 Solution

ID: past-exam-of-the-stochastic-differential-equations-and-brownian-motion-in-complex-analysis-course-of-the-university-of-oxford/2009/a/1/c/solution

Use the valid Gaussian maximum estimate from part (b), with

a_{n} = 2 (n + 1) lo g 2 > 1.

a_{n}

. Bounding the first part by

a_{n}

and applying the Gaussian tail estimate to the second gives

E M_{n} \leq a_{n} + C_{n} \int_{a_{n}}^{\infty} e^{- x^{2} /2} d x \leq a_{n} + \frac{C _{n}}{a _{n}} e^{- a_{n}^{2} /2} = a_{n} + \frac{1}{2 π a _{n}} .

Thus

E M_{n} = O (n)

. The finitely many small indices also have finite expectation; the bound above already covers them.

By Tonelli's theorem, nonnegativity permits interchanging sum and expectation:

E [n \geq 1 \sum 2^{- n /2} M_{n}] = n \geq 1 \sum 2^{- n /2} E M_{n} < \infty.

The final series converges because its exponential decay dominates the factor

n

. This establishes the requested conclusion independently of the false bound printed in (1).

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