Past exam of the martingales and financial mathematics course of the University of Oxford / 2009 / A / 1 / b / i

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- Solution i

Solution

Independent random variables satisfy, for every finite selection of distinct indices

n_{1}, \dots, n_{m}

C_{1}, \dots, C_{m}

,

P (⋂_{j = 1}^{m} {X_{n_{j}} \in C_{j}}) = \prod_{j = 1}^{m} P (X_{n_{j}} \in C_{j}) .

The Second Borel-Cantelli lemma says that if events

E_{n}

are independent and

\sum_{n} P (E_{n}) = \infty

, then

P (n lim sup E_{n}) = 1.

Here

lim sup_{n} E_{n} = ⋂_{N} ⋃_{n \geq N} E_{n}

is the event that infinitely many of the

E_{n}

occur.

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