On the Convergence Rate in the Invariance Principle for Strongly Mixing Sequences

1984 ◽  
Vol 28 (4) ◽  
pp. 816-821 ◽  
Author(s):  
V. V. Gorodetskii
Keyword(s):  
Convergence Rate ◽  
Invariance Principle ◽  
Strongly Mixing ◽  
Mixing Sequences

scholarly journals Limiting behavior of the perturbed empirical distribution functions evaluated at U-statistics for strongly mixing sequences of random variables

1997 ◽  
Vol 10 (1) ◽  
pp. 3-20 ◽  
Author(s):  
Shan Sun ◽  
Ching-Yuan Chiang
Keyword(s):  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Random Variables ◽  
Distribution Functions ◽  
Limiting Behavior ◽  
U Statistics ◽  
Strongly Mixing ◽  
Almost Sure Representation ◽  
Mixing Sequences ◽  
Empirical Distribution Functions

We prove the almost sure representation, a law of the iterated logarithm and an invariance principle for the statistic Fˆn(Un) for a class of strongly mixing sequences of random variables {Xi,i≥1}. Stationarity is not assumed. Here Fˆn is the perturbed empirical distribution function and Un is a U-statistic based on X1,…,Xn.

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 A note on two measures of dependence and mixing sequences

1983 ◽  
Vol 15 (02) ◽  
pp. 461-464
Author(s):  
Magda Peligrad
Keyword(s):  
Invariance Principle ◽  
Random Variables ◽  
Time Process ◽  
Mixing Rate ◽  
Coefficient Of Correlation ◽  
Mixing Coefficient ◽  
Measures Of Dependence ◽  
Mixing Coefficients ◽  
Two Measures ◽  
Mixing Sequences

In this note we establish an inequality between the maximal coefficient of correlation and the φ -mixing coefficient which is symmetric in its arguments. Motivated by this inequality, we introduce a mixing coefficient which is the product of two φ -mixing coefficients. We also study an invariance principle under conditions imposed on this new mixing coefficient. As a consequence of this result it follows that the invariance principle holds when either the direct-time process or its time-reversed process is φ -mixing; when both processes are φ-mixing the invariance principle holds for sequences of L 2-integrable random variables under a mixing rate weaker than that used by Ibragimov.

scholarly journals On the accuracy of bootstrapping sample quantiles of strongly mixing sequences

2007 ◽  
Vol 82 (2) ◽  
pp. 263-282 ◽  
Author(s):  
Shuxia Sun
Keyword(s):  
Rate Of Convergence ◽  
Marginal Distribution ◽  
Regularity Conditions ◽  
One Dimensional ◽  
Strongly Mixing ◽  
Moving Block Bootstrap ◽  
Mixing Sequences ◽  
The One ◽  
Approximations To Distributions ◽  
Sample Quantiles

AbstractIn this paper, we examine the rate of convergence of moving block bootstrap (MBB) approximations to the distributions of normalized sample quantiles based on strongly mixing observations. Under suitable smoothness and regularity conditions on the one-dimensional marginal distribution function, the rate of convergence of the MBB approximations to distributions of centered and scaled sample quantiles is of order O(n−1¼ log logn).

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