(*a*) *Convolution of probability densities*

The addition of independent random variables or vectors leads to the convolution of their probability distributions: if and are two *n*-dimensional random vectors independently distributed with probability densities and , respectively, then their sum has probability density given by*i.e.*

This result can be extended to the case where and are singular measures (distributions of order zero, Section 1.3.2.3.4) and do not have a density with respect to the Lebesgue measure in .

(*b*) *Characteristic functions*

This convolution can be turned into a simple multiplication by considering the Fourier transforms (called the *characteristic functions*) of , and , defined with a slightly different normalization in that there is no factor of in the exponent (see Section 1.3.2.4.5), *e.g.*Then by the convolution theoremso that may be evaluated by Fourier inversion of its characteristic function as(see Section 1.3.2.4.5 for the normalization factors).

It follows from the differentiation theorem that the partial derivatives of the characteristic function at are related to the moments of a distribution* P* by the identitiesfor any *n*-tuple of non-negative integers .

(*c*) *Moment-generating functions*

The above relation can be freed from powers of *i* by defining (at least formally) the *moment-generating function*:which is related to by so that the inversion formula readsThe moment-generating function is well defined, in particular, for any probability distribution with compact support, in which case it may be continued analytically from a function over into an entire function of *n complex* variables by virtue of the Paley–Wiener theorem (Section 1.3.2.4.2.10). Its moment-generating properties are summed up in the following relations:

(*d*) *Cumulant-generating functions*

The multiplication of moment-generating functions may be further simplified into the addition of their logarithms:or equivalently of the coefficients of their Taylor series at , *viz*:These coefficients are called *cumulants*, since they add when the independent random vectors to which they belong are added, and log *M* is called the *cumulant-generating function.* The inversion formula for then reads

(*e*) *Asymptotic expansions and limit theorems*

Consider an *n*-dimensional random vector **X** of the formwhere the *N* summands are independent *n*-dimensional random vectors identically distributed with probability density *P.* Then the distribution of **X** may be written in closed form as a Fourier transform:whereis the moment-generating function common to all the summands.

This an exact expression for , which may be exploited analytically or numerically in certain favourable cases. Supposing for instance that *P* has compact support, then its characteristic function can be sampled finely enough to accommodate the bandwidth of the support of (this sampling rate clearly depends on *n*) so that the above expression for can be used for its numerical evaluation as the *discrete* Fourier transform of . This exact method is practical only for small values of the dimension *n*.

In all other cases some form of approximation must be used in the Fourier inversion of . For this purpose it is customary (Cramér, 1946) to expand the cumulant-generating function around with respect to the carrying variables **t**:where is a multi-index (Section 1.3.2.2.3). The first-order terms may be eliminated by recentring around its vector of first-order cumulantswhere denotes the mathematical expectation of a random vector. The second-order terms may be grouped separately from the terms of third or higher order to givewhere is the covariance matrix of the multivariate distribution *P.* Expanding the exponential gives rise to a series of terms of the formeach of which may now be subjected to a Fourier transformation to yield a Hermite function of **t** (Section 1.3.2.4.4.2) with coefficients involving the cumulants κ of *P.* Taking the transformed terms in natural order gives an asymptotic expansion of *P* for large *N* called the *Gram–Charlier series* of , while grouping the terms according to increasing powers of gives another asymptotic expansion called the *Edgeworth series* of . Both expansions comprise a leading Gaussian term which embodies the *central-limit theorem*:

(*f*) *The saddlepoint approximation*

A limitation of the Edgeworth series is that it gives an accurate estimate of only in the vicinity of , *i.e.* for small values of **E**. These convergence difficulties are easily understood: one is substituting a *local* approximation to log *M* (*viz* a Taylor-series expansion valid near ) into an integral, whereas integration is a *global* process which consults values of log *M* far from .

It is possible, however, to let the point **t** where log *M* is expanded as a Taylor series depend on the particular value of **X** for which an accurate evaluation of is desired. This is the essence of the *saddlepoint method* (Fowler, 1936; Khinchin 1949; Daniels, 1954; de Bruijn, 1970; Bleistein & Handelsman, 1986), which uses an analytical continuation of from a function over to a function over (see Section 1.3.2.4.2.10). Putting then , the version of Cauchy's theorem (Hörmander, 1973) gives rise to the identityfor *any* . By a convexity argument involving the positive-definiteness of covariance matrix **Q**, there is a unique value of **τ** such thatAt the *saddlepoint* , the modulus of the integrand above is a maximum and its phase is stationary with respect to the integration variable **s**: as *N* tends to infinity, all contributions to the integral cancel because of rapid oscillation, except those coming from the immediate vicinity of where there is no oscillation. A Taylor expansion of log to second order with respect to **s** at then givesand henceThe last integral is elementary and gives the `saddlepoint approximation':whereand where

This approximation scheme amounts to using the `conjugate distribution' (Khinchin, 1949)instead of the original distribution for the common distribution of all *N* random vectors . The exponential modulation results from the analytic continuation of the characteristic (or moment-generating) function into , as in Section 1.3.2.4.2.10. The saddlepoint approximation is only the leading term of an asymptotic expansion (called the *saddlepoint expansion*) for , which is actually the Edgeworth expansion associated with .