Stochastic Convergence

2021 ◽  
pp. 400-417
Author(s):  
James Davidson
Keyword(s):  
Almost Sure Convergence ◽  
Convergence In Probability ◽  
Sufficient Condition ◽  
Stochastic Convergence ◽  
Cantelli Lemma ◽  
Laws Of Large Numbers ◽  
Large Numbers

The modes of convergence introduced in Chapter 12 are studied in detail. Conditions for almost‐sure convergence are derived via the Borel–Cantelli lemma. Convergence in probability is contrasted, and then a number of results for convergence of transformed series are given. Convergence in LP‐norm is introduced as a sufficient condition for convergence in probability. Examples are given, and the chapter concludes with a preliminary look at the laws of large numbers.

Convergence in Lp Norm

2021 ◽  
pp. 418-437
Author(s):  
James Davidson
Keyword(s):  
Law Of Large Numbers ◽  
Almost Sure Convergence ◽  
Convergence In Probability ◽  
Martingale Differences ◽  
Lp Norm ◽  
Large Numbers

This chapter looks in detail at proofs of the weak law of large numbers (convergence in probability) using the technique of establishing convergence in Lp‐norm. The extension to a proof of almost‐sure convergence is given, and then special results for martingale differences, mixingales, and approximable processes. These results are proved in array notation to allow general forms of heterogeneity.

scholarly journals On the almost sure convergence of weighted sums of random elements inD[0,1]

1981 ◽  
Vol 4 (4) ◽  
pp. 745-752
Author(s):  
R. L. Taylor ◽  
C. A. Calhoun
Keyword(s):  
Almost Sure Convergence ◽  
Growth Conditions ◽  
Weighted Sums ◽  
Integral Conditions ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Convergence Results ◽  
Large Numbers ◽  
Random Elements

Let{wn}be a sequence of positive constants andWn=w1+…+wnwhereWn→∞andwn/Wn→∞. Let{Wn}be a sequence of independent random elements inD[0,1]. The almost sure convergence ofWn−1∑k=1nwkXkis established under certain integral conditions and growth conditions on the weights{wn}. The results are shown to be substantially stronger than the weighted sums convergence results of Taylor and Daffer (1980) and the strong laws of large numbers of Ranga Rao (1963) and Daffer and Taylor (1979).

scholarly journals Strong laws of large numbers for arrays of rowwise independent random elements

1987 ◽  
Vol 10 (4) ◽  
pp. 805-814 ◽  
Author(s):  
Robert Lee Taylor ◽  
Tien-Chung Hu
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Complete Convergence ◽  
Almost Sure Convergence ◽  
Random Variable ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Random Elements ◽  
Rowwise Independent ◽  
Uniformly Bounded

Let{Xnk}be an array of rowwise independent random elements in a separable Banach space of typep+δwithEXnk=0for allk,n. The complete convergence (and hence almost sure convergence) ofn−1/p∑k=1nXnk to 0,1≤p<2, is obtained when{Xnk}are uniformly bounded by a random variableXwithE|X|2p<∞. When the array{Xnk}consists of i.i.d, random elements, then it is shown thatn−1/p∑k=1nXnkconverges completely to0if and only ifE‖X11‖2p<∞.

scholarly journals Strong laws of large numbers for arrays of row-wise exchangeable random elements

1985 ◽  
Vol 8 (1) ◽  
pp. 135-144 ◽  
Author(s):  
Robert Lee Taylor ◽  
Ronald Frank Patterson
Keyword(s):  
Banach Space ◽  
Almost Sure Convergence ◽  
Kernel Density ◽  
Triangular Array ◽  
Density Estimates ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Kernel Density Estimates ◽  
Large Numbers ◽  
Random Elements

