Convergence of probability measures

If $\mathcal{S}$ is a metric space, then it is well known that $C(\mathcal{S})$, the collection of continuous functions with the metric

is a metric space. Show that if $\mathcal{S}$ is compact, then $C(\mathcal{S})$ is separable.

Suppose $X_{n}$ converges weakly to $X$ and the random variables $Z_{n}$ are such that $d\left(X_{n}, Z_{n}\right)$ converges to 0 in probability. Prove that $Z_{n}$ converges weakly to $X$. This is known as Slutsky's theorem.

Hint: Start with $\mathbb {P}\left(Z_{n} \in F\right) \leq \mathbb {P}\left(X_{n} \in \overline{F_{\delta }}\right)+\mathbb {P}\left(d\left(X_{n}, Z_{n}\right) \geq \delta \right)$.

Suppose $X_{n}$ take values in a normed linear space and converge weakly to $X$. Suppose $c_{n}$ are scalars converging to $c$. Show $c_{n} X_{n}$ converges weakly to $c X$.

Give an example of a sequence $\mathbb {P}_{n}$ converging weakly to $\mathbb {P}$ and a function $f$ that is continuous but not bounded such that $\int f d \mathbb {P}_{n}$ does not converge to $\int f d \mathbb {P}$.

Give an example of a sequence $\mathbb {P}_{n}$ converging weakly to $\mathbb {P}$ and a function $f$ that is bounded but not continuous such that $\int f d \mathbb {P}_{n}$ does not converge to $\int f d \mathbb {P}$.

Show that if $X_{n}$ converges weakly to $X$ and $Y_{n}$ converges in probability to 0 , then $X_{n} Y_{n}$ converges in probability to 0 .

This exercise considers a sequence of probability measures that have densities. Suppose $\mathcal{S}$ is furnished with the Borel $\sigma$-field and $\mu$ is a measure on $\mathcal{S}$. Suppose that $f_{n}: \mathcal{S} \rightarrow [0, \infty )$ and $f: \mathcal{S} \rightarrow [0, \infty )$ are measurable functions, each of whose integral over $\mathcal{S}$ is one, and define $\mathbb {P}_{n}(A)=\int_{A} f_{n}(x) \mu (d x)$ for each $n$ and $\mathbb {P}\left(A\right)=\int_{A} f(x) \mu (d x)$.

Give an example of continuous processes $X_{n}$ and $X$ such that all the finite-dimensional distributions of $X_{n}$ converge weakly to the corresponding finite-dimensional distributions of $X$, but where $X_{n}$ does not converge weakly to $X$ with respect to the topology of $C[0,1]$.

Suppose $X$ is a random variable taking values in a complete separable metric space. If $\varepsilon >0$, show there exists a compact set $K$ such that $\mathbb {P}\left(X \notin K\right)<\varepsilon$.

Hint: For each $n$ choose closed balls $\left\{ B_{n i}, i=1, \ldots , N_{n}\right\}$ such that

Then $K=\cap_{n=1}^{\infty } \cup_{i=1}^{N_{n}} B_{n i}$ is totally bounded, hence compact.

Suppose $X_{n}$ converges weakly to $X$ and the metric space $\mathcal{S}$ is complete and separable. Prove that the sequence $\left\{ X_{n}\right\}$ is tight.

Let $\mathcal{L}$ be the collection of continuous functions on $\mathcal{S}$ such that

1. $\sup_{x \in \mathcal{S}}|f(x)| \leq 1$.

2. $|f(x)-f(y)| \leq d(x, y)$ for all $x, y \in \mathcal{S}$.

Define

Show that $d_{\mathcal{L}}$ is a metric on the collection of probability measures on the Borel $\sigma$-field of $\mathcal{S}$. Prove that a sequence of probability measures $\mathbb {P}_{n}$ converges weakly to $\mathbb {P}$ if and only if $d_{\mathcal{L}}\left(\mathbb {P}_{n}, \mathbb {P}\right) \rightarrow 0$.

Suppose $\mathcal{S}$ is a separable metric space. Show that $\mathcal{M}$ is separable.