Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 201 / 3 / b

Table of contents
- Solution b

Solution

The increment variance is

Var (ξ_{k}) = 1 - (2 p - 1)^{2} = 4 p (1 - p),

so independence gives

Var (S_{n}) = 4 n p (1 - p) .

Linear interpolation makes the supremum of the absolute centered process occur at an integer time. The Doob L2 maximal inequality therefore gives

E 0 \leq t \leq 1 sup ∣ S_{t}^{(n)} - μ t ∣^{2} = \frac{1}{n ^{2}} E 0 \leq k \leq n max ∣ M_{k} ∣^{2} \leq \frac{4}{n ^{2}} E M_{n}^{2} = \frac{16 p ( 1 - p )}{n} \leq \frac{4}{n} .

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