An invariance principle for discrete-time nonlinear systems

In this paper, we derive an invariance principle generalizing LaSalle's invariance principle for discrete-time nonlinear systems. Next, using the invariance principle, we develop a series of results relating stability, observability, and converse Lyapunov theorems for discrete-time nonlinear systems.     

Invariance principles for the law of the iterated logarithm under G-framework

We obtain a general invariance principle of G-Brownian motion for the law of the iterated logarithm(LIL for short). For continuous bounded independent and identically distributed random variables in G-expectation space, we also give an invariance principle for LIL. In some sense, this result is an extension of the classical Strassen's invariance principle to the case where probability measure is no longer additive. Furthermore,we give some examples as applications.     

Entropic measure of the order-disorder character in lattice systems in the representation of coordination Cayley tree graphs

We systematically expound the infodynamical method for analyzing lattice and grid systems. We establish the logic and algorithm for mapping given objects to coordination Cayley tree graphs and present their main properties. Tree graphs of grid systems are complicated objects, and the principle of cluster-type simplicial decomposition can be used to study them. Based on a simplicial decomposition, we construct the enumerating structures, from which we construct entropy-type functionals. We pose the percolation problem on Cayley tree graphs in a nonconventional sense, which may be considered for both enumerating structures and their entropies. The corresponding entropy percolational dependences and their critical indices can be considered sufficiently universal measures of order in lattice systems. The simpliciality also implies an analogy with the fractality principle. We introduce three types of fractal characteristics and give analytic expressions for fractal dimensions for the tangential and streamer representations and for the Mandelbrot shell.     

A strong invariance principle for nonconventional sums

In Kifer and Varadhan (Nonconventional limit theorems in discrete and continuous time via martingales, 2010) we obtained a functional central limit theorem (known also as a weak invariance principle) for sums of the form ${\sum_{n=1}^{[Nt]} F\big(X(n), X(2n), .\, .\, .\, .\, X(kn), X(q_{k+1}(n)), X(q_{k+2}(n)), .\, .\, .\, , X(q_\ell(n))\big)}$ (normalized by ${1/\sqrt N}$ ) where X(n), n ≥ 0 is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polynomial growth and certain regularity properties and q _i , i > k are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when q _i ’s are polynomials of growing degrees. This paper deals with strong invariance principles (known also as strong approximation theorems) for such sums which provide their uniform in time almost sure approximation by processes built out of Brownian motions with error terms growing slower than ${\sqrt N}$ . This yields, in particular, an invariance principle in the law of iterated algorithm for the above sums. Among motivations for such results are their applications to multiple recurrence for stochastic processes and dynamical systems as well, as to some questions in metric number theory and they can be considered as a natural follow up of a series of papers dealing with nonconventional ergodic averages.     

Invariance principles for the law of the iterated logarithm under <Emphasis Type="Italic">G</Emphasis>-framework

We obtain a general invariance principle of G-Brownian motion for the law of the iterated logarithm (LIL for short). For continuous bounded independent and identically distributed random variables in G-expectation space, we also give an invariance principle for LIL. In some sense, this result is an extension of the classical Strassen’s invariance principle to the case where probability measure is no longer additive. Furthermore, we give some examples as applications.     

A new strong invariance principle for sums of independent random vectors

We provide a strong invariance principle for sums of independent, identically distributed random vectors that need not have finite second absolute moments. Various applications are indicated. In particular, we show how one can re-obtain some recent LIL type results from this invariance principle. Bibliography: 16 titles.     

Central limit theorem for linear processes generated by IID random variables under the sub-linear expectation

In this paper, we investigate the central limit theorem and the invariance principle for linear processes generated by a new notion of independently and identically distributed(IID) random variables for sub-linear expectations initiated by Peng [19]. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov's central limit theorem and invariance principle to the case where probability measures are no longer additive.     

