Convergence in probability and the limiting behavior of moments: A useful (elementary) reminder

 Convergence in probability is a stochastic analog of the convergence of a sequence of numbers. Hence, a sequence of random variables converges in probability to a constant   i.e.

if

as

which suggests that for sufficiently large will be sufficiently close to .

 

 

Note that by the Chebychev inequality we get that the sufficient condition for is that tends to in quadratic mean i.e.