Weak Invariance Principle Research Articles

Weak Invariance Principle for the Local Times of Partial Sums of Markov Chains

Let {X n } be an integer-valued Markov chain with finite state space. Let $S_{n}=\sum_{k=0}^{n}X_{k}$ and let L n (x) be the number of times S k hits x∈ℤ up to step n. Define the normalized local time process l n (t,x) by The subject of this paper is to prove a functional weak invariance principle for the normalized sequence l n (t,x), i.e., we prove under the assumption of strong aperiodicity of the Markov chain that the normalized local times converge in distribution to the local time of the Brownian motion.

Invariance Principles for Products of U-Statistics Without Variance

Let be a U-statistic based on a kernel function h(x 1, x 2) and independent and identically distributed samples {X n ; n ≥ 1}. The author obtained a weak invariance principle and a strong functional limit theorem for the products of when may be infinite. Moreover, the author also applies the results to the sample variance and the Gini's mean difference, and obtain the limiting properties of their products.

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 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.

Limit Theorems for Dispersing Billiards with Cusps

Dispersing billiards with cusps are deterministic dynamical systems with a mild degree of chaos, exhibiting “intermittent” behavior that alternates between regular and chaotic patterns. Their statistical properties are therefore weak and delicate. They are characterized by a slow (power-law) decay of correlations, and as a result the classical central limit theorem fails. We prove that a non-classical central limit theorem holds, with a scaling factor of \({\sqrt{n\log n}}\) replacing the standard \({\sqrt{n}}\) . We also derive the respective Weak Invariance Principle, and we identify the class of observables for which the classical CLT still holds.

Sequential testing of gradual changes in the drift of a stochastic process

In this paper, some sequential monitoring procedures are constructed and analyzed for detecting a “gradual” change in the drift parameter of a general stochastic process satisfying a certain (weak) invariance principle. It is shown that the tests can be constructed such that the “false alarm rate” attains a prescribed level (say) α and that the tests have “asymptotic power 1”. A more precise analysis of the procedures under the alternative proves that the stopping times, suitably normalized, have a standard normal limiting distribution. A few results from a small simulation study are also presented in order to give an idea of the finite sample behaviour of the suggested procedures.

SELF-NORMALIZED WEAK LIMIT THEOREMS FOR A ø-MIXING SEQUENCE

Let {<TEX>$X_j,\;j\geq1$</TEX>} be a strictly stationary <TEX>$\phi$</TEX>-mixing sequence of non-degenerate random variables with <TEX>$EX_1$</TEX> = 0. In this paper, we establish a self-normalized weak invariance principle and a central limit theorem for the sequence {<TEX>$X_j$</TEX>} under the condition that L(x) := <TEX>$EX_1^2I{|X_1|{\leq}x}$</TEX> is a slowly varying function at <TEX>$\infty$</TEX>, without any higher moment conditions.

Weak invariance principle for mixing sequences in the domain of attraction of normal law

In this article we prove a weak invariance principle for a strictly stationary φ -mixing sequence { Xj } j≧1 , whose truncated variance function \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$L(x): = EX_1^2 1_{\{ |X_1 | \leqq _x \} }$$ \end{document} is slowly varying at ∞ and mixing coefficients satisfy the logarithmic growth condition: Σ n ≧1φ1/2 (2 n ) &lt; ∞. This will be done under the condition that \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop {\lim }\limits_n Var\left( {\sum\limits_{j = 1}^n {\hat X_j } } \right)/\left[ {\sum\limits_{j = 1}^n {Var (\hat X_j )} } \right] = \beta ^2$$ \end{document} exists in (0, ∞), where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\hat X_j = X_j I_{\{ |X_j | \leqq \eta _j \} }$$ \end{document} and ηn2 ∼ nL ( ηn ).

A weak invariance principle for self-normalized products of sums of mixing sequences

Let {X,X n, n ≥ 1} be a sequence of strictly stationary φ-mixing positive random variables in the domain of attraction of the normal law. Under some suitable conditions the weak invariance principle for self-normalized products of partial sums is obtained.

Comparison between criteria leading to the weak invariance principle

Le but de cet article est de comparer différents critères conduisant au théoreme limite centrale et au principe d’invariance faible. Ces critères sont la décomposition martingale-cobord développée par Gordin dans Dokl. Akad. Nauk SSSR 188 (1969), le critère projectif introduit par Dedecker dans Probab. Theory Related Fields 110 (1998), par la suite amélioré par Dedecker et Rio dans Ann. Inst. H. Poincaré Probab. Statist. 36 (2000) et la condition introduite par Maxwell et Woodroofe dans Ann. Probab. 28 (2000), plus tard améliorée par Peligrad et Utev dans Ann. Probab. 33 (2005). On montre que dans tout système dynamique ergodique d’entropie strictement positive, si l’on considère deux de ces critères, on peut trouver une fonction dans $\mathbb{L}^{2}$ vérifiant le premier mais pas le deuxième.

