Nonconventional large deviations theorems (Q2447285)

Let \(X(0), X(1), X(2), \dots{}\) be a Markov chain defined on a probability space \((\Omega, {\mathcal F}, P)\) and evolving on a Polish measurable space \((M, {\mathcal B})\) as its phase space. The authors assume a ``strong'' Doeblin condition saying for some integer \(n_0\), a constant \(C>0\) and a probability measure \(\nu\) on \(M\), the \(n_0\)-step transition probability \(P(n_0,x,\cdot)\) of the above Markov chain \(X\) satisfies \[ C^{-1}\nu(G)\leq P(n_0,x,\dot)\leq C \nu(G) \] for any \(x\in M\) and every measurable set \(G\subset M\). Let \(F=F(x_1,x_2,\dots,x_l)\) be a bounded measurable function on the \(l\)-times product space \(M^l=M\times\dots\times M\). The setup becomes complete with the introduction of positive increasing functions \(q_j\), \(j=1,\dots,l\) taking integer values on integers and such that \( q_1(n)=n \) while for \(j=2,\dots,l\) and any \(\gamma>0\), \[ \lim_{n\to \infty}(q_j(n)-q_j(n-1))=\infty\, \text{and}\, \liminf_{n\to \infty}(q_j(\gamma n)-q_{j-1}(n-1))>0. \] As usual, one denotes by \(P_x\) the probability conditioned to \(X(0)=0\) and by \(E_x\) the corresponding expectation. For any function \(W\) on \(M^l\), one denotes by \(\hat{W}\) the function on \(M\) defined by \[ \hat{W}:=\int \exp{(W(x,x_2,\dots,x_l))}\,\mu(dx_2)\dots\mu(dx_l), \] where \(\mu\) is a unique invariant measure of the Markov chain \(X\). The authors obtain large deviations theorems for both discrete time expressions of the form \[ S_N=S_N(F):=\sum_{n=1}^{N}F(q_1(n),\dots,X(q_l(n))) \] and similar expressions of the form \[ \int_0^T F(q_1(t),\dots,X(q_l(t)))\, dt \] in continuous time. The first result (for discrete time) is: Theorem. Let \(W_\lambda(x_1,\dots,x_l)\), \(\lambda\in (0,\infty)\) be a family of bounded measurable functions on \(M^l\) which is differentiable in \(\lambda\), such that \(dW_\lambda(x_1,\dots,x_l)/d\lambda\) is bounded for each \(\lambda\) as well. Then, for any \(x\in M\), the limit \[ Q(W_\lambda)=\lim_{N\to \infty}\frac{1}{N}\ln E_x \exp(\sum_{n=1}^{N} W_\lambda(X(q_1(n)),\dots,X(q_l(n)))) \] exists, it is independent of \(x\) and it is differentiable in \(\lambda\). In fact, \(Q(W_\lambda)=\ln r(W_\lambda)\), where \(r(W)\) is the spetral radius of the positive operator \(R(W)\) acting by \[ R(W)g(x):=\int P(x,dy)g(y)\hat{W}(y). \] Furthermore, set \(W_\lambda(x_1,\dots,x_l)=\lambda F(x_1,\dots,x_l)\) and \[ J(u):=\sup_{\lambda} \{\lambda u-r(W_\lambda)\},~~~u\in {\mathbb{R}}. \] Then, for any closed set \(K\subset {\mathbb{R}}\), \[ \limsup_{N\to \infty}\frac{1}{N}\ln P\{\frac{1}{N}S_n\in K\}\leq -\inf_{u\in K}J(u) \] and, for any open set \(U\subset {\mathbb{R}}\), \[ \limsup_{N\to \infty}\frac{1}{N}\ln P\{\frac{1}{N}S_n\in U\}\geq -\inf_{u\in U}J(u). \] Next, the authors obtain the second level of large deviations in the nonconvential setup which deals with occupational measures \(\zeta_N\) on \(M^l\) given by \[ \zeta_N=\frac{1}{N} \sum_{n=1}^{N} \delta_{(X(q_1(n)),\dots{},X(q_l(n)))}. \] Applications to some types of dynamical systems such as mixing subshifts of finite type and hyperbolic and expanding transformations are obtained as well.