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I am reading my lecture notes and our definition of weak convergence or random variables is:

First another definition:

A sequence $\mu_n$ of probability measures on $\mathbb R$ converges weakly to a probability measure $\mu$ if $\int f(x) d\mu_n\rightarrow\int f(x)d\mu$ for all $f\in C_b(\mathbb R)$

1) A sequence of random variables $X_n$ on $\mathbb R$ converges weakly to a real random variable $X$ if $\mathbb P_{X_n} \rightarrow \mathbb P_X$ weakly. Where $\mathbb P_X$ is the push-forward measure of $X$

Here the second definition of weakly convergence of random variables from Durrett p.83

2) A sequence of random variables $X_n$ converges weakly to $X$ if $F_n(x)=P(X_n<x)$ converges weakly.

It may be a very easy question, but I can't see why the two definitions are equivalent..

Can someone help?

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  • $\begingroup$ I think that your second definition needs some clarification. Durrett defines what is meant by weak convergence of distribution functions. A sequence of distribution functions is said to converge weakly to a limit $F$ (written $F_n\Rightarrow F$) if $F_n(y)\to F(y)$ for all $y$ that are continuity points of $F$ (see here, p. 83). $\endgroup$ – Cm7F7Bb Feb 3 '15 at 10:22

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