convergence in probability

Let $\\{X_{i}\\}$ be a sequence of random variables defined on a probability space $(\\Omega,\\mathcal{F},P)$ taking values in a separable metricspace $(Y,d)$ , where $d$ is the metric. Then we say the sequence $X_{i}$ converges in probability or converges in measure to a random variable $X$ iffor 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}\\lx@stackrel{{\\scriptstyle pr}}{{\\longrightarrow}}X.

Equivalently, $X_{i}\\lx@stackrel{{\\scriptstyle pr}}{{\\longrightarrow}}X$ iff every subsequence of $\\{X_{i}\\}$ contains a subsequence which converges to $X$ almost surely.

Remarks.

•
Unlike ordinary convergence, $X_{i}\\lx@stackrel{{\\scriptstyle pr}}{{\\longrightarrow}}X$ and $X_{i}\\lx@stackrel{{\\scriptstyle pr}}{{\\longrightarrow}}Y$ only implies that $X=Y$ almost surely.
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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.

References

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).