An Introduction to Probability and Statistical Inference, 2nd Edition

7.3 Further Limit Theorems

Convergence in probability enjoys some of the familiar properties of the usual pointwise convergence. One such property is stated below in the form of a theorem whose proof is omitted.

Theorem 5

(i) For n ≥ 1, let X_n and X be r.v.'s such that $X_n\underset{n\to \infty }{\overset{p}{\to }}X$, and let g be a continuous real-valued function; that is, $g=\Re \to \Re$ continuous. Then the r.v.'s $g\left(X_n\right),\text{?}n\ge 1$, also converge in probability to g(X);that is, $g\left(X_n\right)\underset{n\to \infty }{\overset{p}{\to }}g\left(X\right)$. More ...