Relation between convergence in distribution and weak convergence
If $(X_n: n\in \mathbb{N}), X$ are a sequence of random variables in $\mathbb{R}$, I wish to show that $X_n \to X$ weakly if and only if $X_n \to X$ in distribution. By 'converging weakly' I mean that $\mu_{X_n}(f) \to \mu_X(f)$ for all continuous bounded functions $f$.
Secondly I want to show that if $(\mu_n: n\in \mathbb{N}), \mu$ are Borel probability measures on $\mathbb{R}^d$, and $\mu_n(f)\to\mu(f)$ for all $C^\infty$ functions on $\mathbb{R}^d$ of compact support, then $\mu_n$ converges weakly to $\mu$ on $\mathbb{R}^d$.
Any help with these questions would be really appreciated. Thanks!
$\endgroup$ 41 Answer
$\begingroup$Assume $X_{n}$ converges weakly to $X$. Let $a,b$ be points of continuity of the CDF for $X$. Define continuous functions $f_{\delta}$ and $g_{\delta}$ so that $f_{\delta}$ is 1 on $[a+\delta,b-\delta]$ and tapers linearly to $0$ on $\mathbb{R}\setminus[a,b]$, and so that $g_{\delta}$ is $1$ on $[a,b]$ and tapers linearly to $0$ on $\mathbb{R}\setminus[a-\delta,b+\delta]$. By the continuity of the distribution of $X$ at $a$ and $b$, for each $\epsilon > 0$, there exists $\delta > 0$ such that $$ \mu_{X}[a,b]-\epsilon < \mu_{X}(f_{\delta}) \le \mu_{X}(g_{\delta}) < \mu_{X}[a,b]+\epsilon. $$ You should be able to take it from there.
For the opposite direction, use behavior distributional behavior of $\mu_{X}$ at $\pm\infty$ and distributional convergence to reduce the problem to a finite interval where you can uniformly approximate a continuous function $f$ by a linear combination of characteristic functions $\chi_{[a,b)}$.
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