Limit theorems in the set up of sumnation of a random number of independent identically distributed random variables

Melamed, J. A.

doi:10.1007/bfb0084174

Cited by 16 publications

(11 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…and the solution is unique. Since T n (x) = cos(n · arccos x) = cosh(n · arccosh x) , the direct plugging gives that the function ϕ(t) = 1/ cosh √ 2t (11) satisfies the system (9), as well as the conditions (10). Hence, the function…”

Section: A New Examplementioning

confidence: 98%

On a Class of Distributions Stable Under Random Summation

et al. 2012

View full text Add to dashboard Cite

We investigate a family of distributions having a property of stability-under-addition, provided that the number ν of added-up random variables in the random sum is also a random variable. We call the corresponding property a ν-stability and investigate the situation with the semigroup generated by the generating function of ν is commutative.Using results from the theory of iterations of analytic functions, we show that the characteristic function of such a ν-stable distribution can be represented in terms of Chebyshev polynomials, and for the case of ν-normal distribution, the resulting characteristic function corresponds to the hyperbolic secant distribution.We discuss some specific properties of the class and present particular examples.

show abstract

Section: A New Examplementioning

confidence: 98%

On a Class of Distributions Stable Under Random Summation

et al. 2012

View full text Add to dashboard Cite

show abstract

“…There exist examples of pairs of commutative functions which are not rational. In this section we provide two classes of such functions: the first was investigated by Melamed [15] and the second appears for the first time here.…”

Section: Examples With Nonrational Generating Functionsmentioning

confidence: 99%

“…In this section we provide two classes of such functions: the first was investigated by Melamed [15] and the second appears for the first time here. [15] for details.) Consider the family of generating functions…”

Section: Examples With Nonrational Generating Functionsmentioning

confidence: 99%

On a Class of Distributions Stable Under Random Summation

Klebanov

Kakosyan

Rachev

et al. 2012

Journal of Applied Probability

View full text Add to dashboard Cite

We study a family of distributions that satisfy the stability-under-addition property, provided that the number ν of random variables in a sum is also a random variable. We call the corresponding property ν-stability and investigate the situation when the semigroup generated by the generating function of ν is commutative. Using results from the theory of iterations of analytic functions, we describe ν-stable distributions generated by summations with rational generating functions. A new case in this class of distributions arises when generating functions are linked with Chebyshev polynomials. The analogue of normal distribution corresponds to the hyperbolic secant distribution.

show abstract

“…Then the distribution of the corresponding sum in (1) is M -divisible, i.e., it can be represented as the convolution of M distribution functions. In this particular case, the fouth central moment μ 4 ( U ) satisfies the inequality [ 14 ]:…”

Section: The Law Of Large Numbers and Random Summationmentioning

confidence: 99%

“…Under mild conditions, these inequalities hold in the case of random ν as well [ 14 ]. If the inequality (15) is met in real biological data, this fact will lend additional support to the presence of signal summation in microarray technology.…”

Section: The Law Of Large Numbers and Random Summationmentioning

confidence: 99%

A nitty-gritty aspect of correlation and network inference from gene expression data

Klebanov

Yakovlev

2008

Biology Direct

View full text Add to dashboard Cite

Background: All currently available methods of network/association inference from microarray gene expression measurements implicitly assume that such measurements represent the actual expression levels of different genes within each cell included in the biological sample under study. Contrary to this common belief, modern microarray technology produces signals aggregated over a random number of individual cells, a "nitty-gritty" aspect of such arrays, thereby causing a random effect that distorts the correlation structure of intra-cellular gene expression levels.Results: This paper provides a theoretical consideration of the random effect of signal aggregation and its implications for correlation analysis and network inference. An attempt is made to quantitatively assess the magnitude of this effect from real data. Some preliminary ideas are offered to mitigate the consequences of random signal aggregation in the analysis of gene expression data. Conclusion:Resulting from the summation of expression intensities over a random number of individual cells, the observed signals may not adequately reflect the true dependence structure of intra-cellular gene expression levels needed as a source of information for network reconstruction. Whether the reported effect is extrime or not, the important point, is to reconize and incorporate such signal source for proper inference. The usefulness of inference on genetic regulatory structures from microarray data depends critically on the ability of investigators to overcome this obstacle in a scientifically sound way. Reviewers: This article was reviewed

show abstract

Limit theorems in the set up of sumnation of a random number of independent identically distributed random variables

Cited by 16 publications

References 5 publications

On a Class of Distributions Stable Under Random Summation

On a Class of Distributions Stable Under Random Summation

On a Class of Distributions Stable Under Random Summation

A nitty-gritty aspect of correlation and network inference from gene expression data

Contact Info

Product

Resources

About