Past exam of the mathematics course of the University of Cambridge / 2018 / ia / Paper 2 / 12F / i

Table of contents
- Solution i

Solution

For

x \geq m

,

X_{n} = x

implies

M_{n} \geq m

. For

x < m

, reflect every step after the first visit to

m

. The reflection principle for simple symmetric random walk bijects such paths ending at

x

with unrestricted paths ending at

2 m - x

. Therefore

P (M_{n} \geq m, X_{n} = x) = {P (X_{n} = x), P (X_{n} = 2 m - x), x \geq m, x < m .

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