Book•
Almost sure convergence
01 Jan 1974-
About: The article was published on 1974-01-01 and is currently open access. It has received 1195 citations till now. The article focuses on the topics: Proofs of convergence of random variables & Convergence of random variables.
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TL;DR: In this article, an exponential ARCH model is proposed to study volatility changes and the risk premium on the CRSP Value-Weighted Market Index from 1962 to 1987, which is an improvement over the widely-used GARCH model.
Abstract: This paper introduces an ARCH model (exponential ARCH) that (1) allows correlation between returns and volatility innovations (an important feature of stock market volatility changes), (2) eliminates the need for inequality constraints on parameters, and (3) allows for a straightforward interpretation of the "persistence" of shocks to volatility. In the above respects, it is an improvement over the widely-used GARCH model. The model is applied to study volatility changes and the risk premium on the CRSP Value-Weighted Market Index from 1962 to 1987. Copyright 1991 by The Econometric Society.
10,019 citations
Book•
01 Jan 1997
TL;DR: In this article, the authors discuss the relationship between Markov Processes and Ergodic properties of Markov processes and their relation with PDEs and potential theory. But their main focus is on the convergence of random processes, measures, and sets.
Abstract: * 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
4,562 citations
Book•
16 Apr 2013
TL;DR: How to Construct Nonparametric Regression Estimates * Lower Bounds * Partitioning Estimates * Kernel Estimates * k-NN Estimates * Splitting the Sample * Cross Validation * Uniform Laws of Large Numbers
Abstract: Why is Nonparametric Regression Important? * How to Construct Nonparametric Regression Estimates * Lower Bounds * Partitioning Estimates * Kernel Estimates * k-NN Estimates * Splitting the Sample * Cross Validation * Uniform Laws of Large Numbers * Least Squares Estimates I: Consistency * Least Squares Estimates II: Rate of Convergence * Least Squares Estimates III: Complexity Regularization * Consistency of Data-Dependent Partitioning Estimates * Univariate Least Squares Spline Estimates * Multivariate Least Squares Spline Estimates * Neural Networks Estimates * Radial Basis Function Networks * Orthogonal Series Estimates * Advanced Techniques from Empirical Process Theory * Penalized Least Squares Estimates I: Consistency * Penalized Least Squares Estimates II: Rate of Convergence * Dimension Reduction Techniques * Strong Consistency of Local Averaging Estimates * Semi-Recursive Estimates * Recursive Estimates * Censored Observations * Dependent Observations
1,931 citations
TL;DR: In this article, necessary and sufficient conditions for the stationarity and ergodicity of the GARCH(l.l) process were established, and it was shown that the IGARCH(1,1) process with no drift converges almost surely to zero.
Abstract: This paper establishes necessary and sufficient conditions for the stationarity and ergodicity of the GARCH(l.l) process. As a special case, it is shown that the IGARCH(1,1) process with no drift converges almost surely to zero, while IGARCH(1,1) with a positive drift is strictly stationary and ergodic. We examine the persistence of shocks to conditional variance in the GARCH(l.l) model, and show that whether these shocks "persist" or not depends crucially on the definition of persistence. We also develop necessary and sufficient conditions for the finiteness of absolute moments of any (including fractional) order.
1,117 citations
Book•
01 Jan 1986
TL;DR: The mathematics of filtering and ee/ise 556: stochastic systems fall 2013 usc search identification and system parameter estimation 1991 gbv is described.
Abstract: stochastic systems estimation identification and adaptive stochastic adaptive control eolss stochastic systems estimation identification and adaptive stochastic systems estimation identification and adaptive control of stochastic systems eolss stochastic systems estimation identification and adaptive stochastic systems: estimation, identification and adaptation in stochastic dynamic systems survey and new identification and stochastic adaptive control (systems identification and adaptive control methods for some robust stochastic adaptive control dspace@mit: home adaptation in stochastic dynamic systems survey and new chapter 1: introduction to adaptive control stochastic systems estimation identification and adaptive stochastic adaptive nash certainty equivalence control coefficient estimation in adaptive control systems maximum likelihood identification and realization of (size 44,85mb) download ebook stable adaptive systems optimal adaptive control of uncertain stochastic discrete 19,42mb file download system identification adaptive on-line identification and adaptive trajectory tracking ece686: filtering and control of stochastic linear systems robustness and convergence of least-squares identification 68,58mb file system identification adaptive control bahram adaptation in stochastic dynamic systems survey and new identification and system parameter estimation 1991 gbv stochastic adaptive control via consistent parameter adaptive control of stochastic sage pub stochastic delay estimation and adaptive control of eece 574 adaptive control basics of system identification robust identification of stochastic linear systems with robust adaptive els-qr algorithm for linear discrete time stochastic systems: the mathematics of filtering and ee/ise 556: stochastic systems fall 2013 usc search identification and system parameter estimation 1991 gbv
1,085 citations