# Characteristic functional

An analogue of the concept of a characteristic function; it is used in the infinite-dimensional case. Let $\mathfrak X$ be a non-empty set, let $\Gamma$ be a vector space of real-valued functions defined on $\mathfrak X$ and let $\widehat{C} ( \mathfrak X , \Gamma )$ be the smallest $\sigma$- algebra of subsets of $\mathfrak X$ relative to which all functions in $\Gamma$ are measurable. The characteristic functional of a probability measure $\mu$ given on $\widehat{C} ( \mathfrak X , \Gamma )$ is defined as the complex-valued functional $\widehat \mu$ on $\Gamma$ given by the equation

$$\widehat \mu ( g) = \ \int\limits _ {\mathfrak X } \mathop{\rm exp} [ ig ( x)] d \mu ( x),\ \ g \in \Gamma .$$

From now on the most important and simplest case when $\mathfrak X$ is a separable real Banach space and $\Gamma$ is its topological dual $\mathfrak X ^ {*}$ is studied. In this case $\widehat{C} ( \mathfrak X , \mathfrak X ^ {*} )$ coincides with the $\sigma$- algebra of Borel sets of $\mathfrak X$. The concept of a characteristic functional for infinite-dimensional Banach spaces was introduced by A.N. Kolmogorov in .

The characteristic functional of a random variable $X$ with values in $\mathfrak X$ is, by definition, that of its probability distribution $\mu _ {X} ( B) = {\mathsf P} \{ X \in B \}$, $B \subset \mathfrak X$.

Main properties of the characteristic functional:

1) $\widehat \mu ( 0) = 1$ and $\widehat \mu$ is positive definite, i.e. $\sum _ {k,l} \alpha _ {k} \overline \alpha \; _ {l} \widehat \mu ( x _ {k} ^ {*} - x _ {l} ^ {*} ) \geq 0$ for any finite set of complex numbers $\alpha _ {i}$ and elements $x _ {i} ^ {*} \in \mathfrak X ^ {*}$;

2) $\widehat \mu$ is continuous in the strong topology and sequentially continuous in the weak $*$ topology of $\mathfrak X ^ {*}$;

3) $| \widehat \mu ( x ^ {*} ) | \leq 1$,

$$| \widehat \mu ( x _ {1} ^ {*} ) - \widehat \mu ( x _ {2} ^ {*} ) | ^ {2} \leq \ 2 [ 1 - \mathop{\rm Re} \widehat \mu ( x _ {1} ^ {*} - x _ {2} ^ {*} )],$$

where $x ^ {*} , x _ {1} ^ {*} , x _ {2} ^ {*} \in \mathfrak X ^ {*}$;

4) $\overline{ {\widehat \mu ( x ^ {*} ) }}\; = \widehat \mu (- x ^ {*} )$; in particular, $\widehat \mu$ takes only real values (and is an even functional) if and only if the measure $\mu$ is symmetric, that is, $\mu ( B) = \mu (- B)$, where $- B = \{ {x } : {- x \in B } \}$;

5) the characteristic functional determines the measure uniquely;

6) the characteristic functional of the convolution of two probability measures (of the sum of two independent random variables) is the product of their characteristic functionals.

In the finite-dimensional case the method of characteristic functionals is based on the theorem about the continuity of the correspondence between measures and their characteristic functionals, and on a theorem concerning the description of the class of characteristic functionals. In the infinite-dimensional case the direct analogues of these theorems do not hold. If a sequence of probability measures $( \mu _ {n} )$ converges weakly to $\mu$, then $( \widehat \mu _ {n} )$ converges pointwise to $\widehat \mu$, and this convergence is uniform on bounded subsets of $\mathfrak X ^ {*}$; if $K$ is a weakly relatively-compact family of probability measures on $\mathfrak X$, then the family $\{ {\widehat \mu } : {\mu \in K } \}$ is equicontinuous in the strong topology of $\mathfrak X ^ {*}$. The converse assertions only hold in the finite-dimensional case. However, the conditions of convergence and of weak relative compactness of families of probability measures can be expressed in terms of characteristic functionals (see ). Furthermore, in contrast to the finite-dimensional case, not every positive-definite normalized (equal to 1 at the origin) continuous functional is a characteristic functional: continuity in the metric topology is not sufficient. A topology in $\mathfrak X ^ {*}$ is called sufficient, or necessary, if in this topology the continuity of a positive-definite normalized functional is sufficient, or necessary, for it to be the characteristic functional of some probability measure on $\mathfrak X$. A necessary and sufficient topology is said to be an $S$- topology. A space $\mathfrak X$ is called an $S$- space if there is an $S$- topology on $\mathfrak X ^ {*}$. A Hilbert space is an $S$- space (see ).

The most important characteristic functionals are those of Gaussian measures. A measure $\mu$ in $\mathfrak X$ is called a centred Gaussian measure if for all $x ^ {*} \in \mathfrak X ^ {*}$,

$$\tag{* } \widehat \mu ( x ^ {*} ) = \ \mathop{\rm exp} \left [ - { \frac{1}{2} } x ^ {*} ( Rx ^ {*} ) \right ] ,$$

where $R$, a bounded linear positive operator from $\mathfrak X ^ {*}$ into $\mathfrak X$, is the covariance operator of the measure $\mu$, defined by the relation

$$x ^ {*} ( Rx ^ {*} ) = \ \int\limits x ^ {*} 2 ( x) d \mu ( x)$$

(see ). In contrast to the finite-dimensional case, not every functional of the form (*) is a characteristic functional: additional restrictions on $R$ are needed, depending on the space $\mathfrak X$. For example, if $\mathfrak X = l _ {p}$, $1 \leq p < \infty$, then an additional (necessary and sufficient) condition is $\sum r _ {kk} ^ {p/2} < + \infty$, where $\| r _ {ij} \|$ is the matrix of the operator $R$ in the natural basis (see ). In particular, in a Hilbert space the additional condition is that the operator $R$ be nuclear.

How to Cite This Entry:
Characteristic functional. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Characteristic_functional&oldid=46321
This article was adapted from an original article by N.N. Vakhania (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article