scholarly journals The Distributional Asymptotics Mod 1 of (logb n)

2019 ◽  
Vol 14 (1) ◽  
pp. 105-122
Author(s):  
Chuang Xu
Keyword(s):  
Rate Of Convergence ◽  
Upper Bound ◽  
Rates Of Convergence ◽  
Limit Set ◽  
Exponential Distributions ◽  
Kantorovich Metric ◽  
Kolmogorov Metric ◽  
Omega Limit Set ◽  
Upper Estimate

AbstractThis paper studies the distributional asymptotics of the slowly changing sequence of logarithms (logb n) with b ∈ 𝕅 \ {1}. It is known that (logbn) is not uniformly distributed modulo one, and its omega limit set is composed of a family of translated exponential distributions with constant log b. An improved upper estimate (\sqrt {\log N} /N) is obtained for the rate of convergence with respect to (w. r. t.)the Kantorovich metric on the circle, compared to the general results on rates of convergence for a class of slowly changing sequences in the author’s companion in-progress work. Moreover, a sharp rate of convergence (log N/N)w. r. t. the Kantorovich metric on the interval [0, 1], is derived. As a byproduct, the rate of convergence w.r.t. the discrepancy metric (or the Kolmogorov metric) turns out to be (log N/N) as well, which verifies that an upper bound for this rate derived in [Ohkubo, Y.—Strauch, O.: Distribution of leading digits of numbers, Unif. Distrib. Theory, 11 (2016), no.1, 23–45.] is sharp.

Poisson approximation for (k1, k2)-events via the Stein-Chen method

2004 ◽  
Vol 41 (4) ◽  
pp. 1081-1092 ◽  
Author(s):  
P. Vellaisamy
Keyword(s):  
Limit Theorem ◽  
Rate Of Convergence ◽  
Upper Bound ◽  
Success Probability ◽  
Random Variable ◽  
Poisson Approximation ◽  
Bernoulli Trials ◽  
Poisson Random Variable ◽  
Markov Dependent Trials ◽  
Special Case

Consider a sequence of independent Bernoulli trials with success probability p. Let N(n; k1, k2) denote the number of times that k1 failures are followed by k2 successes among the first n Bernoulli trials. We employ the Stein-Chen method to obtain a total variation upper bound for the rate of convergence of N(n; k1, k2) to a suitable Poisson random variable. As a special case, the corresponding limit theorem is established. Similar results are obtained for Nk3(n; k1, k2), the number of times that k1 failures followed by k2 successes occur k3 times successively in n Bernoulli trials. The bounds obtained are generally sharper than, and improve upon, some of the already known results. Finally, the technique is adapted to obtain Poisson approximation results for the occurrences of the above-mentioned events under Markov-dependent trials.

On the computability of a construction of Brownian motion

2013 ◽  
Vol 23 (6) ◽  
pp. 1257-1265 ◽  
Author(s):  
GEORGE DAVIE ◽  
WILLEM L. FOUCHÉ
Keyword(s):  
Brownian Motion ◽  
Lower Bound ◽  
Rate Of Convergence ◽  
Upper Bound ◽  
Binary Sequence ◽  
Computable Analysis

We examine a construction due to Fouché in which a Brownian motion is constructed from an algorithmically random infinite binary sequence. We show that although the construction is provably not computable in the sense of computable analysis, a lower bound for the rate of convergence is computable in any upper bound for the compressibilty of the sequence, making the construction layerwise computable.

scholarly journals UNIFORM ASYMPTOTIC NORMALITY IN STATIONARY AND UNIT ROOT AUTOREGRESSION

2011 ◽  
Vol 27 (6) ◽  
pp. 1117-1151 ◽  
Author(s):  
Chirok Han ◽  
Peter C. B. Phillips ◽  
Donggyu Sul
Keyword(s):  
Normal Distribution ◽  
Rate Of Convergence ◽  
Unit Root ◽  
Asymptotic Theory ◽  
Signal Strength ◽  
Rates Of Convergence ◽  
Moment Conditions ◽  
First Order ◽  
The Cost ◽  
Artificial Shrinkage

While differencing transformations can eliminate nonstationarity, they typically reduce signal strength and correspondingly reduce rates of convergence in unit root autoregressions. The present paper shows that aggregating moment conditions that are formulated in differences provides an orderly mechanism for preserving information and signal strength in autoregressions with some very desirable properties. In first order autoregression, a partially aggregated estimator based on moment conditions in differences is shown to have a limiting normal distribution that holds uniformly in the autoregressive coefficient ρ, including stationary and unit root cases. The rate of convergence is $\root \of n $ when $\left| \rho \right| < 1$ and the limit distribution is the same as the Gaussian maximum likelihood estimator (MLE), but when ρ = 1 the rate of convergence to the normal distribution is within a slowly varying factor of n. A fully aggregated estimator (FAE) is shown to have the same limit behavior in the stationary case and to have nonstandard limit distributions in unit root and near integrated cases, which reduce both the bias and the variance of the MLE. This result shows that it is possible to improve on the asymptotic behavior of the MLE without using an artificial shrinkage technique or otherwise accelerating convergence at unity at the cost of performance in the neighborhood of unity. Confidence intervals constructed from the FAE using local asymptotic theory around unity also lead to improvements over the MLE.

