convergence in probability is preserved under continuous transformations

Theorem 1.

Let g:RkRl be a continuous functionMathworldPlanetmathPlanetmath. If {Xn} are Rk-valued random variablesMathworldPlanetmath converging to X in probability, then {g(Xn)} converge in probability to g(X) also.


Suppose first that g is uniformly continuousPlanetmathPlanetmath. Given ϵ>0, there is δ>0 such that g(Xn)-g(X)<ϵ whenever Xn-X<δ. Therefore,


as n.

Now suppose g is not necessarily uniformly continuous on k. But it will be uniformly continuous on any compact set {xk:xm} for m0. Consequently, if Xn and X are boundedPlanetmathPlanetmathPlanetmathPlanetmath (by m), then the proof just given is applicable. Thus we attempt to reduce the general case to the case that Xn and X are bounded.



Clearly, fm:kk is continuous; in fact, it can be verified that fm is uniformly continuous on k. (This is geometrically obvious in the one-dimensional case.)

Set Xnm=fm(Xn) and Xm=fm(X), so that Xnm convergePlanetmathPlanetmath to Xm in probability for each m0.

We now show that g(Xn) converge to g(X) in probability by a four-step estimate. Let ϵ>0 and δ>0 be given. For any m0 (which we will later),


Choose M such that for mM,


(This is possible since limm(Xm)=(m=0{Xm})=()=0.)

In particular, let m=M+1. Since Xnm converge in probability to Xm and Xnm, Xm are bounded, g(Xnm) converge in probability to g(Xm). That means for n large enough,


Finally, since XnXn-X+X, and Xn converge to X in probability, we have


for large enough n.

Collecting the previous inequalitiesMathworldPlanetmath together, we have


for large enough n. ∎

Title convergence in probability is preserved under continuous transformations
Canonical name ConvergenceInProbabilityIsPreservedUnderContinuousTransformations
Date of creation 2013-03-22 16:15:05
Last modified on 2013-03-22 16:15:05
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 10
Author stevecheng (10074)
Entry type Theorem
Classification msc 60A10