scholarly journals Asymptotic distribution for the sum and maximum of Gaussian processes

2000 ◽  
Vol 37 (04) ◽  
pp. 958-971 ◽  
Author(s):  
W. P. McCormick ◽  
Y. Qi
Keyword(s):  
Correlation Function ◽  
Asymptotic Distribution ◽  
Gaussian Processes ◽  
Random Variables ◽  
Stationary Sequence ◽  
Asymptotic Independence ◽  
Standard Normal

Previous work on the joint asymptotic distribution of the sum and maxima of Gaussian processes is extended here. In particular, it is shown that for a stationary sequence of standard normal random variables with correlation function r, the condition r(n)ln n = o(1) as n → ∞ suffices to establish the asymptotic independence of the sum and maximum.

scholarly journals Asymptotic distribution for the sum and maximum of Gaussian processes

2000 ◽  
Vol 37 (4) ◽  
pp. 958-971 ◽  
Author(s):  
W. P. McCormick ◽  
Y. Qi
Keyword(s):  
Correlation Function ◽  
Asymptotic Distribution ◽  
Gaussian Processes ◽  
Random Variables ◽  
Stationary Sequence ◽  
Asymptotic Independence ◽  
Standard Normal

Previous work on the joint asymptotic distribution of the sum and maxima of Gaussian processes is extended here. In particular, it is shown that for a stationary sequence of standard normal random variables with correlation function r, the condition r(n)ln n = o(1) as n → ∞ suffices to establish the asymptotic independence of the sum and maximum.

scholarly journals Asymptotic distribution of sum and maximum for Gaussian processes

1999 ◽  
Vol 36 (4) ◽  
pp. 1031-1044 ◽  
Author(s):  
Hwai-Chung Ho ◽  
William P. McCormick
Keyword(s):  
Asymptotic Distribution ◽  
Gaussian Processes ◽  
Continuous Time ◽  
Joint Distribution ◽  
Covariance Function ◽  
Random Variables ◽  
Regularity Conditions ◽  
Gaussian Sequence ◽  
Stationary Gaussian Sequence ◽  
Standard Normal

Let {Xn, n ≥ 0} be a stationary Gaussian sequence of standard normal random variables with covariance function r(n) = EX0Xn. Let Under some mild regularity conditions on r(n) and the condition that r(n)lnn = o(1) or (r(n)lnn)−1 = O(1), the asymptotic distribution of is obtained. Continuous-time results are also presented as well as a tube formula tail area approximation to the joint distribution of the sum and maximum.

scholarly journals Asymptotic distribution of sum and maximum for Gaussian processes

1999 ◽  
Vol 36 (04) ◽  
pp. 1031-1044 ◽  
Author(s):  
Hwai-Chung Ho ◽  
William P. McCormick
Keyword(s):  
Asymptotic Distribution ◽  
Gaussian Processes ◽  
Continuous Time ◽  
Joint Distribution ◽  
Covariance Function ◽  
Random Variables ◽  
Regularity Conditions ◽  
Gaussian Sequence ◽  
Stationary Gaussian Sequence ◽  
Standard Normal

Let {X n , n ≥ 0} be a stationary Gaussian sequence of standard normal random variables with covariance function r(n) = E X 0 X n . Let Under some mild regularity conditions on r(n) and the condition that r(n)lnn = o(1) or (r(n)lnn)−1 = O(1), the asymptotic distribution of is obtained. Continuous-time results are also presented as well as a tube formula tail area approximation to the joint distribution of the sum and maximum.

scholarly journals Precise Rates in Log Laws for NA Sequences

2007 ◽  
Vol 2007 ◽  
pp. 1-11
Author(s):  
Yuexu Zhao
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Stationary Sequence ◽  
Normal Random Variable ◽  
Negatively Associated ◽  
Standard Normal Random Variable ◽  
Na Random Variables ◽  
Standard Normal ◽  
Na Sequences

