Past exam of the probability course of the University of Oxford / 2014 / 1 / a

Convergence in distribution means that the cumulative distribution functions satisfyat every $x$ where the cumulative distribution function of $X$ is continuous. Values at its discontinuities need not converge. This definition uses only the probability distributions, so the random variables need not be defined on the same probability space.

Let $\mu =\mathbb{E}[X_1]$ and ${\sigma }^2=\mathrm{Var}(X_1)\in (0,\infty )$. For independent and identically distributed random variables, the central limit theorem statesThe limit is the standard normal distribution. If the variance is zero, then $X_i=\mu$ almost surely; the centered sums are zero, and division by $\sigma$ is not defined.

The implication convergence in probability implies convergence in distribution follows by comparing nearby events. Fix a continuity point $x$ of the cumulative distribution function $F_X$ and let $\epsilon >0$. If $X\le x-\epsilon$ and $|X_n-X|\le \epsilon$, then $X_n\le x$. Conversely, $X_n\le x$ and $|X_n-X|\le \epsilon$ imply $X\le x+\epsilon$. ThusThe error probability tends to zero by convergence in probability. Taking lower and upper limits and then letting $\epsilon \downarrow 0$, continuity at $x$ squeezes both bounds to $F_X(x)$. This is exactly convergence in distribution.