Past exam of the martingales and financial mathematics course of the University of Oxford / 2010 / A / 3 / b / iii

Table of contents
- Solution iii

Solution

Since

X_{n} > 0

, the pointwise inequality

1_{{X_{\infty} = 0}} \leq \frac{( X _{n} - X _{\infty} ) ^{2}}{X _{n}^{2}}

holds: on that event both sides equal one, and elsewhere the left side is zero. Take conditional expectation and pull out the positive

F_{n}

-measurable denominator to obtain the requested bound. The supplied variance estimate improves it to

0 \leq P (X_{\infty} = 0 ∣ F_{n}) \leq \frac{1}{( m + n ) X _{n}} = \frac{1}{B _{n}} ⟶ 0

almost surely, by part (ii). These conditional probabilities are bounded by one. The dominated convergence theorem and tower property of conditional expectation give

P (X_{\infty} = 0) = n lim E [P (X_{\infty} = 0 ∣ F_{n})] = E [n lim P (X_{\infty} = 0 ∣ F_{n})] = 0.

This proves Positivity of the limiting Pólya urn proportion without needing its explicit beta distribution.

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