### Abstract

We obtain large deviations theorems for both discrete time expressions of the form ∑_{n=1}
^{N} F(X(q1(n)),..., X(qℓ(n))) and similar expressions of the form ∫_{0}
^{T} F(X(q1(t)),..., X(qℓ(t))) dt in continuous time. Here X(n),n≥ 0 or X(t), t≥ 0 is a Markov process satisfying Doeblin's condition, F is a bounded continuous function and qi(n) = in for i ≤ k while for i>k they are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when qi's are polynomials of increasing degrees. Applications to some types of dynamical systems such as mixing subshifts of finite type and hyperbolic and expanding transformations will be obtained, as well.

Original language | English (US) |
---|---|

Pages (from-to) | 197-224 |

Number of pages | 28 |

Journal | Probability Theory and Related Fields |

Volume | 158 |

Issue number | 1-2 |

DOIs | |

State | Published - 2014 |

### Fingerprint

### Keywords

- Hyperbolic diffeomorphisms
- Large deviations
- Markov processes
- Nonconventional averages

### ASJC Scopus subject areas

- Statistics and Probability
- Analysis
- Statistics, Probability and Uncertainty

### Cite this

*Probability Theory and Related Fields*,

*158*(1-2), 197-224. https://doi.org/10.1007/s00440-013-0481-4

**Nonconventional large deviations theorems.** / Kifer, Yuri; Varadhan, Srinivasa.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 158, no. 1-2, pp. 197-224. https://doi.org/10.1007/s00440-013-0481-4

}

TY - JOUR

T1 - Nonconventional large deviations theorems

AU - Kifer, Yuri

AU - Varadhan, Srinivasa

PY - 2014

Y1 - 2014

N2 - We obtain large deviations theorems for both discrete time expressions of the form ∑n=1 N F(X(q1(n)),..., X(qℓ(n))) and similar expressions of the form ∫0 T F(X(q1(t)),..., X(qℓ(t))) dt in continuous time. Here X(n),n≥ 0 or X(t), t≥ 0 is a Markov process satisfying Doeblin's condition, F is a bounded continuous function and qi(n) = in for i ≤ k while for i>k they are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when qi's are polynomials of increasing degrees. Applications to some types of dynamical systems such as mixing subshifts of finite type and hyperbolic and expanding transformations will be obtained, as well.

AB - We obtain large deviations theorems for both discrete time expressions of the form ∑n=1 N F(X(q1(n)),..., X(qℓ(n))) and similar expressions of the form ∫0 T F(X(q1(t)),..., X(qℓ(t))) dt in continuous time. Here X(n),n≥ 0 or X(t), t≥ 0 is a Markov process satisfying Doeblin's condition, F is a bounded continuous function and qi(n) = in for i ≤ k while for i>k they are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when qi's are polynomials of increasing degrees. Applications to some types of dynamical systems such as mixing subshifts of finite type and hyperbolic and expanding transformations will be obtained, as well.

KW - Hyperbolic diffeomorphisms

KW - Large deviations

KW - Markov processes

KW - Nonconventional averages

UR - http://www.scopus.com/inward/record.url?scp=84892886489&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84892886489&partnerID=8YFLogxK

U2 - 10.1007/s00440-013-0481-4

DO - 10.1007/s00440-013-0481-4

M3 - Article

VL - 158

SP - 197

EP - 224

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 1-2

ER -