Weak convergence of sequences from fractional parts of random variables and applications. (English. Russian original) Zbl 1247.60032

Giuliano, Rita

Theory Probab. Math. Stat. 83, 59-69 (2011); translation from Teor. Jmovirn. Mat. Stat. 83, 49-58 (2010).

Let \((Y_n)_{n\geq 1}\) be a general sequence of real-valued random variables and set \(Z_n=Y_n ~(\text{mod 1}) ~(=\{Y_n\})\), the fractional part of \(Y_n,~n\geq 1\). The author is interested in providing conditions for the weak convergence to the uniform distribution on \([0,1]\) of the sequence \((Z_n)_{n\geq 1}\). As a first main result a “Weyl criterion” for probability laws on \(\mathbb{R}\) is derived, which gives a necessary and sufficient condition for the latter convergence in terms of the Fourier coefficients \(\hat\mu_n(h),~h\in\mathbb{Z}\), belonging to the law \(\mu_n\) of \(Y_n,~n\geq 1\). Some applications of this criterion are presented demonstrating that the result covers much more general situations than just the classical case of partial sums of i.i.d.random variables taking values in \([0,1]\). Two more results are proved providing sufficient conditions for the weak convergence of \((Z_n)_{n\geq 1}\) to the uniform distribution on \([0,1]\), both assuming that the densities of \(Y_n,~n\geq 1,\) exist, the first one then once again relying on the Weyl criterion, whereas the second one is based on a kind of “generalized unimodality” of the underlying densities.

Reviewer: Josef Steinebach (Köln)

MSC:

60F05	Central limit and other weak theorems
60G52	Stable stochastic processes
60G70	Extreme value theory; extremal stochastic processes
11K06	General theory of distribution modulo \(1\)
62G07	Density estimation
42A10	Trigonometric approximation
42A61	Probabilistic methods for one variable harmonic analysis

Keywords:

weak convergence; Weyl criterion; Fourier coefficient; characteristic function; fractional part; partial sum; sample maximum; uniform distribution; Central Limit Theorem; domain of attraction; stable density; stable law; unimodal density; Benford’s law

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