POTW Convergence in Probability

Euge
Gold Member
MHB
POTW Director
Messages
2,072
Reaction score
245
Prove that if ##\{X_n\}_{n = 1}^\infty## is a sequence of real random variables on probability space ##(\Omega, \mathscr{F},\mathbb{P})## such that ##\lim_n \mathbb{E}[X_n] = \mu## and ##\lim_n \operatorname{Var}[X_n] = 0##, then ##X_n## converges to ##\mu## in probability.
 
Physics news on Phys.org
We're going to use
https://en.m.wikipedia.org/wiki/Chebyshev's_inequality

For any ##m##, there exists ##N## such that for ##n>N##, we have ##P(|X_n-\mu | > 1/m) < 1/m^2##, applying Chebyshev's inequality with ##\sigma < 1/m^3## and ##k=m##, and ##N## picked such that ##var(X_n) < 1/m^3## for ##n>N##.

That's pretty much it, since ##m## it's arbitrary.

For any ##\epsilon >0##, for any ##m## such that ##1/m<\epsilon##, and for ##n## large enough, we have ##P(|X_n-\epsilon) |<1/m##. Since ##m## it's arbitrary if we make ##n## big enough, in the limit as n goes to infinity this goes to zero. Hence all the probability weight must be on ##\mu## as desired.
 
Back
Top