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 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}\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.

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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 $X$ .
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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).

Title	convergence in probability
Canonical name	ConvergenceInProbability
Date of creation	2013-03-22 15:01:05
Last modified on	2013-03-22 15:01:05
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	7
Author	CWoo (3771)
Entry type	Definition
Classification	msc 60B10
Synonym	converge in probability
Synonym	converges in measure
Synonym	converge in measure
Synonym	convergence in measure
Defines	converges in probability