This paper presents a parameter covariance matrix estimator which is consistent even when the disturbances of a linear regression model are heteroskedastic. This estimator does not depend on a formal model of the structure of the heteroskedasticity. By comparing the elements of the new estimator to those of the usual covariance estimator, one obtains a direct test for heteroskedasticity, since in the absence of heteroskedasticity, the two estimators will be approximately equal, but will generally diverge otherwise. The test has an appealing least squares interpretation.

A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity

A low-density parity-check code is a code specified by a parity-check matrix with the following properties: each column contains a small fixed number j \geq 3 of l's and each row contains a small fixed number k > j of l's. The typical minimum distance of these codes increases linearly with block length for a fixed rate and fixed j . When used with maximum likelihood decoding on a sufficiently quiet binary-input symmetric channel, the typical probability of decoding error decreases exponentially with block length for a fixed rate and fixed j . A simple but nonoptimum decoding scheme operating directly from the channel a posteriori probabilities is described. Both the equipment complexity and the data-handling capacity in bits per second of this decoder increase approximately linearly with block length. For j > 3 and a sufficiently low rate, the probability of error using this decoder on a binary symmetric channel is shown to decrease at least exponentially with a root of the block length. Some experimental results show that the actual probability of decoding error is much smaller than this theoretical bound.

/pdf/low-density-parity-check-codes-vltmv2dex4.pdf

Low-Density Parity-Check Codes

During the past decade a number of variations in the simple up‐down procedure have been used in psychoacoustic testing. A broad class of these methods is described with due emphasis on the related problems of parameter estimation and the efficient placing of observations. The advantages of up‐down methods are many, including simplicity, high efficiency, robustness, small‐sample reliability, and relative freedom from restrictive assumptions. Several applications of these procedures in psychoacoustics are described, including examples where conventional techniques are inapplicable.

/pdf/transformed-up-down-methods-in-psychoacoustics-31x0jr24ab.pdf

Transformed Up‐Down Methods in Psychoacoustics

This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems.

Probability: Theory and Examples

* Measure Theory-Basic Notions * Measure Theory-Key Results * Processes, Distributions, and Independence * Random Sequences, Series, and Averages * Characteristic Functions and Classical Limit Theorems * Conditioning and Disintegration * Martingales and Optional Times * Markov Processes and Discrete-Time Chains * Random Walks and Renewal Theory * Stationary Processes and Ergodic Theory * Special Notions of Symmetry and Invariance * Poisson and Pure Jump-Type Markov Processes * Gaussian Processes and Brownian Motion * Skorohod Embedding and Invariance Principles * Independent Increments and Infinite Divisibility * Convergence of Random Processes, Measures, and Sets * Stochastic Integrals and Quadratic Variation * Continuous Martingales and Brownian Motion * Feller Processes and Semigroups * Ergodic Properties of Markov Processes * Stochastic Differential Equations and Martingale Problems * Local Time, Excursions, and Additive Functionals * One-Dimensional SDEs and Diffusions * Connections with PDEs and Potential Theory * Predictability, Compensation, and Excessive Functions * Semimartingales and General Stochastic Integration * Large Deviations * Appendix 1: Advanced Measure Theory * Appendix 2: Some Special Spaces * Historical and Bibliographical Notes * Bibliography * Indices

http://cyber.sibsutis.ru:82/Monarev/docs/nauka/PROBABILITY/Kallenberg%20O.%20Foundations%20of%20Modern%20Probability%20(Springer,1997)(ISBN%200387949577)(535s).pdf

Foundations of modern probability

Renewal theory can be and has been viewed in several different ways. One way is to reduce the problems to those concerning the addition of independent, non-negative random variables having a common distribution. Accordingly we introduce the random variables X1, X2, * * *, independent, non-negative, and all possessing the same distribution function F(x) with mean m-=f.r.xdF(x) (O<m =< X), and their successive sums S. = Et=1 X. Either F(x) is purely discontinuous with all its discontinuities located at the multiples of a fixed real number or it is not so; we call the first case the lattice case and the second the nonlattice case. In the lattice case there is no real loss of generality by assuming (as we shall do in the following) the said discontinuities are all located at integers whose greatest common divisor is one. One of the main results of renewal theory can then be stated as follows: (i) In the lattice case, if x runs through integers:

/pdf/an-extension-of-renewal-theory-186qhzj1um.pdf

An extension of renewal theory

The fundam entals of t he tJleory of denumerable Markov cha ins with station a ry trans ition probabilit ies were laid down by Kolmogorov, and fur ther work was don e by Doblin. The t heory of recurrent events of F eller is closely related , if no t coextensive. So me n ew resul ts obtained by T. E. Harris turn out to t ie up very nicely with so me amplifications of Doblin 's work. Harris was led to consid er t he probabilities of hi ttin g one state before another , starting from a third one. This idea of consid erin g t hree states, one initial , one "taboo", and one fin a l, is more full y developed in t he presen t work. The notion of first passage t ime to t he " union" or " in tersection " of two states is also in trodu ced h ere. The in terplay between t hese no t ions is illustra ted.

/pdf/contributions-to-the-theory-of-markov-chains-5bqplx9lsl.pdf

Contributions to the theory of Markov chains

In Section 1, a temporal bound is estimated by a spatial bound, for a Markov process whose transition density satisfies a simple condition. This includes the Brownian motion, for which comparison with a more special method is made. In Section 2, the result is related to the Green operator and examples are given. In Section 3, the result is applied to an old problem of eigenvalues of the Laplacian. In Section 4, recent extensions from the Laplacian to the Schrodinger circle-of-ideas are briefly described. In this case, time is measured by an exponential functional of the process, commonly known under the names Feynman-Kac. This article was written as script for two talks, one general and one technical, given in Taiwan in January 1991. In the spirit of the occasion, the style of exposition is deliberately paced and discursive.

Greenian bounds for Markov processes

1. Let X1 , X2 , • • • , Xn . . . be independent random variables and let Sn = X, . In the so-called law of the iterated logarithm, completely solved by Feller recently, the upper limit of S n as n -4 co is considered and its true order of magnitude is found with probability one . A counterpart to that problem is to consider the lower limit of Sn as n --> oo and to make a statement about its order of magnitude with probability one . THEOREM 1 . Let XI , • • •, Xn , • • be independent random variables with the common distribution : Pr(X,b = 1) = p, Pr(X,,, = 0) = 1 p = q. Let ¢(n) I oo and

On the lower limit of sums of independent random variables

and computed the second moment of the difference of the expressions in (1) and (2). In this paper, by examining the excursions in Brownian motion and using a new formula for the distribution of their maxima, we obtain a direct identification of the limit in (1) without using (2). Let Tx=inf{t: W(t )=x} , Tx=inf{t: Y( t )=x} . For a > 0 let R~=0, R~= T/, + T/~ o Or,~, and for n> 2 let R~= R~,_ 1 + R~ o OR~_ 1 . Here {0~, t>0} is the usual collection of shift operators: W(s, Ot co) = W(s + t, co) and if S is a random variable, Os=O t on {S=t}. If S is a random variable, let d , (S )=sup{n: R~, 0 for ~ < s < fl; { Y(s, co), c~ < s < fl} is called an excursion if

https://math.duke.edu/~rtd/reprints/paper2.pdf

Kai Lai Chung

Papers

An extension of renewal theory

Contributions to the theory of Markov chains

Greenian bounds for Markov processes

On the lower limit of sums of independent random variables

Downcrossings and local time