Let{Xnk,1≤k≤n,n≤1}be a triangular array of row-wise exchangeable random elements in a separable Banach space. The almost sure convergence ofn−1/p∑k=1nXnk,1≤p<2, is obtained under varying moment and distribution conditions on the random elements. In particular, strong laws of large numbers follow for triangular arrays of random elements in(Rademacher) typepseparable Banach spaces. Consistency of the kernel density estimates can be obtained in this setting.

scholarly journals On exact strong laws of large numbers under general dependence conditions

2018 ◽  
Vol 38 (1) ◽  
pp. 103-121 ◽  
Author(s):  
André Adler ◽  
Przemysław Matuła
Keyword(s):  
Random Variables ◽  
Almost Sure Convergence ◽  
Weighted Sums ◽  
Dependent Random Variables ◽  
Laws Of Large Numbers ◽  
Associated Random Variables ◽  
Negatively Associated ◽  
Strong Laws ◽  
Large Numbers ◽  
Dependence Conditions

We study the almost sure convergence of weighted sums of dependent random variables to a positive and finite constant, in the case when the random variables have either mean zero or no mean at all. These are not typical strong laws and they are called exact strong laws of large numbers. We do not assume any particular type of dependence and furthermore consider sequences which are not necessarily identically distributed. The obtained results may be applied to sequences of negatively associated random variables.

Uniform Stochastic Convergence

2021 ◽  
pp. 471-492
Author(s):  
James Davidson
Keyword(s):  
Uniform Convergence ◽  
Metric Spaces ◽  
Main Issue ◽  
Stochastic Convergence ◽  
Convergence Conditions ◽  
Laws Of Large Numbers ◽  
Random Sequences ◽  
Large Numbers ◽  
Stochastic Equicontinuity

This chapter concerns random sequences of functions on metric spaces. The main issue is the distinction between convergence at all points of the space (pointwise) and uniform convergence, where limit points are also taken into account. The role of the stochastic equicontinuity property is highlighted. Generic uniform convergence conditions are given and linked to the question of uniform laws of large numbers.

On the convergence of stochastically monotone sequences of random variables and some applications

1981 ◽  
Vol 18 (3) ◽  
pp. 592-605 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Branching Processes ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Convergent Sequence ◽  
Convergence In Probability ◽  
Sufficient Condition ◽  
Stochastic Monotonicity ◽  
Monotone Sequences ◽  
Necessary And Sufficient ◽  
The Relationship

The paper is concerned with the relationship between various modes of convergence for stochastically monotone sequences of random variables. A necessary and sufficient condition, as well as a sufficient condition, for convergence in probability of a vaguely convergent sequence are given. If, in addition, the sequence is assumed Markovian the same conditions are shown to pertain to almost sure convergence. A counterexample in the case when stochastic monotonicity fails is presented and some applications to branching processes are discussed.

On the convergence of stochastically monotone sequences of random variables and some applications

1981 ◽  
Vol 18 (03) ◽  
pp. 592-605 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Branching Processes ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Convergent Sequence ◽  
Convergence In Probability ◽  
Sufficient Condition ◽  
Stochastic Monotonicity ◽  
Monotone Sequences ◽  
Necessary And Sufficient ◽  
The Relationship

The paper is concerned with the relationship between various modes of convergence for stochastically monotone sequences of random variables. A necessary and sufficient condition, as well as a sufficient condition, for convergence in probability of a vaguely convergent sequence are given. If, in addition, the sequence is assumed Markovian the same conditions are shown to pertain to almost sure convergence. A counterexample in the case when stochastic monotonicity fails is presented and some applications to branching processes are discussed.

On the volume of parallel bodies: a probabilistic derivation of the Steiner formula

1995 ◽  
Vol 27 (1) ◽  
pp. 97-101 ◽  
Author(s):  
Richard A. Vitale
Keyword(s):  
Convex Bodies ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Steiner Formula ◽  
Random Convex

We give a proof of the Steiner formula based on the theory of random convex bodies. In particular, we make use of laws of large numbers for both random volumes and random convex bodies themselves.

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