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

For $0<b<\lambda =2$, integrate the identity $e^{bz}-1={\int }_0^{z}b{e}^{bt}dt$ for $z\ge 0$. The Tonelli theorem and the tail bound in (c) giveFor $b=0$ the expectation is $1$, and for $b<0$ the fact $Z\ge 0$ gives $0<e^{bZ}\le 1$. ThereforeThis 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.