On the Toeplitz Lemma, Convergence in Probability, and Mean Convergence
نویسنده
چکیده
Three examples are provided which demonstrate that “convergence in probability” versions of the Toeplitz lemma, the Cesàro mean convergence theorem, and the Kronecker lemma can fail. “Mean convergence” versions of the Toeplitz lemma, Cesàro mean convergence theorem, and the Kronecker lemma are presented and a general mean convergence theorem for a normed sum of independent random variables is established. Some additional problems are posed.
منابع مشابه
Generalizations of Borel-Cantelli Lemma
The Borel-Cantelli Lemma is very important in the probability theory. In this paper, we first describe the general case of the Borel-Cantelli Lemma. The first part of this lemma, assuming convergence and the second part includes divergence and independence assumptions. In the following, we have brought generalizations of the first and second part of this lemma. In most generalizat...
متن کاملFractional Probability Measure and Its Properties
Based on recent studies by Guy Jumarie [1] which defines probability density of fractional order and fractional moments by using fractional calculus (fractional derivatives and fractional integration), this study expands the concept of probability density of fractional order by defining the fractional probability measure, which leads to a fractional probability theory parallel to the classical ...
متن کاملOn Martingales and Least Squares Linear System Identification
In this paper, a nontrivial generalization of a scalar martingale convergence theorem to the vector case is derived. In particular, a theorem is derived concerning the almxat sure convergence to zero of a (vector) martingale normalized by its (ntatrix) process vaziance. This theorem allows a derivation of almost a~n-e convergence results for least squarea identification algorithms applicable to...
متن کاملOn the convergence of the inverses of Toeplitz matrices and its applications
Many issues in signal processing involve the inverses of Toeplitz matrices. One widely used technique is to replace Toeplitz matrices with their associated circulant matrices, based on the well-known fact that Toeplitz matrices asymptotically converge to their associated circulant matrices in the weak sense. This often leads to considerable simplification. However, it is well known that such a ...
متن کاملANFIS+PID Hybrid Controller Design for Controlling of a 6-DOF Robot Manipulator and its Error Convergence Analysis
In this paper, an ANFIS+PID hybrid control policy has been addressed to control a 6-degree-of freedom (6-DOF) robotic manipulator. Then its error convergence has been also evaluated. The ability to formulate and estimate the system uncertainties and disturbances along with system dynamics and rejecting the disturbances effect are some advantages of the proposed method in comparing with the co...
متن کامل