0.1 Convergence in Probability
A sequence of random variables \(\{X_n\}_{n\in\mathbb{N}}\) on probability space \((\Omega, \mathcal{F}, \mathbb{P})\) converges in probability to \(X\), denoted \(X_n\stackrel{\mathbb{P}}{\rightarrow}X\) \((n\rightarrow\infty)\) if \[\lim_{ n \to \infty } \mathbb{P}(\{\omega \in\Omega:|X_{n}(\omega)-X(\omega)|>\epsilon\})=0\]for every \(\epsilon>0\).
Random variables can also converge almost surely, in norm and in law and the relationships between all types of convergence can be found in the note: Convergence of Random Variables.
Figure: Random Variable Convergence Relationships.
0.2 Convergence in Norm & Convergence in Probability
If \(X_{n}\stackrel{L^1}{\to}X\) then \(X_{n}\stackrel{\mathbb{P}}{\to}X\) as \(n \to \infty\).
\begin{proof} This result follows directly from applying Markov Inequality since, for every \(\epsilon>0\) we have that \[
\mathbb{P}(|X_{n}-X|>\epsilon)\leq \frac{1}{\epsilon}\cdot \mathbb{E}|X_{n}-X|.
\] \end{proof} Note: The converse does not hold in general.
0.3 Convergence Almost Surely & Convergence in Probability
If \(X_{n}\stackrel{a.s}{\to}X\) as \(n \to \infty\) then \(X_{n}\stackrel{\mathbb{P}}{\to}X\) as \(n \to \infty\).
Proof: This result follows from an application of the Monotone Convergence Theorem.
If \(X_{n}\stackrel{{\mathbb{P}}}{\to}X\) as \(n \to \infty\), then \(X_{n}\stackrel{\mathcal{D}}{\to}X\) as \(n\to \infty\).
Proof: For every \(x \in\mathbb{R}\), \(\epsilon>0\) we have that \[ \mathbb{P}(X\leq x-\epsilon)\leq \liminf_{n \to \infty } \mathbb{P}(X_{n}\leq x)\leq \limsup_{ n \to \infty }\mathbb{P}(X_{n}\leq x)\leq \mathbb{P}(X\leq x+\epsilon). \] If \(x\) is a continuity point of \(X\), then as \(\epsilon\downarrow 0\), the left and right sides converge and hence \[ \lim_{ n \to \infty } \mathbb{P}(X_{n}\leq x)=P(X\leq x), \] giving the result.
If \(X_{n} \stackrel{\mathbb{P}}{\to}X\) as \(n \to \infty\), then there exists a subsequence \(\{n_{i}\}\) such that \(X_{n_{i}}\stackrel{a.s.}{\to}X\) as \(i \to \infty\).
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Proof: This results follows from an application of the Borel-Cantelli Lemma.