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

References

[a1]	N.N. Vakhania, "Probability distributions on linear spaces" , North-Holland (1981) (Translated from Russian)