<P> The central limit theorem applies in particular to sums of independent and identically distributed discrete random variables . A sum of discrete random variables is still a discrete random variable, so that we are confronted with a sequence of discrete random variables whose cumulative probability distribution function converges towards a cumulative probability distribution function corresponding to a continuous variable (namely that of the normal distribution). This means that if we build a histogram of the realisations of the sum of n independent identical discrete variables, the curve that joins the centers of the upper faces of the rectangles forming the histogram converges toward a Gaussian curve as n approaches infinity, this relation is known as de Moivre--Laplace theorem . The binomial distribution article details such an application of the central limit theorem in the simple case of a discrete variable taking only two possible values . </P> <P> The law of large numbers as well as the central limit theorem are partial solutions to a general problem: "What is the limiting behaviour of S as n approaches infinity?" In mathematical analysis, asymptotic series are one of the most popular tools employed to approach such questions . </P> <P> Suppose we have an asymptotic expansion of f (n): </P> <Dl> <Dd> f (n) = a 1 φ 1 (n) + a 2 φ 2 (n) + O (φ 3 (n)) (n → ∞). (\ displaystyle f (n) = a_ (1) \ varphi _ (1) (n) + a_ (2) \ varphi _ (2) (n) + O (\ big () \ varphi _ (3) (n) (\ big)) \ qquad (n \ rightarrow \ infty).) </Dd> </Dl>

Central limit theorem for non identically distributed random variables