Implications between Modes of Convergence of Random Elements

In the setting of Convergence of Probability Measures and Random Elements: Standing Notation, let $(S,d)$ be a separable metric space, and let $(X_{n})_{n\in\mathbb{N}}$ and $X$ be random elements of $S$.

(i) Almost sure convergence implies convergence in probability. If $X_{n}\xrightarrow{\text{a.s.}}X$, then $X_{n}\xrightarrow{P}X$.

(ii) Convergence in $L^{p}$ implies convergence in probability. Let $m\in\mathbb{N}$, let $p\ge1$ be real, let $\lVert\cdot\rVert$ be the Euclidean norm and $d_{E}$ the Euclidean distance on $\mathbb{R}^{m}$, and let $(Y_{n})_{n\in\mathbb{N}}$ and $Y$ be random vectors in $\mathbb{R}^{m}$ with $\lVert Y_{n}-Y\rVert\in\mathcal{L}^{p}(P)$ for every $n$. If $Y_{n}\xrightarrow{L^{p}}Y$, then $Y_{n}\xrightarrow{P}Y$ in the separable metric space $(\mathbb{R}^{m},d_{E})$ (The Real Line and Euclidean Space Are Separable (ii)).

(iii) A subsequence converges almost surely. If $X_{n}\xrightarrow{P}X$, then there is a subsequence $(X_{n_{k}})_{k\in\mathbb{N}}$ with $X_{n_{k}}\xrightarrow{\text{a.s.}}X$.

(iv) Convergence in probability implies convergence in distribution. If $X_{n}\xrightarrow{P}X$, then $X_{n}\xrightarrow{d}X$.

(v) Limits in probability are almost surely unique. If $Z$ is a random element of $S$ with $X_{n}\xrightarrow{P}X$ and $X_{n}\xrightarrow{P}Z$, then $X=Z$ almost surely.