similar simple criteria for when a given function φ could be the characteristic function of some random variable. The central result here is Bochner’s theorem, although its usefulness is limited because the main condition of the theorem, non-negative definiteness, is very hard to verify. Other theorems also exist, such as Khinchine’s, Mathias’s, or Cramér’s, although their application is just as difficult. Pólya’s theorem, on the other hand, provides a very simple convexity condition which is sufficient but not necessary. Characteristic functions which satisfy this condition are called Pólya-type. Bochner’s theorem. An arbitrary function φ : Rn → C is the characteristic function of some random variable if and only if φ is positive definite, continuous at the origin, and if φ(0) = 1. Khinchine’s criterion. A complex-valued, absolutely continuous function φ, with φ(0) = 1, is a characteristic function if and only if it admits the representation Mathias’ theorem. A real-valued, even, continuous, absolutely integrable function φ, with φ(0) = 1, is a characteristic function if and only if for n = 0,1,2,..., and all p > 0. Here H2n denotes the Hermite polynomial of degree 2n. Pólya’s theorem. If is a real-valued, even, continuous function which satisfies the conditions , is convex for , , then φ(t) is the characteristic function of an absolutely continuous distribution symmetric about 0. Uses Because of the continuity theorem, characteristic functions are used in the most frequently seen proof of the central limit theorem. The main technique involved in making calculations with a characteristic function is recognizing the function as the characteristic function of a particular distribution. Basic manipulations of distributions Characteristic functions are particularly useful for dealing with linear functions of independent random variables. For example, if is a sequence of independent (and not necessarily identically distributed) random variables, and where the ai are constants, then the characteristic function for Sn is given by In particular, . To see this, write out the definition of characteristic function: The independence of X and Y is required to establish the equality of the third and fourth expressions. Another special case of interest for identically distributed random variables is when and then Sn is the sample mean. In this case, writing for the mean, Moments Characteristic functions can also be used to find moments of a random variable. Provided that the nth moment exists, the characteristic function can be differentiated n times and For example,