convergence in probability

Let {Xi} be a sequence of random variablesMathworldPlanetmath defined on a probability spaceMathworldPlanetmath (Ω,,P) taking values in a separablePlanetmathPlanetmath metric space (Y,d), where d is the metric. Then we say the sequence Xi converges in probability or converges in measure to a random variable X if for every ε>0,


We denote convergence in probability of Xi to X by


Equivalently, XiprX iff every subsequence of {Xi} contains a subsequence which convergesPlanetmathPlanetmath to X almost surely.


  • Unlike ordinary convergence, XiprX and XiprY only implies that X=Y almost surely.

  • The need for separability on Y is to ensure that the metric, d(Xi,X), is a random variable, for all random variables Xi and X.

  • Convergence almost surely implies convergence in probability but not conversely.


  • 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