probability distributions and characteristic functions. That is, for any two random variables X1, X2, both have the same probability distribution if and only if . If a random variable X has moments up to k-th order, then the characteristic function φX is k times continuously differentiable on the entire real line. In this case If a characteristic function φX has a k-th derivative at zero, then the random variable X has all moments up to k if k is even, but only up to if k is odd. If X1, ..., Xn are independent random variables, and a1, ..., an are some constants, then the characteristic function of the linear combination of the Xi 's is One specific case is the sum of two independent random variables X1 and X2 in which case one has Let and be two random variables with characteristic functions and . and are independent if and only if . The tail behavior of the characteristic function determines the smoothness of the corresponding density function. Let the random variable be the linear transformation of a random variable . The characteristic function of is . For random vectors and (where A is a constant matrix and B a constant vector), we have . Continuity The bijection stated above between probability distributions and characteristic functions is sequentially continuous. That is, whenever a sequence of distribution functions Fj(x) converges (weakly) to some distribution F(x), the corresponding sequence of characteristic functions φj(t) will also converge, and the limit φ(t) will correspond to the characteristic function of law F. More formally, this is stated as Lévy’s continuity theorem: A sequence Xj of n-variate random variables converges in distribution to random variable X if and only if the sequence φXj converges pointwise to a function φ which is continuous at the origin. Where φ is the characteristic function of X. This theorem can be used to prove the law of large numbers and the central limit theorem. Inversion formula There is a one-to-one correspondence between cumulative distribution functions and characteristic functions, so it is possible to find one of these functions if we know the other. The formula in the definition of characteristic function allows us to compute φ when we know the distribution function F (or density f). If, on the other hand, we know the characteristic function φ and want to find the corresponding distribution function, then one of the