Almost Sure Invariance Principle Research Articles

Large deviations, moment estimates and almost sure invariance principles for skew products with mixing base maps and expanding-on-average fibers

AbstractIn this paper we show how to apply classical probabilistic tools for partial sums $\sum _{j=0}^{n-1}\varphi \circ \tau ^j$ generated by a skew product $\tau $ , built over a sufficiently well-mixing base map and a random expanding dynamical system. Under certain regularity assumptions on the observable $\varphi $ , we obtain a central limit theorem (CLT) with rates, a functional CLT, an almost sure invariance principle (ASIP), a moderate-deviations principle, several exponential concentration inequalities and Rosenthal-type moment estimates for skew products with $\alpha $ -, $\phi $ - or $\psi $ -mixing base maps and expanding-on-average random fiber maps. All of the results are new even in the uniformly expanding case. The main novelty here (in contrast to [2]) is that the random maps are not independent, they do not preserve the same measure and the observable $\varphi $ depends also on the base space. For stretched exponentially ${\alpha }$ -mixing base maps our proofs are based on multiple correlation estimates, which make the classical method of cumulants applicable. For $\phi $ - or $\psi $ -mixing base maps, we obtain an ASIP and maximal and concentration inequalities by establishing an $L^\infty $ convergence of the iterates ${\mathcal K}^{\,n}$ of a certain transfer operator ${\mathcal K}$ with respect to a certain sub- ${\sigma }$ -algebra, which yields an appropriate (reverse) martingale-coboundary decomposition.

A Vector-Valued Almost Sure Invariance Principle for Random Expanding on Average Cocycles

We obtain a quenched vector-valued almost sure invariance principle (ASIP) for random expanding on average cocycles. This is achieved by combining the adapted version of Gouëzel’s approach for establishing ASIP (developed in Dragičević and Hafouta in Nonlinearity 34:6773–6798, 2021) and the recent construction of the so-called adapted norms (carried out in Dragičević and Sedro in Quenched limit theorems for expanding on average cocycles. arXiv:2105.00548 , 2021), which in some sense eliminate the non-uniformity of the decay of correlations. For real-valued observables, we also show that the martingale approximation technique is applicable in our setup, and that it yields the ASIP with better error rates. Finally, we present an example showing the necessity of the scaling condition (12), answering a question of Dragičević and Sedro in Quenched limit theorems for expanding on average cocycles. arXiv:2105.00548 , 2021.

An almost sure invariance principle for some classes of non-stationary mixing sequences

In this note we (in particular) prove an almost sure invariance principle (ASIP) for non-stationary and uniformly bounded sequences of random variables which are exponentially fast ϕ-mixing. The obtained rate is of order o(Vn14+δ) for an arbitrary δ>0, where Vn is the variance of the underlying partial sums Sn. For certain classes of inhomogeneous Markov chains we also prove a vector-valued ASIP with similar rates.

Almost sure invariance principle for random dynamical systems via Gouëzel's approach

We extend the spectral approach of S Gouëzel for the vector-valued almost sure invariance principle (ASIP) to certain classes of non-stationary sequences with a weaker control over the behaviour of the covariance matrices, assuming only linear growth. Then we apply this extension to obtain a quenched vector-valued ASIP for random perturbations of a fixed Anosov diffeomorphism as well as random perturbations of a billiard map associated to the periodic Lorentz gas. We also consider certain classes of random piecewise expanding maps.

Sharp Second-Order Pointwise Asymptotics for Lossless Compression with Side Information.

The problem of determining the best achievable performance of arbitrary lossless compression algorithms is examined, when correlated side information is available at both the encoder and decoder. For arbitrary source-side information pairs, the conditional information density is shown to provide a sharp asymptotic lower bound for the description lengths achieved by an arbitrary sequence of compressors. This implies that for ergodic source-side information pairs, the conditional entropy rate is the best achievable asymptotic lower bound to the rate, not just in expectation but with probability one. Under appropriate mixing conditions, a central limit theorem and a law of the iterated logarithm are proved, describing the inevitable fluctuations of the second-order asymptotically best possible rate. An idealised version of Lempel-Ziv coding with side information is shown to be universally first- and second-order asymptotically optimal, under the same conditions. These results are in part based on a new almost-sure invariance principle for the conditional information density, which may be of independent interest.

Non-stationary Almost Sure Invariance Principle for Hyperbolic Systems with Singularities

We investigate a wide class of two-dimensional hyperbolic systems with singularities, and prove the almost sure invariance principle (ASIP) for the random process generated by sequences of dynamically H\"older observables. The observables could be unbounded, and the process may be non-stationary and need not have linearly growing variances. Our results apply to Anosov diffeomorphisms, Sinai dispersing billiards and their perturbations. The random processes under consideration are related to the fluctuation of Lyapunov exponents, the shrinking target problem, etc.

