Proof of Convergence in Distribution to a Constant and Convergence in Probability

Throughout, $d$ is symmetric with $d(x,x)=0$ (Metric Space (iv), Metric Space (iii)); the laws $P_{Y_{n}}$ and $P_{c}$ are Borel probability measures on $S$ with $P_{Y_{n}}(B)=P_{n}(Y_{n}^{-1}(B))$ and $P_{c}(B)=P(c^{-1}(B))$ for $B\in\mathcal{B}(S)$ (The Law of a Random Element (i)); and the bounded sequences of probabilities below have a limit superior and a limit inferior, compared by Properties of the Limit Superior and the Limit Inferior: Tail Formula, Order, Comparison, Bounds, Eventual and Frequent Inequalities, Subsequences, Negation and Convergence (ii) and Properties of the Limit Superior and the Limit Inferior: Tail Formula, Order, Comparison, Bounds, Eventual and Frequent Inequalities, Subsequences, Negation and Convergence (iv).

Step 1 (proof of (i)). Fix a real $\varepsilon>0$. The complement of $F_{\varepsilon}$ is the open ball $B_{d}(c,\varepsilon)$, which is open by Open Ball in a Metric Space is Open (i); so $F_{\varepsilon}$ is closed (Closed Subset of a Topological Space (i)) and therefore lies in the Borel $\sigma$-algebra $\mathcal{B}(S)$ (Borel Sets of a Metric Space: Open and Closed Sets, Points, Open Sets of Subspaces, and Subspaces (ii)). The displayed set equals $Y_{n}^{-1}(F_{\varepsilon})$ by the definition of $F_{\varepsilon}$, and it lies in $\mathcal{F}_{n}$ because $Y_{n}$ is a random element (Probability Spaces, Events and Random Elements (iii), Measurable Map, and the Set of Measurable Maps (i)).

Step 2 (the law of the constant). Since $d(c,c)=0<\varepsilon$, $c\notin F_{\varepsilon}$, so $c^{-1}(F_{\varepsilon})=\emptyset$ and $P_{c}(F_{\varepsilon})=0$.

Step 3 (proof of (ii)). By hypothesis and Convergence in Distribution of Random Elements of a Metric Space (i), $P_{Y_{n}}\Rightarrow P_{c}$. The portmanteau theorem, from Portmanteau Theorem on a Metric Space (i) to Portmanteau Theorem on a Metric Space (iii), applied to the closed set $F_{\varepsilon}$, and Step 2 give

$$0\le\liminf_{n}P_{n}\bigl(Y_{n}^{-1}(F_{\varepsilon})\bigr)\le\limsup_{n}P_{n}\bigl(Y_{n}^{-1}(F_{\varepsilon})\bigr)\le P_{c}(F_{\varepsilon})=0 ,$$

so $P_{n}(Y_{n}^{-1}(F_{\varepsilon}))\to0$ by Properties of the Limit Superior and the Limit Inferior: Tail Formula, Order, Comparison, Bounds, Eventual and Frequent Inequalities, Subsequences, Negation and Convergence (xi).

Step 4 (proof of (iii)). Now every $Y_{n}$ is defined on $(\Omega,\mathcal{F},P)$. Let $G=S\setminus\bar{B}_{d}(c,\varepsilon)$, the complement of the closed ball, so that $G=\{x\in S:d(c,x)>\varepsilon\}$. The set $G$ is open: for $x\in G$ let $r=d(c,x)-\varepsilon>0$; if $d(x,y)<r$, then by the triangle inequality (Metric Space (v))

$$d(c,y)\ge d(c,x)-d(x,y)>d(c,x)-r=\varepsilon ,$$

so the open ball $B_{d}(x,r)$ lies in $G$. Hence $G\in\mathcal{B}(S)$ (Borel Sets of a Metric Space: Open and Closed Sets, Points, Open Sets of Subspaces, and Subspaces (i)), and $\{d(Y_{n},c)>\varepsilon\}=Y_{n}^{-1}(G)\in\mathcal{F}$. Since $G\subseteq F_{\varepsilon}$, monotonicity of $P$ (Basic Properties of a Measure (iii)) gives

$$0\le P\bigl(d(Y_{n},c)>\varepsilon\bigr)=P\bigl(Y_{n}^{-1}(G)\bigr)\le P\bigl(Y_{n}^{-1}(F_{\varepsilon})\bigr),$$

and the right side tends to $0$ by (ii); by squeezing (Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (xi)) the left side tends to $0$. As $\varepsilon>0$ was arbitrary, $Y_{n}\xrightarrow{P}c$ by Convergence in Probability (i) (which applies because $(S,d)$ is separable).