Pointwise properties of convergence in probability (Q1061411)

If a sequence of random variables \(X_ n\) converges to X in probability we know little about the pointwise behavior of \(X_ n(\omega)\). In this note we show that if \(X_ n\) converges to X quickly enough (for example, like \(n^{-\alpha}\) for \(\alpha >0)\) then, for almost all \(\omega\), \(X_ n(\omega)\) converges to X(\(\omega)\) outside a set of density zero.