Limit theorems for weighted sums and stochastic approximation processes

Consistent variance estimators for certain stochastic processes are suggested using the fact that (weak or strong) invariance principles may be available. Convergence rates are also derived, the latter being essentially determined by the approximation rates in the corresponding invariance principles. As an application, a change point test in a simple AMOC renewal model is briefly discussed, where variance estimators possessing good enough convergence rates are required.

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 qi, i > k are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when qi’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.

Chapter 6 - Limiting null distributions

A successful method to describe the asymptotic behavior of various deterministic and stochastic processes such as asymptotically autonomous differential equations or stochastic approximation processes is to relate them to an appropriately chosen limit semiflow. Benaïm and Schreiber (2000) define a general class of such stochastic processes, which they call weak asymptotic pseudotrajectories and study their ergodic behavior. In particular, they prove that the weak* limit points of the empirical measures associated to such processes are almost surely invariant for the associated deterministic semiflow. Continuing a program started by Benaïm, Hofbauer and Sorin (2005), we generalize the ergodic results mentioned above to weak asymptotic pseudotrajectories relative to set-valued dynamical systems.

Widening the scope of an eigenvector stochastic approximation process and application to streaming PCA and related methods

A strictly stationary [formula omitted]-mixing process satisfying the central limit theorem but not the weak invariance principle

In this paper we shall refer to some problems involving stochastic calculus, diffusion approximation and Markov processes. Finally a problem of stochastic approximation in genetics systems is discussed.

7 - Random Limit Theorems via Strong Invariance Principles

We prove an estimate for the Prohorov-distance in the central limit theorem for strong mixing Banach space valued random variables. Using a recent variant of an approximation theorem of Berkes and Philipp (1979) we obtain as a corollary a strong invariance principle for absolutely regular sequences with error term \(t^{\tfrac{1}{2} - \gamma }\). For strong mixing sequences we prove a strong invariance principle with error term o((t log logt)1/2).

A strong invariance principle and some of its applications

The functional law of the iterated logarithm for the empirical process of some long-range dependent sequences

The statistical properties of endomorphisms under the assumption that the associated Perron–Frobenius operator is quasicompact are considered. In particular, the central limit theorem, weak invariance principle and law of the iterated logarithm for sufficiently regular observations are examined. The approach clarifies the role of the usual assumptions of ergodicity, weak mixing, and exactness. Sufficient conditions are given for quasicompactness of the Perron–Frobenius operator to lift to the corresponding equivariant operator on a compact group extension of the base. This leads to statistical limit theorems for equivariant observations on compact group extensions. Examples considered include compact group extensions of piecewise uniformly expanding maps (for example Lasota–Yorke maps), and subshifts of finite type, as well as systems that are nonuniformly expanding or nonuniformly hyperbolic.

If V is a Lyapunov function of an equation du/dt = u′ = Zu in a Banach space then asymptotic stability of an equilibrium point may be easily proved if it is known that sup(V′) < 0 on sufficiently small spheres centered at the equilibrium point. In this paper weak asymptotic stability is proved for a bounded infinitesimal generator Z under a weaker assumption V′ ≤ 0 (which alone implies ordinary stability only) if some observability condition, involving Z and the Frechet derivative of V′, is satisfied. The proof is based on an extension of LaSalle′s invariance principle, which yields convergence in a weak topology and uses a strongly continuous Lyapunov function. The theory is illustrated with an example of an integro‐differential equation of interest in the theory of chemical processes. In this case strong asymptotic stability is deduced from the weak one and explicit sufficient conditions for stability are given. In the case of a normal infinitesimal generator Z in a Hilbert space, strong asymptotic stability is proved under the following assumptions Z*+Z is weakly negative definite and Ker Z = {0}. The proof is based on spectral theory.

We consider a system of interacting diffusions on the integer lattice. By letting the mesh size go to zero and by using a suitable scaling, we show that the system converges (in a strong sense) to a solution of the stochastic heat equation on the real line. As a consequence, we obtain comparison inequalities for product moments of the stochastic heat equation with different nonlinearities.

On the central limit theorem for weakly dependent sequences with a decomposed strong mixing coefficient