Math Problem Statement

If, for a ≥ 1, Ta = min{n ≥ 1 : Sn = a} and G1(t) is the generat- ing function for T1, find the generating function for Ta for a > 1

Solution

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Math Problem Analysis

Mathematical Concepts

Generating Functions
Stochastic Processes
Random Walks
Markov Property

Formulas

G_1(t) = \mathbb{E}(t^{T_1}) = \sum_{n=1}^{\infty} \mathbb{P}(T_1 = n) t^n
G_a(t) = G_1(t) G_{a-1}(t)
G_a(t) = G_1(t)^a

Theorems

Markov Property
Recurrence Relation for Random Walks

Suitable Grade Level

Undergraduate Mathematics