Tightness of random variales

Question

If $\{X_n\}$ is a tight family of positive r.v.s. can we say something about $\{f(X_n)\}$ where $f$ is a continuous function?

Answer 1

Write for a positive $R$, $$\mathbb P(|f(X_n)|\gt R)=\mathbb P(|f(X_n)|\gt R,|X_n|\leqslant M)+\mathbb P(|X_n|\gt M).$$ For a fixed $\varepsilon$, take $M$ such that $\mathbb P(|X_n|\gt M)\lt\varepsilon$. Once this $M$ is chosen, notice that the function $f$ is bounded on $[-M,M]$, hence for $R$ greater than $\sup_{t\in[-M,M]}|f(t)|$, we obtain $$\mathbb P(|f(X_n)|\gt R)\lt\varepsilon.$$