STATISTICAL PROPERTIES OF ENDOMORPHISMS AND COMPACT GROUP EXTENSIONS

A Non-Uniform Estimate for the Convergence Speed in the Multi-Dimensional Central Theorem

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

ABSTRACTFrequentists’ inference often delivers point estimators associated with confidence intervals or sets for parameters of interest. Constructing the confidence intervals or sets requires understanding the sampling distributions of the point estimators, which, in many but not all cases, are related to asymptotic Normal distributions ensured by central limit theorems. Although previous literature has established various forms of central limit theorems for statistical inference in super population models, we still need general and convenient forms of central limit theorems for some randomization-based causal analyses of experimental data, where the parameters of interests are functions of a finite population and randomness comes solely from the treatment assignment. We use central limit theorems for sample surveys and rank statistics to establish general forms of the finite population central limit theorems that are particularly useful for proving asymptotic distributions of randomization tests under the sharp null hypothesis of zero individual causal effects, and for obtaining the asymptotic repeated sampling distributions of the causal effect estimators. The new central limit theorems hold for general experimental designs with multiple treatment levels, multiple treatment factors and vector outcomes, and are immediately applicable for studying the asymptotic properties of many methods in causal inference, including instrumental variable, regression adjustment, rerandomization, cluster-randomized experiments, and so on. Previously, the asymptotic properties of these problems are often based on heuristic arguments, which in fact rely on general forms of finite population central limit theorems that have not been established before. Our new theorems fill this gap by providing more solid theoretical foundation for asymptotic randomization-based causal inference. Supplementary materials for this article are available online.

In this note we give generalisations of Livsic's result that a priori measurable solutions to cocycle equations must in fact be more regular. We go beyond the original continuous hyperbolic examples of Livsic to consider examples of this phenomenon in the context of: (a) 0-transformations; (b) rational maps; and (c) planar maps with indifferent periodic points. Such examples are not immediately covered by Livsic's original approach either due to a lack of continuity or hyperbolicity. 0. INTRODUCTION In 1972 Livsic showed that given a mixing subshift of finite type T: X -* X and a H6lder continuous function c: X -* IR then any solution to the equation (1) c(x) u(Tx) u(x) with u: X ---1iR measurable and essentially bounded (with respect to an equilibrium measure for a H6lder continuous function) must have a continuous version i.e. eu': X -* R continuous such that u(x) = u'(x) a.e. [2], [3] This is an elementary type of rigidity result. Livsic also applied this result to Anosov systems. Another approach to this result for subshifts of finite type and using Perron-Frobenius type operators appeared in [6]. In this note we shall discuss some simple generalisations of Livsic's result to other examples of dynamical systems. We shall be interested in systems with discontinuities or which are not uniformly hyperbolic and thus are not covered by Livsic's original results. In this context the Perron-Frobenius type operator method adapts easily to prove the generalisations of Livsic's original results. To illustrate this approach we shall apply this method to 3-transformations (section 1), rational maps (section 2) and certain multi-dimensional maps (section 3). 1. }5-TRANSFORMATION We begin by recalling the definition of the well-known 3-transformation on the unit interval. Let / > 1 and define T: [0,1) -* [0,1) by T(x) = fx (mod 1). Let v denote Lebesgue measure on the interval [0,1] and Ll ([0, 1], 13, v) denotes the space of integrable functions on [0,1]. Received by the editors August 5, 1996. 1991 Mathematics Subject Classification. Primary 58Fxx. (?)1999 American Mathematical Society 559 This content downloaded from 157.55.39.255 on Wed, 25 May 2016 04:56:50 UTC All use subject to http://about.jstor.org/terms 560 M. POLLICOTT AND M. YURI Definition. We define a Perron-Frobenius operator Li : Ll ([O, 1], 6,B v) --Ll ([O, 1], 6,B v) by Ty= 13 where the summation is over the [,3] or [a1] + 1 pre-images of x. We denote by BBV the space of functions of bounded variation i.e. if f E BBV then we have that var(f) :=sup{ZEif(Xi+1)-f(xi)I: O xoX 0 for x E [0,1). We shall also need the following more general operator. Definition. Given a Lipschitz function c: [0,11 --? R. We define a c-weighted Perron-Frobenius operator L2: L' ([O, 1], B, v) -* L1 ([O, 1], 13, v) by L2 f (x) = E ecp f (y) TY=X (When c= 0 then L2 clearly reduces to Lp.) It is convenient to denote cn (x) = c(x) + c(Tx) +.. + c(Tn lx) for n > 1. The following result is important in proving Theorem 1. Lemma 2. (i) Vf E L1([0, 1], B, v) we have I2nf h(f fdv) in the L1 norm, (ii) Let c: [0,11 -? IR be a Lipschitz function. There exists w E BBV with w > 0 and A > 0 such that the sequence of functions An (T-y=x BBV. The convergence Ijf -* h(J f dv), as n -4 +oo is proved in [9, Theorem 10] when f is uniformly continuous. Since any function in L1 ([O,1], B, v) can be approximated from above and below by such functions, this is sufficient. This content downloaded from 157.55.39.255 on Wed, 25 May 2016 04:56:50 UTC All use subject to http://about.jstor.org/terms REGULARITY OF SOLUTIONS l'O THE MEASURABLE LIVSIC EQUATION 561 (ii) The operator L2: BBV -BBV has a simple maximal eigenvalue A, say, with positive eigenvector w E BBV ([9, Lemma 9]). Moreover, there exists a measure ,t on [0, 1] such that fn =~ --> w f dti, as n +oo [9, Theorem 10]. 1] Our main result on the measurable Livsic equation for fl-transformations is the following. Theorem 1. If c: [0, 1t -* IR is Lipschitz then for any measurable and essentially bounded solution u to (1') c(x) = u(Tx) u(x) there exists uo(x) = v(x) log h(x), where v(x) E BBV such that u == uo a.e. (nt). Proof. We can make the choice f 1 and then iterating the operator 12 introduced above we can write (L n1)(X)= ASn fn (X) Tny3/X In particular, substituting from the identity (1') we get that f Ix eCn(y) fn (x) An E , n

