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Nonconventional limit theorems in discrete and continuous time via martingales. (English) Zbl 1304.60041

This paper is concerned with functional central limit theorems for expressions of the type \[ \xi_t^N:= N^{-1/2}\sum_{n=1}^{\left\lfloor Nt \right\rfloor} \left(F(X_{q_1(n)}, \dots, X_{q_{l}(n)}) - \bar{F}\right), \] where \((X_{n})_{n \geq 0}\) is a process with some stationary properties, moment conditions and a sufficiently fast mixing property, \(F\) is a continuous integer-valued function on integers with polynomial growth, \(q_i\) are integer-valued on integers with \(q_{i}(n) = in\) for \(i \leq k \leq n\) and some growth conditions, and \(\bar{F} = \mathcal E [F(X_0, \dots, X_0)]\). These results are inspired by nonconventional ergodic theorems in a similar setting and generalize results in [Y. Kifer. Probab. Theory Relat. Fields 148, No. 1–2, 71–106 (2010; Zbl 1205.60047)]. Moreover, the results are extended to central limit theorems for the continuous analogue \[ \xi_t^N:= N^{-1/2}\int_{0}^{\left\lfloor Nt \right\rfloor} \left(F(X_{q_1(t)}, \dots, X_{q_{l}(t)}) - \bar{F}\right)dt. \] The limits are Gaussian processes with stationary and not sufficiently independent increments. The proofs are based on martingale differences and a subtle construction of martingale approximations for the expressions above which overcomes the correlation structure due to \((X_{n})_{n \geq 0}\) and the \(q_i\). Examples of applications are \(X_i(n)= T^n f_i\), where \(f_i\) are bounded (and sufficiently regular) functions and \(T\) are some dynamical systems, including mixing shifts of finite type, hyperbolic diffeomorphisms, or some expanding transformations. Other examples are \(X_i(t)= f_i(Y_t)\), where \((Y_t)_{t \geq 0}\) are some finite state Markov chains or diffusion processes.

MSC:

60F17 Functional limit theorems; invariance principles
60G42 Martingales with discrete parameter
37D99 Dynamical systems with hyperbolic behavior
60G15 Gaussian processes

Citations:

Zbl 1205.60047
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References:

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