Overview

In 1945, very early in the history of the development of a rigorous analytical theory of probability, Feller (1945) wrote a paper called “The fundamental limit theorems in probability” in which he set out what he considered to be “the two most important limit theorems in the modern theory of probability: the central limit theorem and the recently discovered … ‘Kolmogoroff’s cel­ ebrated law of the iterated logarithm’ ”. A little later in the article he added to these, via a charming description, the “little brother (of the central limit theo­ rem), the weak law of large numbers”, and also the strong law of large num­ bers, which he considers as a close relative of the law of the iterated logarithm. Feller might well have added to these also the beautiful and highly applicable results of renewal theory, which at the time he himself together with eminent colleagues were vigorously producing. Feller’s introductory remarks include the visionary: “The history of probability shows that our problems must be treated in their greatest generality: only in this way can we hope to discover the most natural tools and to open channels for new progress. This remark leads naturally to that characteristic of our theory which makes it attractive beyond its importance for various applications: a combination of an amazing generality with algebraic precision.

Full Product Details

Author:   Ross Maller ,  Ishwar Basawa ,  Peter Hall ,  Eugene Seneta
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Edition:   Softcover reprint of the original 1st ed. 2010
Weight:   1.286kg
ISBN:  

9781493940592

ISBN 10:   1493940597
Pages:   463
Publication Date:   23 August 2016
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
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Table of Contents

Author’s Pick.- Chris Heyde’s Contribution to Inference in Stochastic Processes.- Chris Heyde’s Work on Rates of Convergence in the Central Limit Theorem.- Chris Heyde’s Work in Probability Theory, with an Emphasis on the LIL.- Chris Heyde on Branching Processes and Population Genetics.- On a Property of the Lognormal Distribution.- Two Probability Theorems and Their Application to Some First Passage Problems.- Some Renewal Theorems with Application to a First Passage Problem.- Some Results on Small-Deviation Probability Convergence Rates for Sums of Independent Random Variables.- A Contribution to the Theory of Large Deviations for Sums of Independent Random Variables.- On Large Deviation Problems for Sums of Random Variables which are not Attracted to the Normal Law.- On the Influence of Moments on the Rate of Convergence to the Normal Distribution.- On Large Deviation Probabilities in the Case of Attraction to a Non-Normal Stable Law.- On the Converse to the Iterated Logarithm Law.- A Note Concerning Behaviour of Iterated Logarithm Type.- On Extended Rate of Convergence Results for the Invariance Principle.- On the Maximum of Sums of Random Variables and the Supremum Functional for Stable Processes.- Some Properties of Metrics in a Study on Convergence to Normality.- Extension of a Result of Seneta for the Super-Critical Galton–Watson Process.- On the Implication of a Certain Rate of Convergence to Normality.- A Rate of Convergence Result for the Super-Critical Galton-Watson Process.- On the Departure from Normality of a Certain Class of Martingales.- Some Almost Sure Convergence Theorems for Branching Processes.- Some Central Limit Analogues for Supercritical Galton-Watson Processes.- An Invariance Principle and Some Convergence Rate Results for BranchingProcesses.- Improved classical limit analogues for Galton-Watson processes with or without immigration.- Analogues of Classical Limit Theorems for the Supercritical Galton-Watson Process with Immigration.- On Limit Theorems for Quadratic Functions of Discrete Time Series.- Martingales: A Case for a Place in the Statistician’s Repertoire.- On the Influence of Moments on Approximations by Portion of a Chebyshev Series in Central Limit Convergence.- Estimation Theory for Growth and Immigration Rates in a Multiplicative Process.- An Iterated Logarithm Result for Martingales and its Application in Estimation Theory for Autoregressive Processes.- On the Uniform Metric in the Context of Convergence to Normality.- Invariance Principles for the Law of the Iterated Logarithm for Martingales and Processes with Stationary Increments.- An Iterated Logarithm Result for Autocorrelations of a Stationary Linear Process.- On Estimating the Variance of the Offspring Distribution in a Simple Branching Process.- A Nonuniform Bound on Convergence to Normality.- Remarks on efficiency in estimation for branching processes.- The Genetic Balance between Random Sampling and Random Population Size.- On a unified approach to the law of the iterated logarithm for martingales.- The Effect of Selection on Genetic Balance when the Population Size is Varying.- On Central Limit and Iterated Logarithm Supplements to the Martingale Convergence Theorem.- A Log Log Improvement to the Riemann Hypothesis for the Hawkins Random Sieve.- On an Optimal Asymptotic Property of the Maximum Likelihood Estimator of a Parameter from a Stochastic Process.- On Asymptotic Posterior Normality for Stochastic Processes.- On the Survival of a Gene Represented in a Founder Population.- An alternative approach to asymptoticresults on genetic composition when the population size is varying.- On the Asymptotic Equivalence of Lp Metrics for Convergence to Normality.- Quasi-likelihood and Optimal Estimation.- Fisher Lecture.- On Best Asymptotic Confidence Intervals for Parameters of Stochastic Processes.- A quasi-likelihood approach to estimating parameters in diffusion-type processes.- Asymptotic Optimality.- On Defining Long-Range Dependence.- A Risky Asset Model with Strong Dependence through Fractal Activity Time.- Statistical estimation of nonstationary Gaussian processes with long-range dependence and intermittency.

Reviews

From the reviews: This work would be a welcome shelf volume for research workers in probability and statistics and should certainly be a reference available in departmental libraries. ... offers scientists and scholars the opportunity of assembling and commenting upon major classical works in probability and statistics. ... The volume contains 50 original papers, a chronological listing of all publications, as well as individual commentary on particular facets of the research by each of the editors. (Roger Gay, International Statistical Review, Vol. 79 (2), 2011) Each editor provided a nice commentary on their respective topics. ... The volume showcased an interesting and informative biography on Heyde and includes a complimentary introduction by the editors and offers a comprehensive bibliography of Heyde's work. ... This is a valuable collection with several useful pieces of statistical and mathematical works and gives the historical developments of such work in their respective fields. Indeed, this collection will be of interest to research scholars in probability, statistics and related fields. (Technometrics, Vol. 53 (2), May, 2011)

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