Weighted Invariance Principle for Banach Space-Valued Random Variables

A weighted weak invariance principle for nonseparable Banach space-valued functions is described via asymptotic behavior of a weighted Wiener process. It is proved that, unlike the usual weak invariance principle, the weighted variant cannot be characterized via validity of a central limit theorem in a Banach space. A strong invariance principle is introduced in the present context and used to prove the weighted weak invariance principle that we seek herewith. The result then is applied to empirical processes.     

The Invariance Principle for Linear Processes Generated by a Negatively Associated Sequence and Its Applications

In this paper, we obtain the invariance principle for linear processes generated by a negatively associated sequence.     

The invariance principle for ϕ-mixing sequences

Summary In this paper we investigate the invariance principle for -mixing sequences, satisfying restrictions on the variances which are a weak form of stationarity. No mixing rate is assumed. For -mixing strictly stationary sequences we give a necessary and sufficient condition for the invariance principle.     

Discontinuous local semiflows for Kurzweil equations leading to LaSalle's invariance principle for differential systems with impulses at variable times

In this paper, we consider an initial value problem for a class of generalized ODEs, also known as Kurzweil equations, and we prove the existence of a local semidynamical system there. Under certain perturbation conditions, we also show that this class of generalized ODEs admits a discontinuous semiflow which we shall refer to as an impulsive semidynamical system. As a consequence, we obtain LaSalle's invariance principle for such a class of generalized ODEs. Due to the importance of LaSalle's invariance principle in studying stability of differential systems, we include an application to autonomous ordinary differential systems with impulse action at variable times.     

Convergence of a class of discrete-time semiflows with application to neutral delay differential equations

In this paper, we introduce a class of pseudo-monotone maps on ordered topological spaces. By exploiting monotonicity methods and the invariance of the omega limit set, we establish a convergence principle for discrete-time semiflows generated by the maps introduced. The convergence principle is then applied to a class of periodic neutral delay differential equations, which leads to some novel and sharper results.     

Invariance principles for iterated maps that contract on average

We consider iterated function schemes that contract on average. Using a transfer operator approach, we prove a version of the almost sure invariance principle. This allows the system to be modelled by a Brownian motion, up to some error term. It follows that many classical statistical properties hold for such systems, such as the weak invariance principle and the law of the iterated logarithm.

     

Strassen-type invariance principles for exchangeable sequences

We prove strong invariance principles for exchangeable sequences of real random variables. Under the same conditions as those used for the central limit theorem, we obtain Strassen's invariance principle.     

On stochastic and chaotic motion

In this paper we prove results regarding certain precise relationships between random motion and chaotic motion. In particular we prove a strong invariance principle for smooth functions of certain chaotic dynamical systems, and show that solutions of dynamical systems which are coupled to such chaotic systems may be approximated by solutions of stochastic differential equations     

Strong approximation of partial sums under dependence conditions with application to dynamical systems

In this paper, we obtain precise rates of convergence in the strong invariance principle for stationary sequences of real-valued random variables satisfying weak dependence conditions including strong mixing in the sense of Rosenblatt (1956) [30] as a special case. Applications to unbounded functions of intermittent maps are given.     

A useful estimate in the multidimensional invariance principle

Summary An estimate of the convergence speed in the multidimensional invariance principle is obtained. Using this estimate, we can prove strong invariance principles for partial sums of independent not necessarily identically distributed multidimensional random vectors.     

Asymptotically efficient estimation of the sparsity function at a point

The sparsity function is important in nonparametric inference based on order statistics. In this paper, we consider kernel estimation of the sparsity function. We establish an invariance principle for the kernel estimator and then construct a simple adaptive estimator which we show is asymptotically efficient in the mean squared error sense.     

Hölderian Invariance Principle for Triangular Arrays of Random Variables

In this paper, we extend the Hölderian invariance principle of Lamperti [6] to the case of partial-sum processes based on a triangular array of row-wise independent random variables. As an application, we obtain necessary and sufficient conditions for the almost sure (resp. in probability) weak Hölder convergence of partial-sum processes based on bootstrapped samples.