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.

scholarly journals The law of the iterated logarithm for exchangeable random variables

1995 ◽  
Vol 18 (2) ◽  
pp. 391-396
Author(s):  
Hu-Ming Zhang ◽  
Robert L. Taylor
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Exchangeable Random Variables ◽  
Iterated Logarithm ◽  
The Law ◽  
Necessary And Sufficient

In this note, necessary and sufficient conditions for laws of the iterated logarithm are developed for exchangeable random variables.

scholarly journals Asymptotic normality for the number of records from general distributions

2011 ◽  
Vol 43 (02) ◽  
pp. 422-436 ◽  
Author(s):  
Raul Gouet ◽  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Asymptotic Normality ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
General Distribution ◽  
Continuous Components ◽  
General Distributions ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

We provide necessary and sufficient conditions for the asymptotic normality of N n , the number of records among the first n observations from a sequence of independent and identically distributed random variables, with general distribution F. In the case of normality we identify the centering and scaling sequences. Also, we characterize distributions for which the limit is not normal in terms of their discrete and continuous components.

scholarly journals Recurrence properties of Processes with stationary independent increments

1964 ◽  
Vol 4 (2) ◽  
pp. 223-228 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Random Walk ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Independent Increments ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

Let X1, X2,…Xn, … be independent and identically distributed random variables, and write . In [2] Chung and Fuchs have established necessary and sufficient conditions for the random walk {Zn} to be recurrent, i.e. for Zn to return infinitely often to every neighbourhood of the origin. The object of this paper is to obtain similar results for the corresponding process in continuous time.

scholarly journals Asymptotic normality for the number of records from general distributions

2011 ◽  
Vol 43 (2) ◽  
pp. 422-436 ◽  
Author(s):  
Raul Gouet ◽  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Asymptotic Normality ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
General Distribution ◽  
Continuous Components ◽  
General Distributions ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

We provide necessary and sufficient conditions for the asymptotic normality of Nn, the number of records among the first n observations from a sequence of independent and identically distributed random variables, with general distribution F. In the case of normality we identify the centering and scaling sequences. Also, we characterize distributions for which the limit is not normal in terms of their discrete and continuous components.

On the Ordering Property and Law of Importation in Fuzzy Logic

2010 ◽  
Vol 1 (3) ◽  
pp. 17-30
Author(s):  
Huiwen Deng ◽  
Huan Jiang
Keyword(s):  
Fuzzy Logic ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Fuzzy Implications ◽  
Fuzzy Implication ◽  
The Law ◽  
Necessary And Sufficient ◽  
Triple I Method

In this paper, the authors investigate the ordering property (OP), , together with the general form of the law of importation(LI), i.e., , whereis a t-norm andis a fuzzy implication for the four main classes of fuzzy implications. The authors give necessary and sufficient conditions under which both (OP) and (LI) holds for S-, R-implications and some specific families of QL-, D-implications. Following this, the paper proposes the sufficient condition under which the equivalence between CRI and triple I method for FMP can be established. Moreover, this conclusion can be viewed as a unified triple I method, a generalized form of the known results proposed by Wang and Pei.

Stability of maxima of random variables defined on a Markov chain

1972 ◽  
Vol 4 (2) ◽  
pp. 285-295 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Markov Chain ◽  
Regularity Condition ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Finite Product ◽  
Normalizing Constants ◽  
Necessary And Sufficient

Consider maxima Mn of a sequence of random variables defined on a finite Markov chain. Necessary and sufficient conditions for the existence of normalizing constants Bn such that are given. The problem can be reduced to studying maxima of i.i.d. random variables drawn from a finite product of distributions πi=1mHi(x). The effect of each factor Hi(x) on the behavior of maxima from πi=1mHi is analyzed. Under a mild regularity condition, Bn can be chosen to be the maximum of the m quantiles of order (1 - n-1) of the H's.

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 < ∞, the law of the iterated logarithm (Hartman and Wintner [1]) tells us that

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