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.
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.
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
does not imply that
The convergence in probability claims that for large 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.