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.

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 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.

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.

scholarly journals Strong Laws of Large Numbers for𝔹-Valued Random Fields

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Zbigniew A. Lagodowski
Keyword(s):  
Banach Space ◽  
Random Fields ◽  
Law Of Large Numbers ◽  
Random Vectors ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers ◽  
Geometry Of Banach Space ◽  
Necessary And Sufficient

We extend to random fields case, the results of Woyczynski, who proved Brunk's type strong law of large numbers (SLLNs) for𝔹-valued random vectors under geometric assumptions. Also, we give probabilistic requirements for above-mentioned SLLN, related to results obtained by Acosta as well as necessary and sufficient probabilistic conditions for the geometry of Banach space associated to the strong and weak law of large numbers for multidimensionally indexed random vectors.

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.

scholarly journals On the strong law of large numbers for ϕ-sub-Gaussian random variables

2021 ◽  
Vol 73 (3) ◽  
pp. 431-436
Author(s):  
K. Zajkowski
Keyword(s):  
Natural Number ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Random Variable ◽  
Gaussian Random Variables ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers

UDC 517.9 For let if and if . For a random variable ξ let denote ; is a norm in a space - subgaussian random variables. We prove that if for a sequence there exist positive constants and such that for every natural number the following inequality holds then converges almost surely to zero as . This result is a generalization of the strong law of large numbers for independent sub-Gaussian random variables [see R. L. Taylor, T.-C. Hu, <em>Sub-Gaussian techniques in proving strong laws of large numbers</em>, Amer. Math. Monthly, <strong>94</strong>, 295 – 299 (1987)] to the case of dependent -sub-Gaussian random variables.

scholarly journals Strong laws of large numbers for the sequence of the maximum of partial sums of i.i.d. random variables

2019 ◽  
Vol 39 (1) ◽  
pp. 19-38
Author(s):  
Shuhua Chang ◽  
Deli Li ◽  
Andrew Rosalsky
Keyword(s):  
Law Of Large Numbers ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Random Variable ◽  
Partial Sums ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers ◽  
Necessary And Sufficient

Let 0 < p ≤ 2, let {Xn; n ≥ 1} be a sequence of independent copies of a real-valued random variable X, and set Sn = X1 + . . . + Xn, n ≥ ­ 1. Motivated by a theorem of Mikosch 1984, this note is devoted to establishing a strong law of large numbers for the sequence {max1≤k≤n |Sk| ; n ≥ ­ 1}. More specifically, necessary and sufficient conditions are given forlimn→∞ max1≤k≤n |Sk|log n−1 = e1/p a.s.,where log x = loge max{e, x}, x ≥­ 0.

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