Convergence of Probability Measures and Random Elements: Standing Notation

(i) Base. This setting extends Probability Spaces: Notation and Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity: their conventions and notation are in force.

(ii) Metric spaces and Borel measures. $(S,d)$ and $(T,\rho)$ denote metric spaces, with Borel $\sigma$-algebras $\mathcal{B}(S)$ and $\mathcal{B}(T)$. A finite Borel measure on $S$ is a finite measure on $(S,\mathcal{B}(S))$, and a Borel probability measure on $S$ is a probability measure on $(S,\mathcal{B}(S))$; likewise for $T$. For $A\subseteq S$, $\operatorname{cl}A$, $\operatorname{int}A$ and $\partial A$ are its closure, interior and boundary. For nonempty $A\subseteq S$ and $x\in S$, $\operatorname{dist}(x,A)$ is the distance from $x$ to $A$. Background: Closure and Interior in a Metric Space: Containments, the Closure is Closed, the Interior is Open, The Boundary of a Subset of a Metric Space Is Closed and Borel and The Distance to a Subset of a Metric Space: Bounds, the Lipschitz Estimate, Continuity, and Vanishing Exactly on the Closure.

(iii) Bounded continuous functions. $C_{b}(S)$ is the set of bounded continuous functions $S\to\mathbb{R}$.

(iv) Weak convergence. For finite Borel measures $\mu_{n}$ ($n\in\mathbb{N}$) and $\mu$ on $S$, $\mu_{n}\Rightarrow\mu$ means that $(\mu_{n})$ converges weakly to $\mu$.

(v) Modes of convergence. For random elements $X_{n}$ ($n\in\mathbb{N}$) and $X$ of $S$ on $(\Omega,\mathcal{F},P)$, $X_{n}\xrightarrow{\text{a.s.}}X$ means that $(X_{n})$ converges to $X$ almost surely, and, when $(S,d)$ is separable, $X_{n}\xrightarrow{P}X$ means that $(X_{n})$ converges to $X$ in probability. For $k\in\mathbb{N}$, real $p\ge1$ and random vectors $Y_{n}$ ($n\in\mathbb{N}$) and $Y$ in $\mathbb{R}^{k}$ as in Convergence in Lᵖ of Random Vectors (i), $Y_{n}\xrightarrow{L^{p}}Y$ means that $(Y_{n})$ converges to $Y$ in $L^{p}$. For random elements $X_{n}$ of $S$ on probability spaces $(\Omega_{n},\mathcal{F}_{n},P_{n})$ ($n\in\mathbb{N}$) and $X$ of $S$ on $(\Omega,\mathcal{F},P)$, $X_{n}\xrightarrow{d}X$ means that $(X_{n})$ converges to $X$ in distribution.