Estimates of semiinvariants and centered moments of stochastic processes with mixing. I (Q1116525)

Let \(X_ t\), \(t=1,2,..\). be a random process, defined on the probability space (\(\Omega\),\({\mathcal F},{\mathbb{P}})\). Denote by \(\Gamma_ K(S_ n)\) the K th order semiinvariant of the sum \(S_ n=X_ 1+...+X_ n\), and a mixed semiinvariant by \[ \Gamma (X_{t_ 1},...,X_{t_ k})=i^{- k}(\partial^ k/\partial u_ 1...\partial u_ k)\ln E e^{i(u,X)}|_{u=0}, \] where \(u\in R^ k\), \(X=(X_{t_ 1},...,X_{t_ k})\). This work is devoted to obtaining more exact upper estimates for \(\Gamma (X_{t_ 1},...,X_{t_ k})\) and \(\Gamma_ k(S_ n)\) in terms of \(\alpha\)-, \(\phi\)- and \(\psi\)-mixing functions. In this part all theorems and the bibliography are presented; for part II see the following review, Zbl 0666.60028.