Let $\lbrace X_i \rbrace$ be a sequence of random variables defined on a probability space $(\Omega,\mathcal{F},P)$ taking values in a separable metric space $(Y,d)$ where $d$ is the metric. Then we say the sequence $X_i$ converges in probability or converges in measure to a random variable $X$ if for every $\varepsilon>0$ $$\lim_{i\rightarrow\infty}P(d(X_i,X)\geq\varepsilon)=0.$$ We denote convergence in probability of $X_i$ to $X$ by $$X_i\stackrel{pr}{\longrightarrow} X.$$ Equivalently, $X_i\stackrel{pr}{\longrightarrow} X$ iff every subsequence of $\lbrace X_i\rbrace$ contains a subsequence which converges to $X$ almost surely.

Remarks.

• Unlike ordinary convergence, $X_i\stackrel{pr}{\longrightarrow} X$ and $X_i\stackrel{pr}{\longrightarrow} Y$ only implies that $X=Y$ almost surely.
• The need for separability on $Y$ is to ensure that the metric, $d(X_i,X)$ is a random variable, for all random variables $X_i$ and $X$
• Convergence almost surely implies convergence in probability but not conversely.

Bibliography

1

R. M. Dudley, Real Analysis and Probability, Cambridge University Press (2002).

2

W. Feller, An Introduction to Probability Theory and Its Applications. Vol. 1, Wiley, 3rd ed. (1968).