scholarly journals A converse to the log-log law for Martingales

1974 ◽  
Vol 17 (4) ◽  
pp. 496-499
Author(s):  
W. L. Steiger
Keyword(s):  
Random Variables ◽  
Difference Sequence ◽  
Martingale Difference ◽  
Martingale Difference Sequence ◽  
Iterated Logarithm ◽  
Present Context ◽  
Log Law ◽  
Independent And Identically Distributed

For sums of independent and identically distributed random variables xn, the HartmanWintner law of the iterated logarithm is equivalent to xn ∈ L2. We show that this is also true when the xn, form a stationary, ergodic martingale difference sequence. This is accomplished by extending a theorem of Volker Strassen to the present context.

Exponential Inequalities for Bounded Random Variables

2015 ◽  
Vol 65 (6) ◽  
Author(s):  
Guangyue Huang ◽  
Xin Guo ◽  
Hongxia Du ◽  
Yi He ◽  
Yu Miao
Keyword(s):  
Markov Chains ◽  
Random Variables ◽  
Independent Random Variables ◽  
Difference Sequence ◽  
Martingale Difference ◽  
Exponential Inequalities ◽  
Martingale Difference Sequence ◽  
Associated Random Variables ◽  
Negatively Associated ◽  
Negatively Associated Random Variables

AbstractIn the paper, several precise exponential inequalities for the sums of bounded or semi-bounded random variables are established, which involve independent random variables, martingale difference sequence, negatively associated random variables, Markov chains.

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.

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 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.

scholarly journals ON THE CENTRAL AND LOCAL LIMIT THEOREM FOR MARTINGALE DIFFERENCE SEQUENCES

2004 ◽  
Vol 04 (02) ◽  
pp. 153-173
Author(s):  
MOHAMED EL MACHKOURI ◽  
DALIBOR VOLNÝ
Keyword(s):  
Limit Theorem ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Ergodic Measure ◽  
Difference Sequence ◽  
Martingale Difference ◽  
Lattice Distribution ◽  
Martingale Difference Sequence ◽  
The Central Limit Theorem ◽  
Difference Sequences

Let [Formula: see text] be a Lebesgue space and T: Ω→Ω an ergodic measure-preserving automorphism with positive entropy. We show that there is a bounded and strictly stationary martingale difference sequence defined on Ω with a common nondegenerate lattice distribution satisfying the central limit theorem with an arbitrarily slow rate of convergence and not satisfying the local limit theorem. A similar result is established for martingale difference sequences with densities provided the entropy is infinite. In addition, the martingale difference sequence may be chosen to be strongly mixing.

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.

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