Acceptability ratings cannot be taken at face value

2020 ◽  
pp. 189-214
Author(s):  
Carson T. Schütze
Keyword(s):  
Rate Of Convergence ◽  
Experimental Work ◽  
Rates Of Convergence ◽  
Original Method ◽  
Journal Articles ◽  
True Rate ◽  
Acceptability Judgments ◽  
Critical Words ◽  
Recent Experimental Work

This chapter addresses how linguists’ empirical (syntactic) claims should be tested with non-linguists. Recent experimental work attempts to measure rates of convergence between data presented in journal articles and the results of large surveys. Three follow-up experiments to one such study are presented. It is argued that the original method may underestimate the true rate of convergence because it leaves considerable room for naïve subjects to give ratings that do not reflect their true acceptability judgments of the relevant structures. To understand what can go wrong, the experiments were conducted in two parts. The first part had visually presented sentences rated on a computer, replicating previous work. The second part was an interview where the experimenter asked the participants about the ratings they gave to particular items, in order to determine what interpretation or parse they had assigned, whether they had missed any critical words, and so on.

The rate of convergence of extremes of stationary normal sequences

1983 ◽  
Vol 15 (01) ◽  
pp. 54-80 ◽  
Author(s):  
Holger Rootzén
Keyword(s):  
Rate Of Convergence ◽  
Joint Distribution ◽  
Point Processes ◽  
Rates Of Convergence ◽  
Joint Distribution Function ◽  
Maximal Correlation ◽  
Standard Normal ◽  
Image Position ◽  
The Right ◽  
Right Order

Let {ξ; t = 1, 2, …} be a stationary normal sequence with zero means, unit variances, and covariances let be independent and standard normal, and write . In this paper we find bounds on which are roughly of the order where ρ is the maximal correlation, ρ =sup {0, r 1 , r 2, …}. It is further shown that, at least for m-dependent sequences, the bounds are of the right order and, in a simple example, the errors are evaluated numerically. Bounds of the same order on the rate of convergence of the point processes of exceedances of one or several levels are obtained using a ‘representation' approach (which seems to be of rather wide applicability). As corollaries we obtain rates of convergence of several functionals of the point processes, including the joint distribution function of the k largest values amongst ξ1, …, ξn.

scholarly journals Continued fractions, the Chen–Stein method and extreme value theory

2019 ◽  
Vol 41 (2) ◽  
pp. 461-470
Author(s):  
ANISH GHOSH ◽  
MAXIM SØLUND KIRSEBOM ◽  
PARTHANIL ROY
Keyword(s):  
Rate Of Convergence ◽  
Extreme Value Theory ◽  
Upper Bound ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Value Theory ◽  
Extreme Value ◽  
Real Analysis ◽  
Continued Fraction Expansion ◽  
Stein Method

In this work we deal with extreme value theory in the context of continued fractions using techniques from probability theory, ergodic theory and real analysis. We give an upper bound for the rate of convergence in the Doeblin–Iosifescu asymptotics for the exceedances of digits obtained from the regular continued fraction expansion of a number chosen randomly from $(0,1)$ according to the Gauss measure. As a consequence, we significantly improve the best known upper bound on the rate of convergence of the maxima in this case. We observe that the asymptotics of order statistics and the extremal point process can also be investigated using our methods.

Adding machines and wild attractors

1997 ◽  
Vol 17 (6) ◽  
pp. 1267-1287 ◽  
Author(s):  
HENK BRUIN ◽  
GERHARD KELLER ◽  
MATTHIAS ST. PIERRE
Keyword(s):  
Dynamical System ◽  
Critical Point ◽  
Cantor Set ◽  
Limit Set ◽  
Unimodal Maps ◽  
Irrational Rotations ◽  
Circle Rotation ◽  
Omega Limit Set ◽  
Adding Machines

We investigate the dynamics of unimodal maps $f$ of the interval restricted to the omega limit set $X$ of the critical point for cases where $X$ is a Cantor set. In particular, many cases where $X$ is a measure attractor of $f$ are included. We give two classes of examples of such maps, both generalizing unimodal Fibonacci maps [LM, BKNS]. In all cases $f_{|X}$ is a continuous factor of a generalized odometer (an adding machine-like dynamical system), and at the same time $f_{|X}$ factors onto an irrational circle rotation. In some of the examples we obtain irrational rotations on more complicated groups as factors.

Local contractivity of the process of a player rating variation in the Elo model with one adversary

2015 ◽  
Vol 25 (5) ◽  
Author(s):  
Vadim A. Avdeev
Keyword(s):  
Rate Of Convergence ◽  
Stationary Distribution ◽  
Infinite Series ◽  
Limit Distribution ◽  
Rating Model ◽  
Upper Estimate

AbstractWe study the process of variation of a player rating in an infinite series of games with the same adversary in the Elo rating model. This process is shown to have a stationary distribution, an upper estimate of the rate of convergence to which is established. In a previous paper by the author, the existence of a limit distribution was proved under more stringent assumptions on the parameters of a rating model.

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