A Nonconventional Local Limit Theorem

Local limit theorems have their origin in the classical De Moivre–Laplace theorem, and they study the asymptotic behavior as $$N\rightarrow \infty $$ N → ∞ of probabilities having the form $$P\{ S_N=k\}$$ P { S N = k } where $$S_N=\sum ^N_{n=1}F(\xi _n)$$ S N = ∑ n = 1 N F ( ξ n ) is a sum of an integer-valued function F taken on i.i.d. or Markov-dependent sequence of random variables $$\{\xi _j\}$$ { ξ j } . Corresponding results for lattice-valued and general functions F were obtained, as well. We extend here this type of results to nonconventional sums of the form $$S_N=\sum _{n=1}^NF(\xi _n,\xi _{2n}, \ldots ,\xi _{\ell n})$$ S N = ∑ n = 1 N F ( ξ n , ξ 2 n , … , ξ ℓ n ) which continues the recent line of research studying various limit theorems for such expressions.