## 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 $Y_n$ converges in probability to a constant $c$  i.e.

$Y_n \rightarrow^P c$

if $\forall \epsilon >0$

$P ( |Y_n - c | < \epsilon ) \rightarrow 1$ as $n \rightarrow \infty$

which suggests that for $n$ sufficiently large $Y_n$ will be sufficiently close to $c$.

Note that by the Chebychev inequality we get that the sufficient condition for $Y_n \rightarrow^P c$ is that $Y_n$ tends to $c$ in quadratic mean i.e.

$E(Y_n-c)^2 \rightarrow 0$

The Chebychev inequality also implies that if a random variable has a small variance, then its distribution is closely concentrated about the mean. However the converse is not true*: The fact that a random variable is concentrated about its mean tell us nothing about its moments! Which means that

$Y_n \rightarrow^P c$

does not imply that

$E(Y_n-c)^i \rightarrow 0$    $i=1,2,..$

The convergence in probability claims that for large n, $Y_n$ is very likely to be close to c. However, it suggests nothing about where the remaining small probability mass which is away of c is located. This small mass can affect significantly the value of the mean and of other moments.

*Unless the Y’s are uniformly bounded i.e.

$\exists M$   s.th.   $P[|Y_n - c | < M] =1$,   $\forall n$