6.1 The Four Definitions
Definition 6.1.1 (Convergence in probability). \(X_n \underset {P}{\longrightarrow } X\) if for every \(\varepsilon >0\) \[\lim _{n\rightarrow \infty } P\left (\left |X_n - X\right | \geq \varepsilon \right ) = 0 .\]
Definition 6.1.2 (Convergence in distribution). \(X_n \underset {D}{\longrightarrow } X\) if \[\lim _{n\rightarrow \infty } F_{X_n}(x) = F_X(x)\] at every point \(x\) where \(F_X\) is continuous.
Definition 6.1.3 (Convergence in quadratic mean). \(X_n \underset {QM}{\longrightarrow } X\) if \[\lim _{n\rightarrow \infty } E\left [\left (X_n - X\right )^{2}\right ] = 0 .\] More generally \(X_n \underset {L^p}{\longrightarrow } X\) if \(E\left |X_n-X\right |^{p}\rightarrow 0\); the case \(p=2\) is the one used here.
Definition 6.1.4 (Almost sure convergence). \(X_n \underset {a.s.}{\longrightarrow } X\) if \[P\left (\lim _{n\rightarrow \infty } X_n = X\right ) = 1 .\]
Note. The difference between the first and the last is worth dwelling on, because it is the one students most often collapse. Convergence in probability says that at each large \(n\), the chance of \(X_n\) being far from \(X\) is small — but it allows \(X_n\) to stray far infinitely often, provided the occasions become rare. Almost sure convergence says that for almost every outcome the sequence eventually settles and stays. The first is a statement about each \(n\) separately; the second is a statement about whole paths.
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.