Speed of convergence for laws of rare events and escape rates

We obtain error terms on the rate of convergence to Extreme Value Laws, and to the asymptotic Hitting Time Statistics, for a general class of weakly dependent stochastic processes. The dependence of the error terms on the ‘time’ and ‘length’ scales is very explicit. Specialising to data derived from a class of dynamical systems we find even more detailed error terms, one application of which is to consider escape rates through small holes in these systems.

[1]  J. Freitas.  Extremal behaviour of chaotic dynamics , 2013 .

[2]  G. Keller.  Rare events, exponential hitting times and extremal indices via spectral perturbation† , 2012, 1202.3900.

[3]  Matthew Nicol,et al.  Speed of convergence to an extreme value distribution for non-uniformly hyperbolic dynamical systems , 2015 .

[4]  M. R. Leadbetter,et al.  Extremes and Related Properties of Random Sequences and Processes: Springer Series in Statistics , 1983 .

[5]  B. Saussol,et al.  Hitting and returning into rare events for all alpha-mixing processes , 2010, 1003.4856.

[6]  O. Sarig.  Thermodynamic formalism for null recurrent potentials , 2001 .

[7]  Ana Cristina Moreira Freitas,et al.  Extreme Value Laws in Dynamical Systems for Non-smooth Observations , 2010, 1006.3276.

[8]  Richard L. Smith.  Uniform rates of convergence in extreme-value theory , 1982, Advances in Applied Probability.

[9]  Marek R Rychlik,et al.  Bounded variation and invariant measures , 1983 .

[10]  M. Pollicott,et al.  Escape rates for Gibbs measures , 2010, Ergodic Theory and Dynamical Systems.

[11]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[12]  O. Sarig.  Existence of gibbs measures for countable Markov shifts , 2003 .

[13]  Miguel Abadi,et al.  Sharp error terms and neccessary conditions for exponential hitting times in mixing processes , 2004 .

[14]  Rates of Convergence and Approximations to the Distribution of the Maximum of Chain-Dependent Sequences , 2001 .

[15]  Antonio Galves,et al.  Inequalities for the occurrence times of rare events in mixing processes. The state of the art , 2000 .

[16]  Tailen Hsing,et al.  Calculating the extremal index for a class of stationary sequences , 1991, Advances in Applied Probability.

[17]  Pierre Collet,et al.  Statistics of closest return for some non-uniformly hyperbolic systems , 1999, Ergodic Theory and Dynamical Systems.

[18]  Jorge Milhazes Freitas,et al.  On the link between dependence and independence in extreme value theory for dynamical systems , 2008 .

[19]  Miguel Abadi,et al.  Exponential approximation for hitting times in mixing processes. , 2001 .

[20]  M. R. Leadbetter,et al.  On Exceedance Point Processes for Stationary Sequences under Mild Oscillation Restrictions , 1989 .

[21]  Serguei Novak,et al.  On extreme values in stationary sequences , 1996 .

[22]  A. C. Freitas,et al.  The Compound Poisson Limit Ruling Periodic Extreme Behaviour of Non-Uniformly Hyperbolic Dynamics , 2012, 1204.2304.

[23]  Antonio Galves,et al.  Inequalities For Hitting Times In Mixing Dynamical Systems , 1997 .

[25]  B. Saussol,et al.  Absolutely continuous invariant measures for multidimensional expanding maps , 2000 .

[26]  D. Aldous.  Markov chains with almost exponential hitting times , 1982 .

[27]  Miguel Abadi,et al.  Sharp Error Terms for Return Time Statistics under Mixing Conditions , 2008, 0812.1016.

[28]  Sandro Vaienti,et al.  Statistics of Return Times:¶A General Framework and New Applications , 1999 .

[29]  S. Vaienti,et al.  The Rényi entropy function and the large deviation of short return times , 2009, Ergodic Theory and Dynamical Systems.

[30]  S. Vaienti,et al.  LARGE DEVIATIONS FOR SHORT RECURRENCE , 2008 .

[31]  Ana Cristina Moreira Freitas,et al.  Hitting time statistics and extreme value theory , 2008, 0804.2887.

[32]  R. Mauldin,et al.  Gibbs states on the symbolic space over an infinite alphabet , 2001 .

[33]  Lei Si Ni Ke Resnick.S.I..  Extreme values. regular variation. and point processes , 2011 .

[34]  S. Vaienti,et al.  Laws of rare events for deterministic and random dynamical systems , 2012, 1207.5188.

[35]  A. C. Freitas,et al.  The extremal index, hitting time statistics and periodicity , 2010, 1008.1350.

[36]  S. Troubetzkoy,et al.  Recurrence, Dimensions, and Lyapunov Exponents , 2001 .

[37]  S. Resnick.  Extreme Values, Regular Variation, and Point Processes , 1987 .

[38]  From rates of mixing to recurrence times via large deviations , 2010, 1007.3771.

[39]  Mark F. Demers,et al.  Escape rates and conditionally invariant measures , 2006 .

[40]  G. Keller,et al.  Rare Events, Escape Rates and Quasistationarity: Some Exact Formulae , 2008, 0810.2229.

[41]  The rate of convergence in law of the maximum of an exponential sample , 1979 .

[42]  D. Dolgopyat.  On decay of correlations in Anosov flows , 1998 .

[43]  Valentin Afraimovich,et al.  Local dimensions for Poincaré recurrences , 2000 .

[44]  Andrei Török,et al.  Extreme value theory for non-uniformly expanding dynamical systems , 2012 .