Probabilistic characterizations of the Generalized Domain of Attraction of the multivariate normal law

1994 ◽  
Vol 7 (4) ◽  
pp. 857-866 ◽  
Author(s):  
Steven J. Sepanski
Keyword(s):  
Domain Of Attraction ◽  
Multivariate Normal ◽  
Normal Law ◽  
Generalized Domain Of Attraction

Weak invariance principle for mixing sequences in the domain of attraction of normal law

2009 ◽  
Vol 46 (3) ◽  
pp. 329-343
Author(s):  
Raluca Balan ◽  
Rafał Kulik
Keyword(s):  
Invariance Principle ◽  
Domain Of Attraction ◽  
Weak Invariance Principle ◽  
Weak Invariance ◽  
Mixing Coefficients ◽  
Beta 2 ◽  
Mixing Sequences ◽  
Normal Law ◽  
Mixing Sequence ◽  
Logarithmic Growth

In this article we prove a weak invariance principle for a strictly stationary φ -mixing sequence { Xj } j≧1 , whose truncated variance function \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$L(x): = EX_1^2 1_{\{ |X_1 | \leqq _x \} }$$ \end{document} is slowly varying at ∞ and mixing coefficients satisfy the logarithmic growth condition: Σ n ≧1φ1/2 (2 n ) < ∞. This will be done under the condition that \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop {\lim }\limits_n Var\left( {\sum\limits_{j = 1}^n {\hat X_j } } \right)/\left[ {\sum\limits_{j = 1}^n {Var (\hat X_j )} } \right] = \beta ^2$$ \end{document} exists in (0, ∞), where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\hat X_j = X_j I_{\{ |X_j | \leqq \eta _j \} }$$ \end{document} and ηn2 ∼ nL ( ηn ).

scholarly journals On a conjecture of Seneta on regular variation of truncated moments

2021 ◽  
Vol 109 (123) ◽  
pp. 77-82
Author(s):  
Péter Kevei
Keyword(s):  
Distribution Function ◽  
Relative Stability ◽  
Regular Variation ◽  
Domain Of Attraction ◽  
Random Variable ◽  
Nonnegative Random Variable ◽  
Equivalent Characterization ◽  
Regularly Varying ◽  
Normal Law

We prove that h?(x) = ??x0 y??1F?(y)dy is regularly varying with index ? [0, ?) if and only if V?(x) = ?[0,x] y?dF(y) is regularly varying with the same index, where ? > 0, F(x) is a distribution function of a nonnegative random variable, and F?(x) = 1?F(x). This contains at ? = 0, ?= 1 a result of Rogozin [8] on relative stability, and at ? = 0, ? = 2 a new, equivalent characterization of the domain of attraction of the normal law. For ? = 0 and ? > 0 our result implies a recent conjecture by Seneta [9].

scholarly journals Lower bound for the number of real roots of a random algebraic polynomial

1983 ◽  
Vol 35 (1) ◽  
pp. 18-27 ◽  
Author(s):  
M. N. Mishra ◽  
N. N. Nayak ◽  
S. Pattanayak
Keyword(s):  
Lower Bound ◽  
Algebraic Polynomial ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Independent Random Variables ◽  
Real Numbers ◽  
Exceptional Set ◽  
Real Roots ◽  
Normal Law ◽  
Number Of Real Roots

AbstractLet X1, X2, …, Xn be identically distributed independent random variables belonging to the domain of attraction of the normal law, have zero means and Pr{Xr ≠ 0} > 0. Suppose a0, a1, …, an are non-zero real numbers and max and εn is such that as n → ∞, εn. If Nn be the number of real roots of the equation then for n > n0, Nn > εn log n outside an exceptional set of measure at most provided limn→∞ (kn/tn) is finite.

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