| Abstract: |
| The study of nonconventional sums $S_N=\sum_{n=1}^NF(X(n),X(2n),...,X(\ell n))$, where $X(n)=g\circ T^n$
for a measure preserving transformation $T$, has a 40 years history after Furstenberg showed that they
are related to the ergodic theory proof of Szemer\` edi`s theorem about arithmetic progressions in sets
of integers of positive density. Recently, it turned out that various limit theorems of probability
theory can be successfully studied for sums $S_N$ when $X(n),\, n=1,2,...$ are weakly dependent random
variables. I will talk about a more general situation of nonconventional arrays of the form $S_N=
\sum_{n=1}^NF(X(p_1n+q_1N),X(p_2n+q_2N),...,X(p_\ell n+q_\ell N))$ and how this is related to an extended version of Szemer\` edi`s theorem. I`ll discuss also ergodic and limit theorems for such and more general nonconventional arrays. |
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