Limit Theorems:

In certain situations the distribution of a statistic might be difficult to derive or even if the distribution is derivable, it might be too complex to be used in practice. In such cases approximations of distributions are helpful and a good approximation frequent serves the practical reasons. Here, we shall consider certain fundamental   results in the form of theorems of probability that form the basis of some of these' approximations.

Before we consider these fundamental theorems, known as the limit theorems of probability, we introduce the concept of convergence in probability, A sequence of random variables X_I, X₂, ... are said to converge to the random variable X in probability, if

lim P {¦Xn-X¦> ε } =0, for every ε >0

^n→∞

A practical interpretation of the definition of convergence in probability, is that for large n, the difference among the realizations of X_n, and X is negligibly small along with high probability and so the distribution of X_n, might be approximated through the distribution of X.

The notion of convergence in probability is most frequent used in the case while the limiting random variable X has a degenerate distribution P (X = a) = 1 for a certain ' a ' and X_n = x¯, the mean of the elements of a random vector X of n components where the components are not essentially independently and identically distributed. In that cases if

lim P {¦ x¯ - a¦> ε } =0, for every ε >0

^n→∞

then we say that the sequence X_I, X₂, ... satisfies the weak law of large numbers. It is easy to deduce the law follows from a fundamental result in probability called Chebyshev's inequality or Chebyshev's lemma.