Convergence In Probability

Why

Laws of large numbers.

Definition

A sequence of random variables convergences in probability if it converges in measure.

Notation

Let $(X, \mathcal{A} , \mu )$ be a measure space. Let $\seq{f}$ a sequence of real-valued measurable functions on $X$. Let $f: X \to R$ be measurable function. If $f_n$ converges in measure to $f$ we write: $f_n \goesto f$ in probability, read aloud as “f n goes to f in probability.”

Suppose $f_n \goesto f$ in probability. Then for every $\epsilon > 0$,

\[ \lim_{n \to \infty} \mu (\Set*{x \in X}{\abs{f_n(x) - f(x)} > \epsilon }) = 0. \]