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.

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 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.

Laws of large numbers for q-dependent random variables and nonextensive statistical mechanics

2017 ◽  
Vol 31 (15) ◽  
pp. 1750117
Author(s):  
Marco A. S. Trindade
Keyword(s):  
Statistical Mechanics ◽  
Stochastic Processes ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Nonextensive Statistical Mechanics ◽  
Dependent Random Variables ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Large Numbers

In this work, we prove a weak law and a strong law of large numbers through the concept of [Formula: see text]-product for dependent random variables, in the context of nonextensive statistical mechanics. Applications for the consistency of estimators are presented and connections with stochastic processes are discussed.

scholarly journals On Chung-Teicher type strong law for arrays of vector-valued random variables

2004 ◽  
Vol 2004 (9) ◽  
pp. 443-458
Author(s):  
Anna Kuczmaszewska
Keyword(s):  
Banach Space ◽  
Random Variables ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers ◽  
Random Elements ◽  
Vector Valued

We study the equivalence between the weak and strong laws of large numbers for arrays of row-wise independent random elements with values in a Banach spaceℬ. The conditions under which this equivalence holds are of the Chung or Chung-Teicher types. These conditions are expressed in terms of convergence of specific series ando(1)requirements on specific weighted row-wise sums. Moreover, there are not any conditions assumed on the geometry of the underlying Banach space.

scholarly journals Negatively dependent bounded random variable probability inequalities and the strong law of large numbers

2000 ◽  
Vol 13 (3) ◽  
pp. 261-267 ◽  
Author(s):  
M. Amini ◽  
A. Bozorgnia
Keyword(s):  
Law Of Large Numbers ◽  
Random Variables ◽  
Random Variable ◽  
Strong Law ◽  
Probability Inequalities ◽  
Large Numbers ◽  
Bounded Random Variable ◽  
Strongly Consistent ◽  
Uniformly Bounded ◽  
Variable Probability

Let X1,…,Xn be negatively dependent uniformly bounded random variables with d.f. F(x). In this paper we obtain bounds for the probabilities P(|∑i=1nXi|≥nt) and P(|ξˆpn−ξp|>ϵ) where ξˆpn is the sample pth quantile and ξp is the pth quantile of F(x). Moreover, we show that ξˆpn is a strongly consistent estimator of ξp under mild restrictions on F(x) in the neighborhood of ξp. We also show that ξˆpn converges completely to ξp.

LAWS OF LARGE NUMBERS FOR WEIGHTED SUMS OF FUZZY SET-VALUED RANDOM VARIABLES

2004 ◽  
Vol 12 (06) ◽  
pp. 811-825 ◽  
Author(s):  
LI GUAN ◽  
SHOUMEI LI
Keyword(s):  
Fuzzy Set ◽  
Hausdorff Metric ◽  
Random Variables ◽  
Random Variable ◽  
Weighted Sums ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers

In this paper, we shall present weak and strong laws of large numbers (WLLN's and SLLN's) for weighted sums of independent (not necessarily identically distributed) fuzzy set-valued random variables in the sense of the extended Hausdorff metric [Formula: see text], based on the result of set-valued random variable obtained by Taylor and Inoue32,33. This work is a continuation of Li and Ogura20.

scholarly journals On the strong law for arrays and for the bootstrap mean and variance

1997 ◽  
Vol 20 (2) ◽  
pp. 375-382 ◽  
Author(s):  
Tien-Chung Hu ◽  
R. L. Taylor
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Interesting Application ◽  
Strong Law ◽  
Moment Conditions ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers ◽  
Mean And Variance ◽  
Rowwise Independent

Chung type strong laws of large numbers are obtained for arrays of rowwise independent random variables under various moment conditions. An interesting application of these results is the consistency of the bootstrap mean and variance.

Sign in / Sign up

Export Citation Format

Share Document