# 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 [1].

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 [2]). 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 [3]).

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 [4]). 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 [5]). In particular, in a Hilbert space the additional condition is that the operator $R$ be nuclear.

#### References

 [1] A.N. Kolmogorov, C.R. Acad. Sci. Paris , 200 (1935) pp. 1717–1718 [2] Yu.V. Prokhorov, "Convergence of random processes and limit theorems in probability theory" Theory Probab. Appl. , 1 (1956) pp. 157–214 Teor. Veroyatnost. i Primen. , 1 : 2 (1956) pp. 177–238 [3] V.V. Sazonov, "A remark on characteristic functionals" Theory Probab. Appl. , 3 (1958) pp. 188–192 Teor. Veroyatnost. i Primen. , 3 : 2 (1958) pp. 201–205 [4] N.N. Vakhania, V.I. Tarieladze, S.A. Chobanyan, "Probability distributions on Banach spaces" , Reidel (1987) (Translated from Russian) [5] N.N. Vakhania, "Sur les répartitions de probabilités dans les espaces de suites numériques" C.R. Acad. Sci. Paris , 260 (1965) pp. 1560–1562