Past exam of the martingales and financial mathematics course of the University of Oxford / 2008 / A / 2 / b

Table of contents
- i b
  - Solution i
- ii b
  - Solution ii

i

Solution

The variables are independent and identically distributed if they all have one common probability distribution and every finite selection satisfies

P (X_{n_{1}} \in B_{1}, \dots, X_{n_{k}} \in B_{k}) = \prod_{j = 1}^{k} P (X_{n_{j}} \in B_{j})

for distinct indices and arbitrary Borel sets. The condition concerns joint distributions, not merely pairwise correlations.

ii

Solution

The strong law of large numbers states that if

X_{n}

are independent and identically distributed real random variables with

E ∣ X_{1} ∣ < \infty

, then

\frac{1}{N} n = 1 \sum N X_{n} ⟶ E X_{1} almost surely .

The first absolute moment condition is part of the standard statement.

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