scholarly journals A Matrix-Based Regularity Measure for Symbolic Sequences

2020 ◽  
Author(s):  
Trevor Wine
Keyword(s):  
Time Series ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Probability Distributions ◽  
Matrix Product ◽  
Strict Adherence ◽  
Different Types ◽  
Symbolic Sequences ◽  
Potential Applications ◽  
Z Scores

A set of statistics is developed for defining and determining the regularity of symbolic sequences. This is achieved by testing a given sequence against a template set with fixed asymptotic symbol proportions $p_i$, $\sum_i p_i = 1$. The process centers on casting the sequence into matrix product form, and defining a parametrized probability distribution via the entrywise norms. The parameter allows varying the weighting between strict adherence to the template sequences, and a generalized Bernoulli randomness. Numerical methods for estimating the entropy of the resulting probability distributions are also developed. The logarithms of the norms of the sequences under test are further shown to satisfy a central limit theorem. This allows the assignment of z-scores, and rigorous comparison of the regularity between sequences of different types. Potential applications are explored, including time series and ergodic systems.

The asymptotic theory of linear time-series models

1973 ◽  
Vol 10 (01) ◽  
pp. 130-145 ◽  
Author(s):  
E. J. Hannan
Keyword(s):  
Time Series ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Theory ◽  
Linear Time ◽  
Stationary Time Series ◽  
Linear Predictor ◽  
The Central Limit Theorem ◽  
Best Linear Predictor

A linear time-series model is considered to be one for which a stationary time series, which is purely non-deterministic, has the best linear predictor equal to the best predictor. A general inferential theory is constructed for such models and various estimation procedures are shown to be equivalent. The treatment is considerably more general than previous treatments. The case where the series has mean which is a linear function of very general kinds of regressor variables is also discussed and a rather general form of central limit theorem for regression is proved. The central limit results depend upon forms of the central limit theorem for martingales.

scholarly journals Understanding the Central Limit Theorem the Easy Way: A Simulation Experiment

Proceedings ◽  
2018 ◽  
Vol 2 (21) ◽  
pp. 1322
Author(s):  
Monica E. Brussolo
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Electronic Device ◽  
Probability Distributions ◽  
Online Simulation ◽  
The Central Limit Theorem ◽  
Sampling Distributions ◽  
Initial Stage ◽  
Random Samples

Using a simulation approach, and with collaboration among peers, this paper is intended to improve the understanding of sampling distributions (SD) and the Central Limit Theorem (CLT) as the main concepts behind inferential statistics. By demonstrating with a hands-on approach how a simulated sampling distribution performs when the data used has different probability distributions, we expect to clarify the notion of the Central Limit Theorem, and the use of samples in the hypothesis testing process for populations. This paper will discuss an initial stage to create random samples from a given population (using Excel) with collaboration of the students, which has been tested in the classroom. Then, based on that experience, a second stage in which we created an online simulation, controlled by the professor, and in which the students will participate during class time using an electronic device connected to internet. Students will create simple random samples from a variety of probability distributions simulated online in a collaborative way. Once the samples are generated, the instructor will combine and summarize the resulting sample statistics using histograms and the results will be discussed with the students. The objective is to teach some of the central topics of introductory statistics, the Central Limit Theorem and sampling distributions with an interactive and engaging approach.

THE EXQUISITE GEOMETRIC STRUCTURE OF A CENTRAL LIMIT THEOREM

Fractals ◽  
2003 ◽  
Vol 11 (01) ◽  
pp. 39-52 ◽  
Author(s):  
CARLOS E. PUENTE
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Geometric Structure ◽  
Limit Theorems ◽  
Probability Distributions ◽  
Space Filling ◽  
Gaussian Distributions ◽  
Bivariate Gaussian Distribution ◽  
Universal Constructions

Universal constructions of univariate and bivariate Gaussian distributions, as transformations of diffuse probability distributions via, respectively, plane- and space-filling fractal interpolating functions and the central limit theorems that they imply, are reviewed. It is illustrated that the construction of the bivariate Gaussian distribution yields exotic kaleidoscopic decompositions of the bell in terms of exquisite geometric structures which include non-trivial crystalline patterns having arbitrary n-fold symmetry, for any n > 2. It is shown that these results also hold when fractal interpolating functions are replaced by a more general class of attractors that are not functions.

Illustrating the Central Limit Theorem

2016 ◽  
Vol 109 (6) ◽  
pp. 456-462 ◽  
Author(s):  
Mimi Corcoran
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
The Central Limit Theorem ◽  
Hands On ◽  
Different Types

Sampling experiments with different types of beads give students a memorable hands-on experience.

scholarly journals A Theory Of Errors in Quantum Measurement

2003 ◽  
Vol 18 (33n35) ◽  
pp. 2439-2450
Author(s):  
S. G. Rajeev
Keyword(s):  
Quantum Theory ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Probability Distribution ◽  
Quantum Measurement ◽  
Probability Distributions ◽  
Matrix Elements ◽  
Random Errors ◽  
The Central Limit Theorem ◽  
The Matrix

It is common to model the random errors in a classical measurement by the normal (Gaussian) distribution, because of the central limit theorem. In the quantum theory, the analogous hypothesis is that the matrix elements of the error in an observable are distributed normally. We obtain the probability distribution this implies for the outcome of a measurement, exactly for the case of traceless 2 × 2 matrices and in the steepest descent approximation in general. Due to the phenomenon of 'level repulsion', the probability distributions obtained are quite different from the Gaussian.

Sign in / Sign up

Export Citation Format

Share Document