Question

For \( \alpha>0 \), let \[ \{ X^{(\alpha)}_n \}_{n \geq 1} \text{ be a sequence of independent random variables such that} \quad P(X^{(\alpha)}_n = 1) = \frac{1}{n^{2\alpha}} = 1 - P(X^{(\alpha)}_n = 0). \] Let \( S = \{ \alpha>0 : X^{(\alpha)}_n \text{ converges to } 0 \text{ almost surely as } n \to \infty \}. \) \text{Then the infimum of \( S \) equals} _________ \text{ (round off to 2 decimal places).}

Hint:

For sequences of independent random variables, the infimum for convergence is determined by examining the sum of the probabilities and using convergence criteria.

Answer 1

Correct Answer: 0.5

Given that \( P(X^{(\alpha)}_n = 1) = \frac{1}{n^{2\alpha}} \), the condition for convergence to 0 almost surely is determined by the behavior of the series. Using the criteria for almost sure convergence of independent random variables, we conclude that: \[ \inf S = 0.50. \] Thus, the value is \( 0.50 \).