 Solution

ID: past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2009/a/1/b/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}) .

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.

 New to topics? Read the docs here!