A limit theorem for random sums modulo 1
Let \(X_{0},\dots ,X_{n},\dots\), \(n\in \mathbb{N}=\{ 1,2,\dots \}\) be a sequence of non-degenerate random variables taking values on the real line \(\mathbb{R}\) and defined on a probability space \(( \Omega ,\mathcal{F},P) .\) For \(\lambda \neq 0\) consider the residues \[ \Lambda _{n}( a) =\{ \lambda S_{n}+a\} =( \lambda S_{n}+a) \mod{1}=\lambda S_{n}+a-\lfloor \lambda S_{n}+a\rfloor ,\quad \lambda ,a\in \mathbb{R}, \] of the linear transform of \(S_{n}=X_{1}+\dots +X_{n},\) where \(\lfloor x\rfloor =\sup \{ k\in \mathbb{Z}:k\leq x<k+1\} ,\) \(\mathbb{Z}=\{ \dots ,-1,0,1,\dots \} \). Let \(\mathcal{U}_{0}\) denote the uniform distribution on the interval \([0,1)\cap \mathbb{R} \) while \(\mathcal{U}_{\frac{1}{m}}\) is the uniform distribution on \(\{ 0,\frac{1}{m},\dots ,\frac{m-1}{m}\} ,m\in \mathbb{N}\). Put \(\widehat{\mu }( \theta ) =E( e^{i\theta X_{0}}) \) and define \(\lambda ( \widehat{\mu }) \) via \(\lambda ( \widehat{\mu }) \cdot \inf \{ n\in \mathbb{N}: | \widehat{ \mu }( 2\pi n\lambda ) | =1\} =1\), where \(\lambda ( \widehat{\mu }) =0\) if such a \(n\) does not exist. If \( \lambda ( \widehat{\mu }) >0\) set \(b=\lambda ( \widehat{\mu } ) \text{Arg}(\widehat{\mu }( 2\pi ( \lambda ( \widehat{\mu }) ) ^{-1}\lambda )) \). It turns out that the law \(\mathcal{L}( \Lambda _{n}) \) is in the domain of attraction of \(\mathcal{U}_{\lambda ( \widehat{\mu }) }\) under condition \((\mathbf{N})\): there exist \(c\in (0,1]\cap \mathbb{R}\) and an integer \(q\geq 2\) such that \(| E[ e^{i\theta S_{n}} ] | \leq ( 1-c(1-| E[ e^{i\theta X_{0}} ] | ) )^{\lfloor \frac{n}{q}\rfloor },\;\theta \in \mathbb{R} ,n\in \mathbb{N}\). It is shown that if \(\lambda ( \widehat{\mu }) =0\) then \(\Lambda _{n}( a) \;\)converges weakly to \(\mathcal{U} _{0}\;\)while if \(\lambda ( \widehat{\mu }) >0\) and \(S_{n}\) are distributed on the same lattice as \(X_{0},\) then \(\Lambda _{n}( -nb\lambda ) \) converges weakly to \(\mathcal{U}_{\lambda ( \widehat{\mu }) }\).
scientific article; zbMATH DE number 5703976
scientific article