Strong Laws for Dependent Heterogeneous Processes

1991 ◽  
Vol 7 (2) ◽  
pp. 213-221 ◽  
Author(s):  
Bruce E. Hansen
Keyword(s):  
Law Of Large Numbers ◽  
Strong Mixing ◽  
Strong Law ◽  
Strong Laws ◽  
Maximal Inequalities ◽  
Large Numbers ◽  
Heterogeneous Processes ◽  
Mixing Sequences ◽  
Strong Mixing Sequences ◽  
Weakly Dependent

This paper presents maximal inequalities and strong law of large numbers for weakly dependent heterogeneous random variables. Specifically considered are Lr mixingales for r > 1, strong mixing sequences, and near epoch dependent (NED) sequences. We provide the first strong law for Lr-bounded Lr mixingales and NED sequences for 1 > r > 2. The strong laws presented for α-mixing sequences are less restrictive than the laws of McLeish [8].

scholarly journals Laws of Large Numbers for Dependent Heterogeneous Processes

1995 ◽  
Vol 11 (2) ◽  
pp. 347-358 ◽  
Author(s):  
R.M. de Jong
Keyword(s):  
Decay Rate ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers ◽  
Heterogeneous Processes ◽  
Weak Laws ◽  
Weakly Dependent

This paper provides weak and strong laws of large numbers for weakly dependent heterogeneous random variables. The weak laws of large numbers presented extend known results to the case of trended random variables. The main feature of our strong law of large numbers for mixingale sequences is the less strict decay rate that is imposed on the mixingale numbers as compared to previous results.

Two-parameter strong laws and maximal inequalities for U-statistics

1987 ◽  
Vol 107 (1-2) ◽  
pp. 133-151 ◽  
Author(s):  
Terry R. McConnell
Keyword(s):  
Law Of Large Numbers ◽  
Weak Type ◽  
Sufficient Conditions ◽  
Strong Law ◽  
Strong Laws ◽  
Maximal Inequalities ◽  
U Statistics ◽  
Large Numbers ◽  
Necessary And Sufficient ◽  
Two Parameter

SynopsisWe provide necessary and sufficient conditions for two-parameter convergence in the strong law of large numbers for U-statistics. We also obtain weak-type (1,1) inequalities for one and two-sample U-statistics of order 2 which are, in a sense, best possible.

The Strong Law of Large Numbers

2021 ◽  
pp. 438-470
Author(s):  
James Davidson
Keyword(s):  
Law Of Large Numbers ◽  
Mixing Processes ◽  
Strong Law ◽  
Strong Laws ◽  
Large Numbers ◽  
Random Weighting

This chapter focuses largely on methods of proof of the strong law, building on the fundamental convergence lemma. It covers Kolmogorov's three‐series theorem, strong laws for martingales, and random weighting. Then a range of strong laws are proved for mixingales and for near‐epoch dependent and mixing processes.

Maximal inequalities for N-demimartingale and strong law of large numbers

2011 ◽  
Vol 81 (9) ◽  
pp. 1348-1353 ◽  
Author(s):  
Xuejun Wang ◽  
Shuhe Hu ◽  
B.L.S. Prakasa Rao ◽  
Wenzhi Yang
Keyword(s):  
Law Of Large Numbers ◽  
Strong Law ◽  
Maximal Inequalities ◽  
Large Numbers

Explicit Stable Strong Laws of Large Numbers

1994 ◽  
Vol 44 (3-4) ◽  
pp. 141-150 ◽  
Author(s):  
André Adler
Keyword(s):  
Law Of Large Numbers ◽  
Random Variables ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers ◽  
Interesting Class

In this article it is shown, through a very interesting class of random variables, how one may go about explicitly obtaining constants in order to obtain a stable strong law of large numbers. The question at hand is, not when we can find constants an and bn so that our sequence of i. i.d. random variables obeys this type of strong law of large numbers, but how one goes about constructing these constants so that [Formula: see text] almost surely, even though { X, Xn} are i.i.d. with either [Formula: see text] There are three possible cases. We exhibit all three via a particular family of random variables.

scholarly journals On Chung's strong law of large numbers in general Banach spaces

1988 ◽  
Vol 37 (1) ◽  
pp. 93-100 ◽  
Author(s):  
Bong Dae Choi ◽  
Soo Hak Sung
Keyword(s):  
Banach Spaces ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Real Numbers ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers

Let { Xn, n ≥ 1 } be a sequence of independent Banach valued random variables and { an, n, ≥ 1 } a sequence of real numbers such that 0 < an ↑ ∞. It is shown that, under the assumption with some restrictions on φ, Sn/an → 0 a.s. if and only if Sn/an → 0 in probability if and only if Sn/an → 0 in L1. From this result several known strong laws of large numbers in Banach spaces are easily derived.

scholarly journals Strong law of large numbers forρ*-mixing sequences with different distributions

2006 ◽  
Vol 2006 ◽  
pp. 1-7 ◽  
Author(s):  
Guang-Hui Cai
Keyword(s):  
Complete Convergence ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Large Numbers ◽  
Mixing Sequences

Strong law of large numbers and complete convergence forρ*-mixing sequences with different distributions are investigated. The results obtained improve the relevant results by Utev and Peligrad (2003).

scholarly journals On Chung-Teicher Type Strong Law of Large Numbers for -Mixing Random Variables

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
Anna Kuczmaszewska
Keyword(s):  
Law Of Large Numbers ◽  
Random Variables ◽  
Independent Random Variables ◽  
Strong Law ◽  
Strong Laws ◽  
Large Numbers ◽  
Mixing Sequence ◽  
Mixing Random Variables

In this paper the classical strong laws of large number of Kolmogorov, Chung, and Teicher for independent random variables were generalized on the case of -mixing sequence. The main result was applied to obtain a Marcinkiewicz SLLN.

On the second Borel-Cantelli lemma for strongly mixing sequences of events

1997 ◽  
Vol 34 (02) ◽  
pp. 381-394 ◽  
Author(s):  
Dirk Tasche
Keyword(s):  
Law Of Large Numbers ◽  
Exponential Rate ◽  
Cantelli Lemma ◽  
Mixing Rate ◽  
Strong Law ◽  
Moment Conditions ◽  
Sequence Of Events ◽  
Large Numbers ◽  
Strongly Mixing ◽  
Mixing Sequences

Assume a given sequence of events to be strongly mixing at a polynomial or exponential rate. We show that the conclusion of the second Borel-Cantelli lemma holds if the series of the probabilities of the events diverges at a certain rate depending on the mixing rate of the events. An application to necessary moment conditions for the strong law of large numbers is given.

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