 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-33/4/d/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 33 / 4 / d / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-07

For

0 < b < λ = 2

, integrate the identity

e^{b z} - 1 = \int_{0}^{z} b e^{b t} d t

for

z \geq 0

. The Tonelli theorem and the tail bound in (c) give

E e^{b Z} = 1 + b \int_{0}^{\infty} e^{b t} P (Z > t) d t \leq 1 + b \int_{0}^{\infty} e^{- (2 - b) t} d t = \frac{2}{2 - b} .

For

b = 0

the expectation is

1

, and for

b < 0

the fact

Z \geq 0

gives

0 < e^{b Z} \leq 1

. Therefore

E e^{b Z} < \infty for every b < 2.

This is an exponential moment bound for a Gaussian random-walk maximum. The upper-tail bound proves the requested sufficient range; it alone does not establish a converse at the endpoint.

 Read the full article

 New to topics? Read the docs here!