Construction of a class of steadily increasing continuous functions which are not absolutely continuous

Summary Startin with a Gilman's non-dense perfect set of measure zero and three convergent series of positive terms satisfying certain prescribed conditions, a class of steadily increasing continuous functions have been constructed which are not absolutely continuous.

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Riassunto Partendo da un insieme perfetto e non denso di Gilman, di misura nulla, e da tre serie convergenti a termini positivi, soddisfacenti a certe condizioni prestabilite si costruisce una classe di funzioni continue strettamente crescenti che non sono assolutamente continue

     

Absolutely continuous functions of several variables and diffeomorphisms

In [4], a class of absolutely continuous functions of d-variables, motivated by applications to change of variables in an integral, has been introduced. The main result of this paper states that absolutely continuous functions in the sense of [4] are not stable under diffeomorphisms. We also show an example of a function which is absolutely continuous with respect cubes but not with respect to balls.     

Approximately continuous functions which are continuous almost everywhere

Acta Mathematica Hungarica -     

Continuous functions representable as sums of independent random variables

Summary We prove that the only continuous functions representable as a series of Rademacher functions on the unit interval are the linear functions. A generalization using general k-expansions of real numbers is also proved.     

Positive definite rational functions of two variables which are not the sum of three squares

Cassels, Ellison, and Pfister have shown that there is a positive semidefinite function of R(x, y) that is not the sum of three squares. In this paper positive definite functions of R(x, y) are found having the same property. The proof involves showing the nonexistence of points on some elliptic curves defined over C(x), and extends the methods of [1].     

A class of bornological barrelled spaces which are not ultrabornological

N. Bourbaki, [1, p. 35], notices that it is not known if every bornological barrelled space is ultrabornological. In this paper we prove that if E is the topological product of an infinite family of bornological barrelled spaces, of non-zero dimension, there exists an infinite number of bornological barrelled subspaces ofE, which are not ultrabornological. We also give some examples of barrelled normable spaces which are not ultrabornological.Supported in part by the Patronato para el Fomento de la Investigación en la Universidad.     

Limit theorems of general functions of independent and identically distributed random variables

In this paper we derive limit theorems of some general functions of independent and identically distributed random variables. A stability property is used to derive the limit theory for general functions. A procedure followed in de Haan (1976) and Leadbetter et al. (1983) is used to prove the main result. The limit theorems for the maximum, minimum and sum of fixed sample sizes and random sample sizes are derived as special cases of the main result.     

The value distribution of arithmetic functions and maximum of independent random variables

Vilnius University. Published in Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 31, No. 1, pp. 180–186, January–March, 1991.     

A local limit theorem for independent random variables

A local limit theorem is given for independent noninteger random variables under a condition which is more general than one previously given, and which reduces, in the case of identically distributed random variables, to a well-known result.     

Inequalities for the distribution of a sum of functions of independent random variables

Let where ₁,..., n are independent random variables and the are functions (e.g., taking the values 0 and 1). For cases when almost all the summands forming are equal to 0 with a probability close to 1, estimates from above and below are obtained for the quantity P{=0}, as well as upper estimates for the distance in variation between the distribution , and the distribution of the approximating sum of independent random variables.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 745–758, November, 1977.The author is grateful to V. G. Mikhailov for numerous discussions of the results of this paper and for his help in carrying out the tedious auxiliary calculations.