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Conditionally on $Y_1,\dots ,Y_m$, the cumulative distribution function of $X^{\ast }$ is the weighted empirical distributionFor fixed $t$, the strong law of large numbers gives, almost surely,whileThe ratio therefore converges almost surely to $F(t)$. This is precisely the claimed convergence in distribution of the self-normalized importance sampling resample.

Assume first that $X_n\stackrel{d}{\to }X$. Because all variables take values in the compact interval $[-C,C]$, each monomial $x^k$ agrees there with a bounded continuous function on $\mathbb{R}$. The bounded moment criterion for weak convergence in this case begins withfor every natural number $k$.

Conversely, suppose all moments converge. Let $f$ be bounded and continuous. By the Weierstrass approximation theorem, for every $\epsilon >0$ there is a polynomial $P$ withMoment convergence implies $\mathbb{E}[P(X_n)]\to \mathbb{E}[P(X)]$, whileTaking the limit superior and then $\epsilon \downarrow 0$ proves $\mathbb{E}[f(X_n)]\to \mathbb{E}[f(X)]$, which is convergence in distribution.

The sequence $X_n$ converges in distribution to $X$ when $\mathbb{E}f(X_n)\to \mathbb{E}f(X)$ for every bounded continuous function $f$.

In a uniform balls-in-bins allocation with $m\to \infty$ and $n/m\to \lambda \in (0,\infty )$, a fixed bin count has convergence in distribution to a Poisson distribution variable of mean $\lambda$ by the Poisson limit theorem. If $N_j$ counts bins containing exactly $j$ balls, symmetry and the linearity of expectation give $\mathbb{E}[N_j]/m=\mathbb{P}(B_1=j)\to e^{-\lambda }{\lambda }^j/j!$ for each fixed $j$. In particular the expected empty fraction tends to $e^{-\lambda }$, and the expected fraction with at least two balls tends to $1-(1+\lambda )e^{-\lambda }$. These are statements about expectations; convergence of the random fractions requires an additional argument.

For a random walk with centered independent identically distributed increments of variance one, the maximum of its polygonal interpolation is the maximum at its grid vertices. The maximum functional is Lipschitz for the supremum norm, so the continuous mapping theorem applied to the Donsker invariance principle proves the displayed convergence in distribution. The Brownian reflection principle then gives upper-tail limits $2(1-\Phi (a))$ at all $a>0$, since the limiting maximum has no atom there.

For convergence in distribution of random elements in a separable complete metric space, one can construct copies of the sequence and its limit on one probability space which converge almost surely. The copies preserve the individual laws; they need not preserve the original dependence between sequence elements. Suitable more general versions permit a separably supported limit in a metric space.

If independent and identically distributed random variables have characteristic function $\phi (t)=1+iat+o(|t|)$ at zero, then their sample mean has convergence in distribution to $a$: its characteristic function is $\phi (t/n{)}^n\to e^{iat}$. This is also convergence in probability by convergence in distribution to a constant implies convergence in probability.