Asymptotic self-similarity and order-two ergodic theorems for renewal flows

We prove a log average almost-sure invariance principle (abbreviated log asip) for renewal processes with positive i.i.d. gaps in the domain of attraction of an α-stable law with 0 < α < 1. Dynamically, this means that renewal and Mittag-Leffler paths are forward asymptotic in the scaling flow, up to a time average. This strengthens the almost-sure invariance principle in log density we proved in [FT11]. The scaling flow is a Bernoulli flow on a probability space. We study a second flow, the increment flow, transverse to the scaling flow, which preserves an infinite invariant measure constructed using singular cocycles. A cocycle version of the Hopf Ratio Ergodic Theorem leads to an order-two ergodic theorem for the Mittag-Leffler increment flow. Via the log asip, this result then passes to a second increment flow, associated to the renewal process. As corollaries, we have new proofs of theorems of [ADF92] and of [CE51], motivated by fractal geometry.

Statistical Properties of Lorenz-like Flows, Recent Developments and Perspectives

We comment on the mathematical results about the statistical behavior of Lorenz equations and its attractor, and more generally on the class of singular hyperbolic systems. The mathematical theory of such kind of systems turned out to be surprisingly difficult. It is remarkable that a rigorous proof of the existence of the Lorenz attractor was presented only around the year 2000 with a computer-assisted proof together with an extension of the hyperbolic theory developed to encompass attractors robustly containing equilibria. We present some of the main results on the statistical behavior of such systems. We show that for attractors of three-dimensional flows, robust chaotic behavior is equivalent to the existence of certain hyperbolic structures, known as singular-hyperbolicity. These structures, in turn, are associated with the existence of physical measures: in low dimensions, robust chaotic behavior for flows ensures the existence of a physical measure. We then give more details on recent results on the dynamics of singular-hyperbolic (Lorenz-like) attractors: (1) there exists an invariant foliation whose leaves are forward contracted by the flow (and further properties which are useful to understand the statistical properties of the dynamics); (2) there exists a positive Lyapunov exponent at every orbit; (3) there is a unique physical measure whose support is the whole attractor and which is the equilibrium state with respect to the center-unstable Jacobian; (4) this measure is exact dimensional; (5) the induced measure on a suitable family of cross-sections has exponential decay of correlations for Lipschitz observables with respect to a suitable Poincaré return time map; (6) the hitting time associated to Lorenz-like attractors satisfy a logarithm law; (7) the geometric Lorenz flow satisfies the Almost Sure Invariance Principle (ASIP) and the Central Limit Theorem (CLT); (8) the rate of decay of large deviations for the volume measure on the ergodic basin of a geometric Lorenz attractor is exponential; (9) a class of geometric Lorenz flows exhibits robust exponential decay of correlations; (10) all geometric Lorenz flows are rapidly mixing and their time-1 map satisfies both ASIP and CLT.

Spectral analysis of hyperbolic systems with singularities

We study the statistical properties of a general class of two-dimensional hyperbolic systems with singularities by constructing Banach spaces on which the associated transfer operators are quasi-compact. When the map is mixing, the transfer operator has a spectral gap and many related statistical properties follow, such as exponential decay of correlations, the central limit theorem, the identification of Ruelle resonances, large deviation estimates and an almost-sure invariance principle. To demonstrate the utility of this approach, we give two applications to specific systems: dispersing billiards with corner points and the reduced maps for certain billiards with focusing boundaries.

Dynamical attraction to stable processes

En appliquant des idees venues des systemes dynamiques aux probabilites, nous prouvons un principe d’invariance presque sur au sens de la densite logarithmique pour des processus stables. L’auto-similarite d’un processus stable revet une expression plus forte, celle de la Bernoullicite du flot d’echelle agissant sur l’espace de Skorokhod des trajectoires. Nous montrons qu’il existe un couplage de la marche aleatoire a accroissements i.i.d. dans le domaine d’attraction d’une loi stable et d’un processus stable tel que presque surement, apres un changement de temps deterministe et a variation reguliere, sous l’action du flot d’echelle, les deux processus soient asymptotiques dans le futur sauf pour un ensemble de temps de densite nulle. Il en decoule que presque toute marche (a un changement de temps pres) est un point generique du flot. Dans le cas brownien, compte-tenu de resultats bien connus dans la litterature, nous avons un resultat plus fort : sous l’action du flot, les trajectoires de la marche et du brownien sont asymptotiques dans le futur avec une vitesse exponentielle donnee par l’hypothese de moment.

Spectral analysis of the transfer operator for the Lorentz gas

We study the billiard map associated with both the finite- andinfinite-horizon Lorentz gases having smooth scatterers with strictlypositive curvature. We introduce generalized function spaces (Banach spacesof distributions) on which the transfer operator is quasicompact. Themixing properties of the billiard map then imply the existence of aspectral gap and related statistical properties such as exponential decayof correlations and the Central Limit Theorem. Finer statistical propertiesof the map such as the identification of Ruelle resonances, large deviationestimates and an almost-sure invariance principle follow immediately oncethe spectral picture is established.

Advanced Statistical Properties of Dispersing Billiards

A new approach to statistical properties of hyperbolic dynamical systems emerged recently; it was introduced by L.-S. Young and modified by D. Dolgopyat. It is based on coupling method borrowed from probability theory. We apply it here to one of the most physically interesting models—Sinai billiards. It allows us to derive a series of new results, as well as make significant improvements in the existing results. First we establish sharp bounds on correlations (including multiple correlations). Then we use our correlation bounds to obtain the central limit theorem (CLT), the almost sure invariance principle (ASIP), the law of iterated logarithms, and integral tests.