On the Weak Invariance Principle for Non-Adapted Sequences under Projective Criteria

In this paper we study the central limit theorem and its weak invariance principle for sums of non-adapted stationary sequences, under different normalizations. Our conditions involve the conditional expectation of the variables with respect to a given σ-algebra, as done in Gordin (Dokl. Akad. Nauk SSSR 188, 739–741, 1969) and Heyde (Z. Wahrsch. verw. Gebiete 30, 315–320, 1974). These conditions are well adapted to a large variety of examples, including linear processes with dependent innovations or regular functions of linear processes.

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.

On the Weak Invariance Principle for Stationary Sequences under Projective Criteria

In this paper, we study the central limit theorem and its weak invariance principle for sums of a stationary sequence of random variables, via a martingale decomposition. Our conditions involve the conditional expectation of sums of random variables with respect to the distant past. The results contribute to the clarification of the central limit question for stationary sequences.

Estimation of the stationary distribution of semi-Markov processes with Borel state space

We present an empirical estimator of the stationary distribution of continuous time semi-Markov processes with Borel state space. It comes as a particular case of an estimator of a linear functional of the stationary distribution. We give asymptotic results for strong consistency, and the weak and strong invariance principles.

Recent advances in invariance principles for stationary sequences

In this paper we survey some recent results on the central limit theorem and its weak invariance principle for stationary sequences. We also describe several maximal inequalities that are the main tool for obtaining the invariance principles, and also they have interest in themselves. The classes of dependent random variables considered will be martingale-like sequences, mixing sequences, linear processes, additive functionals of ergodic Markov chains.

Weak Invariance Principles for Regression Rank Statistics

Weak invariance principles are proved for regression rank statistics. As a consequence limit theorems for max- and L p -functionals of partial sums of vectors of simple linear rank statistics are obtained. The results are useful in change point analysis, particularly in justification of application of permutation arguments, see Antoch and Hušková [Antoch, J.; Hušková, M. Detection of Structural Changes in Regression. Tatra Mountains Publications, 2003, 26, 1–15] and Hušková and Picek [Hušková, M.; Picek, J. M-tests for detection of structural changes in regression. In Statistical Data Analysis Based on the L 1-Norm and Related Methods; Dodge, Y., Ed.; Birhäuser: Basel, 2002; 213–229]. The results of Hušková [Hušková, M. Limit theorems for rank statistics. Statist. Probab. Letters 1997, 32, 45–55] are generalized.

FLUCTUATIONS OF THE METRIC ENTROPY FOR MIXING MEASURES

It is known from the theorem of Shannon–McMillan–Breiman that the measure of cylinder sets decays in the limit exponentially with a rate which is given by the metric entropy. Here we prove that the measure of cylinders satisfies a Central Limit Theorem for a wide class of mixing measures which do not necessarily have the Gibbs property. We, moreover, provide the rate of convergence (which is algebraic). As a consequence we can then also prove that the distribution of the first return time in cylinder sets is log-normally distributed and give the speed of convergence. We also show that the weak invariance principle and the law of the iterated logarithm hold for the convergence to the entropy in the Shannon–McMillan–Breiman and for the distribution of the first return time.

STATISTICAL PROPERTIES OF ENDOMORPHISMS AND COMPACT GROUP EXTENSIONS

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.

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.

STATISTICAL LIMIT LAWS FOR EQUIVARIANT OBSERVATIONS

We show that statistical limit laws for ergodic stationary sequences of G-equivariant observations ϕ on a probability space Ω×G are inherited by sequences of observations ϕ(·,g0) on the probability space Ω for each fixed g0. The statistical limit laws we consider are the central limit theorem, weak invariance principle and the law of the iterated logarithm.

Delay time in sequential detection of change

We consider a sequential test procedure which detects possible changes in the mean of observations satisfying a weak invariance principle. Our test statistic is based on weighted CUSUMs of the underlying random variables. In this paper, we study the asymptotic behaviour of the delay time if a change has occurred in the sample after a training period of size m in which the observations stay in control. It turns out that in this situation the limiting distribution of the delay time for m→∞ is normal under a suitable standardization provided the change appeared sufficiently soon after m.