14.15 Recall Section 14.5.3.1 on page 454 and the definition of the hitting time T given there. Let M(θ) be the moment generating function of the Xi’s, φ(θ) =

log(M(θ)), and Xθ

n = eθSn−nφ(θ)

.

a) Calculate M(θ) and φ(θ) for θ ∈ R, and show that φ(θ) ≥ 0 for all θ > 0.

b) For θ > 0, show that E )

M(θ)

−T 1

= e−θ.

c) Let γ = M(θ)−1; then let y = e−θ to obtain

γ(1 − p)y2 − y + γp = 0.

Solve the equation and deduce that for p < 1 E )

γT 1

= 1 − #1 − 4p(1 − p)γ2 2(1 − p)γ .

What is the value of E )

γT 1 when p = 1?

d) Derive from the previous question that E[T] < ∞ if and only if p > 1/2.