Arithmetic of Probability Distributions, and by G. M. Feldman

By G. M. Feldman

This ebook stories the matter of the decomposition of a given random variable right into a sum of self reliant random variables (components). ranging from the recognized Cramér theorem, which says that each one parts of a regular random variable also are basic random variables, the principal function of the publication is Fel'dman's use of robust analytical innovations. within the algebraic case, one can't at once use analytic tools a result of absence of a average analytic constitution at the twin team, that's the area of attribute features. however, the tools built during this booklet permit one to use analytic innovations within the algebraic environment. the 1st a part of the e-book provides effects at the mathematics of chance distributions of random variables with values in a in the community compact abelian workforce. the second one half experiences difficulties of characterization of a Gaussian distribution of a in the neighborhood compact abelian staff by means of the independence or exact distribution of its linear statistics.

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Example text

Fixing S E Ro , let us consider the function (t, s) = exp{i(t, s)}, t E Rc'o (1) on the group R°° . This function is a character of the group R°° since it is a continuous homomorphism of the group IE°° into T. As can easily be seen, any character of the group R°° possesses the form (1). (*) Editor's note. , infinite sequences with finitely many nonzero terms. II. ARITHMETIC OF DISTRIBUTIONS 38 Let (R°°) be the a-algebra of Borel sets in R°° , that is, the minimal aalgebra containing all sets of the form Uni + ROO" , where J = (31' ...

Denote by (t, s) = E0° 1 t js j . Fixing S E Ro , let us consider the function (t, s) = exp{i(t, s)}, t E Rc'o (1) on the group R°° . This function is a character of the group R°° since it is a continuous homomorphism of the group IE°° into T. As can easily be seen, any character of the group R°° possesses the form (1). (*) Editor's note. , infinite sequences with finitely many nonzero terms. II. ARITHMETIC OF DISTRIBUTIONS 38 Let (R°°) be the a-algebra of Borel sets in R°° , that is, the minimal aalgebra containing all sets of the form Uni + ROO" , where J = (31' ...

Fixing S E Ro , let us consider the function (t, s) = exp{i(t, s)}, t E Rc'o (1) on the group R°° . This function is a character of the group R°° since it is a continuous homomorphism of the group IE°° into T. As can easily be seen, any character of the group R°° possesses the form (1). (*) Editor's note. , infinite sequences with finitely many nonzero terms. II. ARITHMETIC OF DISTRIBUTIONS 38 Let (R°°) be the a-algebra of Borel sets in R°° , that is, the minimal aalgebra containing all sets of the form Uni + ROO" , where J = (31' ...

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