Large deviations for products of empirical measures of dependent sequences (Q2758437)

Let \(\{X_i\}_{i\in \text{N}}\) be a sequence of random variables taking values in a Polish space \(S\). The statistics NEWLINE\[NEWLINE L_n^{\otimes m}={1\over {n^m}}\sum_{i_1,\dots , i_m =1}^n \delta (X_{i_1}, \dots , X_{i_m}),\quad L_n^{ m}={1\over {n_{(m)}}}\sum_{(i_1,\dots , i_m)\in I_{m,n} } \delta (X_{i_1}, \dots , X_{i_m}) NEWLINE\]NEWLINE are considered. Here \(\delta (s_1, \dots , s_m)\subset S^m\) denotes the probability measure concentrated at \((s_1, \dots , s_m)\in\)\(S^m\), \(n_{(m)}=\prod_{k=1}^{m-1}(n-k)\), \(I_{m,n}\subset \{1,\dots , n\}^m \) is the set of all \(m\)-tuples with pairwise different components. The large deviations inequalities for statistics \(L_n^{\otimes m}\), \(L_n^{ m}\) are proved. The sequence \(\{X_i\}_{i\in \text{N}}\) can be a special Markov chain, an exchangeable sequence, mixing sequence, a sequence of independent, but not identically distributed variables.