Introduction. This is a sequel to my paper [1]. The present developments are largely independent of the previous results except in so far as given in the Appendix. Theorem 1 shows a kind of solidarity among the states of a recurrent class; it generalizes a classical result due to Kolmogorov and permits a classification of recurrent states and classes. In ?2 some relations involving the mean recurrence and first passage times are given. In ??3-5 sequences of random variables associated in a natural way with a Markov chain are studied. Theorem 2 is a generalized ergodic theorem which applies to any recurrent class, positive or null. It turns out that in a null class there is a set of numbers which plays the role of stationary absolute probabilities. In the case of a recurrent random walk with independent, stationary steps these numbers are all equal to one and the result is particularly simple. Theorem 3 shows that the kind of solidarity exhibited in Theorem 1 persists in such a sequence; it leads to the clarification of certain conditions stated by Doblin(2) in connection with his central limit theorem. Using a fundamental idea due to Doblin, the weak and strong laws of large numbers, the central limit theorem, the law of the iterated logarithm, and the limit theorems for the maxima of the associated sequence are proved very simply. Owing to the great simplicity of the method it is the conditions of validity of these limit theorems that should deserve attention. Among other things, we shall show by an example that a certain set of conditions, attributed to Kolmogorov, is in reality not sufficient for the validity of the central limit theorem. Furthermore, conditions of validity for the strong limit theorems and the limit theorems for the maxima are obtained by a rather natural strengthening of corresponding conditions for the weak limit theorems. A word about the connection of these conditions with martingale theory closes the paper. 1. The sequence of random variables {Xn}, n=0, 1, 2, * , forms a denumerable Markov chain with stationary transition probabilities. The states will be denoted by the non-negative integers(3) 0, 1, 2, * * . The n-step transition probability from the state i to the state j will be denoted by P(n) (P(l) = Pij). Thus we have

We consider subshifts of finite type on the symbolic space generated by incidence matrices over a countably infinite alphabet. We extend the definition of topological pressure to this context and, as our main result, we construct a new class of Gibbs states of Holder continuous potentials on these symbol spaces. We establish some basic stochastic properties of these Gibbs states: exponential decay of correlations, central limit theorem and an a.s. invariance principle. This is accomplished via detailed studies of the associated Perron-Frobenius operator and its conjugate operator.

The central limit theorem (CLT) is one of the most important theorems in statistics, and it is often introduced to social sciences researchers in an introductory statistics course. However, the recent replication crisis in the social sciences prompts us to investigate just how common certain misconceptions of statistical concepts are. The main purposes of this article are to investigate the misconceptions of the CLT among social sciences researchers and to address these misconceptions by clarifying the definition and properties of the CLT in a manner that is approachable to social science researchers. As part of our article, we conducted a survey to examine the misconceptions of the CLT among graduate students and researchers in the social sciences. We found that the most common misconception of the CLT is that researchers think the CLT is about the convergence of sample data to the normal distribution. We also found that most researchers did not realize that the CLT applies to both sample means and sample sums, and that the CLT has implications for many common statistical concepts and techniques. Our article addresses these misconceptions of the CLT by explaining the preliminaries needed to understand the CLT, introducing the formal definition of the CLT, and elaborating on the implications of the CLT. We hope that through this article, researchers can obtain a more accurate and nuanced understanding of how the CLT operates as well as its role in a variety of statistical concepts and techniques. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

In Chapter 3 we studied the conditions under which the distributions of sums of independent random variables will converge to a limit distribution. We were interested in the weak convergence of distributions; the corresponding limit theorems are called the weak limit theorems, contrary to the strong limit theorems making use of the convergence almost surely. The necessary and sufficient conditions were found for the weak convergence of distributions of sums of independent random variables to a given infinitely divisible distribution. These conditions are quite complicated but are very helpful when we are looking for simpler conditions of convergence to a prescribed infinitely divisible distribution such as a normal, a degenerate, or a Poisson distribution.

