On a functional limit theorem for additive functions

Kh. Kh. Usmanov

Čebyševskij sbornik, Tome 15 (2014) no. 3, pp. 86-99.

Résumé

By means of additive arithmetic functions on a sequence of the shifted prime numbers the processes with realizations from a space of functions without ruptures of the second sort are based. In this space with a topology of Skorokhod and $\sigma$-algebra of the borelean multitudes a sequence of the measures corresponding to constructed arithmetic processes is entered. Exactly, the relative frequency of prime numbers is accepted to a measure of the borelean multitudes. These numbers don't surpass natural number of $n$ to which there a correspond realization of the constructed processes getting to this multitude. Necessary and sufficient conditions of weak convergence of sequence of the entered measures to the measure corresponding to a process are found. Thus process with the independent increments, which distributions are not expressed, is limited. Necessary and sufficient conditions represent two limit ratios the first of which is an infinite of a sequence of the set sums. The proofs of need of performance of this ratio for weak convergence of sequence of measures are the main part of all proofs of the theorem. This proof is carried out by consideration of distributions of increments of arithmetic processes on the intervals close to a unit and a transition to characteristic functions, corresponding to these distributions. Further, using an independence of increments of a limit process and a weak compactness of a sequence of measures (taken from Yu. Prokhorov's known theorem of weak convergence of probability measures), by an asymptotic formula for average values of multiplicative functions on sequence of the shifted prime numbers of N. Timofeev, we receive the first condition of the theorem. At the proof of sufficiency of both conditions for weak convergence of sequence of measures characteristic functions are applied again. It allows, in a particular, to use early the limit theorems received by the author in functional spaces for additive functions on “rare” multitudes. The sequence $\{p+1\}$ is included in a class of the sequences considered in these theorems. However, in them the condition similar to the first condition considered here, isn't necessary, but is sufficient. It allows, applying the specified theorems to a considered case to receive a weak convergence of sequence of measures. A representation for a characteristic function of a limit process is also received. Bibliography: 16 titles.

Détail

Keywords: additive function, characteristic function, stochastic process, measure.

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