scholarly journals On the nonexistence of a law of the iterated logarithm for weighted sums of identically distributed random variables

1990 ◽  
Vol 3 (2) ◽  
pp. 135-140 ◽  
Author(s):  
André Adler
Keyword(s):  
Random Variables ◽  
Weighted Sums ◽  
Iterated Logarithm ◽  
Independent And Identically Distributed

For weighted sums of independent and identically distributed random variables, conditions are placed under which a generalized law of the iterated logarithm cannot hold, thereby extending the usual nonweighted situation.

Law of the iterated logarithm for self-normalized sums and their increments

2006 ◽  
Vol 43 (1) ◽  
pp. 79-114
Author(s):  
Han-Ying Liang ◽  
Jong-Il Baek ◽  
Josef Steinebach
Keyword(s):  
Random Variables ◽  
Domain Of Attraction ◽  
Weighted Sums ◽  
Partial Sums ◽  
Stable Law ◽  
Iterated Logarithm ◽  
Independent And Identically Distributed

Let X1, X2,… be independent, but not necessarily identically distributed random variables in the domain of attraction of a stable law with index 0<a<2. This paper uses Mn=max 1?i?n|Xi| to establish a self-normalized law of the iterated logarithm (LIL) for partial sums. Similarly self-normalized increments of partial sums are studied as well. In particular, the results of self-normalized sums of Horváth and Shao[9]under independent and identically distributed random variables are extended and complemented. As applications, some corresponding results for self-normalized weighted sums of iid random variables are also concluded.

scholarly journals On the law of the iterated logarithm in the infinite variance case

1980 ◽  
Vol 30 (1) ◽  
pp. 5-14 ◽  
Author(s):  
R. A. Maller
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Infinite Variance ◽  
Iterated Logarithm ◽  
The Law ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

AbstractThe main purpose of the paper is to give necessary and sufficient conditions for the almost sure boundedness of (Sn – αn)/B(n), where Sn = X1 + X2 + … + XmXi being independent and identically distributed random variables, and αnand B(n) being centering and norming constants. The conditions take the form of the convergence or divergence of a series of a geometric subsequence of the sequence P(Sn − αn > a B(n)), where a is a constant. The theorem is distinguished from previous similar results by the comparative weakness of the subsidiary conditions and the simplicity of the calculations. As an application, a law of the iterated logarithm general enough to include a result of Feller is derived.

On the converse to the iterated logarithm law

1968 ◽  
Vol 5 (01) ◽  
pp. 210-215 ◽  
Author(s):  
C. C. Heyde
Keyword(s):  
Random Variables ◽  
Iterated Logarithm ◽  
Logarithm Law ◽  
The Law ◽  
Image Position ◽  
Iterated Logarithm Law ◽  
Independent And Identically Distributed

Let Xi, i = 1, 2, 3,… be a sequence of independent and identically distributed random variables with law ℓ(X) and write. if EX = 0 and EX2 = σ2 &lt; ∞, the law of the iterated logarithm (Hartman and Wintner [1]) tells us that

scholarly journals On the rate of convergence in the strong law of large numbers for arrays

1992 ◽  
Vol 45 (3) ◽  
pp. 479-482 ◽  
Author(s):  
Tien-Chung Hu ◽  
N.C. Weber
Keyword(s):  
Rate Of Convergence ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Strong Law ◽  
Second Moment ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
The Law ◽  
Independent And Identically Distributed

For sequences of independent and identically distributed random variables it is well known that the existence of the second moment implies the law of the iterated logarithm. We show that the law of the iterated logarithm does not extend to arrays of independent and identically distributed random variables and we develop an analogous rate result for such arrays under finite fourth moments.

scholarly journals An extension of kesten's generalised law of the iterated Logarithm

1980 ◽  
Vol 21 (3) ◽  
pp. 393-406 ◽  
Author(s):  
R. A. Maller
Keyword(s):  
Normal Distribution ◽  
Random Variables ◽  
Classic Result ◽  
Iterated Logarithm ◽  
Norming Sequence ◽  
Domain Of Partial Attraction ◽  
Limit Points ◽  
Image Position ◽  
Independent And Identically Distributed

Let Xi be independent and identically distributed random variables with Sn = X1 + X2 + … + Xn. We extend a classic result of Kesten, by showing that if Xiare in the domain of partial attraction of the normal distribution, there are sequences αn and B(n) for whichalmost surely, and the almost sure limit points of (sn−αn)/b(n) coincide with the interval [−1, l]. The norming sequence B(n) is slightly different to that used by Kesten, and has properties that are less desirable. The converse to the above result is known to be true by results of Heyde and Rogozin.

scholarly journals Exact laws for sums of ratios of order statistics from the Pareto distribution

2006 ◽  
Vol 4 (1) ◽  
pp. 1-4 ◽  
Author(s):  
André Adler
Keyword(s):  
Order Statistics ◽  
Pareto Distribution ◽  
Random Variables ◽  
Weighted Sums ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Independent And Identically Distributed

AbstractConsider independent and identically distributed random variables {X nk, 1 ≤ k ≤ m, n ≤ 1} from the Pareto distribution. We select two order statistics from each row, X n(i) ≤ X n(j), for 1 ≤ i < j ≤ = m. Then we test to see whether or not Laws of Large Numbers with nonzero limits exist for weighted sums of the random variables R ij = X n(j)/X n(i).

scholarly journals Chover-type laws of the iterated logarithm for weighted sums of ρ∗-mixing sequences

2006 ◽  
Vol 2006 ◽  
pp. 1-7 ◽  
Author(s):  
Guang-hui Cai
Keyword(s):  
Random Variables ◽  
Weighted Sums ◽  
Type Result ◽  
Stable Law ◽  
Iterated Logarithm ◽  
Mixing Sequences

To derive a Baum-Katz-type result, we establish a Chover-type law of the iterated logarithm for the weighted sums of ρ∗-mixing and identically distributed random variables with a distribution in the domain of a stable law. Our result obtained not only generalizes the main results of Peng and Qi (2003) and Qi and Cheng (1996) to ρ∗-mixing sequences of random variables, but also improves them.

scholarly journals On the laws of the iterated logarithm under sub-linear expectations

2021 ◽  
Vol 6 (4) ◽  
pp. 409
Author(s):  
Li-Xin Zhang
Keyword(s):  
Necessary Conditions ◽  
Random Variables ◽  
Independent Random Variables ◽  
Exponential Inequalities ◽  
Iterated Logarithm ◽  
Sufficient And Necessary Conditions ◽  
The Law ◽  
Independent And Identically Distributed

<p style='text-indent:20px;'>In this paper, we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space, where the random variables are not necessarily identically distributed. Exponential inequalities for the maximum sum of independent random variables and Kolmogorov’s converse exponential inequalities are established as tools for showing the law of the iterated logarithm. As an application, the sufficient and necessary conditions of the law of the iterated logarithm for independent and identically distributed random variables under the sub-linear expectation are obtained. In the paper, it is also shown that if the sub-linear expectation space is rich enough, it will have no continuous capacity. The laws of the iterated logarithm are established without the assumption on the continuity of capacities.</p>

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