A nonconventional local limit theorem (Q501827)

Let \({\xi _0},{\xi _1},{\xi _2}\dots\) be a stationary Markov chain evolving on a Polish space. Let also \(\mu \) be its stationary distribution and let \(F\) be a function such that \(\int {F({x_1},\dots,{x_l})d\mu ({x_1})\dots d\mu ({x_l})} = 0\) and \(\int {{F^2}({x_1},\dots,{x_l})d\mu ({x_1})\dots d\mu ({x_l})}< \infty \). The paper establishes a local limit theorem for nonconventional sums of the form \({S_N} = \sum\nolimits_{n = 1}^N {F({\xi _n},{\xi _{2n}},\dots,{\xi _{mn}})} \).