - 1$ given that it is at height $a$ at time $n \geq 1$, we can use some classical results from the theory of random walks and generating functions.

Let's denote:

• $T_{a-1}$ as the time to reach height $a-1$ starting from height $a$.

The generating function for the time $T_{a-1}$ before reaching height $a-1$ can be determined as follows:

1. Define the generating function: Let $G_a(z) = \mathbb{E}[z^{T_{a-1}}]$ be the generating function for the time to hit $a-1$ starting from $a$.

2. Use the recurrence relation: Given the symmetry of the simple random walk, the probability of moving up or down is equal (each with probability 0.5). The recurrence relation for the generating function can be set up as: $G_a(z) = \frac{1}{2} z G_{a+1}(z) + \frac{1}{2} z G_{a-1}(z)$ for $a > a-1$.

3. Boundary condition: The boundary condition is when the random walk reaches $a-1$. At that point, the generating function $G_{a-1}(z)$ should equal 1 because the time to reach $a-1$ from $a-1$ is 0 (we're already there): $G_{a-1}(z) = 1$

4. Solve the recurrence relation: To solve the recurrence, let’s assume a solution of the form: $G_a(z) = A \left( \frac{1 - \sqrt{1 - z^2}}{z} \right)^a + B \left( \frac{1 + \sqrt{1 - z^2}}{z} \right)^a$

   Using the boundary condition $G_{a-1}(z) = 1$, we can solve for the constants $A$ and $B$. However, this form might be complex to deal with directly, so we often use symmetry and the nature of generating functions to simplify further.

5. Result: After solving the above, we arrive at: $G_a(z) = \left( \frac{1 - \sqrt{1 - z^2}}{z} \right)^a$

This formula arises due to the symmetrical nature and specific characteristics of simple random walks.

Summary:

The generating function $G_a(z)$ for the random time before the process reaches height $a-1$ starting from height $a$ is given by: $G_a(z) = \left( \frac{1 - \sqrt{1 - z^2}}{z} \right)^a$

Next Steps and Questions:

1. Do you want a detailed explanation on solving the recurrence relation?
2. Are you interested in understanding how boundary conditions affect the solution?
3. Would you like a discussion on the properties of generating functions?
4. Do you need help with applying this result to specific random walk problems?
5. Would you like to explore similar problems in random walk theory?

Tip: When working with generating functions, always ensure to check and correctly apply boundary conditions as they significantly affect the solution.