Limit theorems for cumulative processes

Central and local limit theorems for the coefficients of polynomials of binomial type

We deal with all the mappings $f_\lambda(z)=\lambda e^z$ that have an attracting periodic orbit. We consider the set $J_r(f_\lambda)$ consisting of those points of the Julia set of $f_\lambda$ that do not escape to infinity under positive iterates of $f_\lambda$. Our ultimate result is that the function $\lambda\mapsto{\rm HD}(J_r(f_\lambda))$ is real-analytic. In order to prove it we develop the thermodynamic formalism of potentials of the form $-t\log|F_\lambda'|$, where $F_\lambda$ is the natural map associated with $f_\lambda$ closely related to the corresponding map introduced in Urbański and Zdunik (2001). The formalism includes appropriately defined topological pressure, Perron–Frobenius operators, and geometric and invariant generalized conformal measures (Gibbs states). We show that our Perron–Frobenius operators are quasicompact and that they embed into a family of operators depending holomorphically on an appropriate parameter, and we obtain several other properties of these operators. We prove an appropriate version of Bowen's formula that the Hausdorff dimension of the set $J_r(f_\lambda)$ is equal to the unique zero of the pressure function. Since the formula for the topological pressure is independent of the set Jr(f), Bowen's formula also indicates that Jr(f) is the right set to deal with. We also study in detail the properties of quasiconformal conjugacies between the maps $f_\lambda$. As a byproduct of our main course of reasoning we prove stochastic properties of the dynamical system generated by $F_\lambda$ and the invariant Gibbs states $\mu_t$ such as the Central Limit Theorem and exponential decay of correlations.

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.

The overuse of lecture-based approaches for instruction in university courses may have limited student access to knowledge, particularly the transfer of complex concepts, such as central limit theorem in statistics. This study seeks to contribute to empirical research regarding the effectiveness of serious educational games (SEGs) to increase undergraduates’ conceptual understanding and affective interest in statistics. An experimental design was used to test the efficacy of an SEG, Deadly Distribution, which simulates a real-world context to learn and interact with statistics concepts, compared to traditional notes and homework problems, as supplements to instruction in addition to class lectures. Students who played the game had similar increases in academic growth of conceptual knowledge as students who studied traditional course material. Furthermore, this treatment group had a significant increase in affective outcomes compared to the control group. These findings extend the current literature, which is mixed and sparse, on the effectiveness of SEGs in the undergraduate classroom. In an undergraduate introductory statistics course, an SEG might be an effective substitute for traditional study time of course materials outside of class to increase their affect toward the subject matter and produce similar gains for students who might not otherwise study.

This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for probabilities $\Pr(n^{-1/2}\sum_{i=1}^n X_i\in A)$ where $X_1,\dots,X_n$ are independent random vectors in $\mathbb{R}^p$ and $A$ is a hyperrectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if $p=p_n\to \infty$ as $n \to \infty$ and $p \gg n$; in particular, $p$ can be as large as $O(e^{Cn^c})$ for some constants $c,C>0$. The result holds uniformly over all hyperrectangles, or more generally, sparsely convex sets, and does not require any restriction on the correlation structure among coordinates of $X_i$. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend only on a small subset of their arguments, with hyperrectangles being a special case.

In this paper we provide a general framework for the study of the central limit theorem (CLT) for empirical processes indexed by uniformly bounded families of functions $\\mathscr{F}$. From this we obtain essentially all known results for the CLT in this case; we improve Dudley's (1982) theorem on entropy with bracketing and Kolcinskii's (1981) CLT under random entropy conditions. One of our main results is that a combinatorial condition together with the existence of the limiting Gaussian process are necessary and sufficient for the CLT for a class of sets (modulo a measurability condition). The case of unbounded $\\mathscr{F}$ is also considered; a general CLT as well as necessary and sufficient conditions for the law of large numbers are obtained in this case. The results for empiricals also yield some new CLT's in $C\\lbrack 0, 1\\rbrack$ and $D\\lbrack 0, 1\\rbrack$.