LetX1,X2,…be a strictly stationary sequence of negatively associated (NA) random variables withEX1=0, setSn=X1+⋯+Xn, suppose thatσ2=EX12+2∑n=2∞EX1Xn>0andEX12<∞,if−1<α≤1;EX12(log|X1|)α<∞, ifα>1. We provelimε↓0ε2α+2∑n=1∞((logn)α/n)P(|Sn|≥σ(ε+κn)2nlogn)=2−(α+1)(α+1)−1E|N|2α+2, whereκn=O(1/logn)and N is the standard normal random variable.

scholarly journals Estimates for the distribution of Hölder semi-norms of real stationary Gaussian processes with a stable correlation function

2020 ◽  
pp. 25-30
Author(s):  
D. Zatula
Keyword(s):  
Correlation Function ◽  
Present Article ◽  
Gaussian Processes ◽  
Random Variables ◽  
Random Processes ◽  
Hölder Spaces ◽  
Stationary Gaussian Processes ◽  
Holder Spaces

Complex random variables and processes with a vanishing pseudo-correlation are called proper. There is a class of stationary proper complex random processes that have a stable correlation function. In the present article we consider real stationary Gaussian processes with a stable correlation function. It is shown that the trajectories of stationary Gaussian proper complex random processes with zero mean belong to the Orlich space generated by the function $U(x) = e^{x^2/2}-1$. Estimates are obtained for the distribution of semi-norms of sample functions of Gaussian proper complex random processes with a stable correlation function, defined on the compact $\mathbb{T} = [0,T]$, in Hölder spaces.

scholarly journals Asymptotic Conditional Distribution of Exceedance Counts

2012 ◽  
Vol 44 (01) ◽  
pp. 270-291 ◽  
Author(s):  
Michael Falk ◽  
Diana Tichy
Keyword(s):  
Asymptotic Distribution ◽  
Conditional Distribution ◽  
Random Variables ◽  
Extremal Index ◽  
Stationary Sequence ◽  
Independent Random Variables ◽  
Fragility Index ◽  
Expected Number

We investigate the asymptotic distribution of the number of exceedances among d identically distributed but not necessarily independent random variables (RVs) above a sequence of increasing thresholds, conditional on the assumption that there is at least one exceedance. Our results enable the computation of the fragility index, which represents the expected number of exceedances, given that there is at least one exceedance. Computed from the first d RVs of a strictly stationary sequence, we show that, under appropriate conditions, the reciprocal of the fragility index converges to the extremal index corresponding to the stationary sequence as d increases.

Asymptotic properties for the loglog laws under positive association

2012 ◽  
Vol 62 (5) ◽  
Author(s):  
Xiao-Rong Yang ◽  
Ke-Ang Fu
Keyword(s):  
Gamma Function ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Positive Association ◽  
Random Variable ◽  
Stationary Sequence ◽  
Normal Random Variable ◽  
Finite Variance ◽  
Associated Random Variables ◽  
Standard Normal

AbstractLet {X n: n ≥ 1} be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set $$S_n = \sum\limits_{k = 1}^n {X_k }$$, $$Mn = \mathop {\max }\limits_{k \leqslant n} \left| {S_k } \right|$$, n ≥ 1. Suppose that $$0 < \sigma ^2 = EX_1^2 + 2\sum\limits_{k = 2}^\infty {EX_1 X_k < \infty }$$. In this paper, we prove that if E|X 1|2+δ < for some δ ∈ (0, 1], and $$\sum\limits_{j = n + 1}^\infty {Cov\left( {X_1 ,X_j } \right) = O\left( {n^{ - \alpha } } \right)}$$ for some α > 1, then for any b > −1/2 $$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2b + 1} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^{b - 1/2} }} {{n^{3/2} \log n}}} E\left\{ {M_n - \sigma \varepsilon \sqrt {2n\log \log n} } \right\}_ + = \frac{{2^{ - 1/2 - b} E\left| N \right|^{2(b + 1)} }} {{(b + 1)(2b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2(b + 1)} }}}$$ and $$\mathop {\lim }\limits_{\varepsilon \nearrow \infty } \varepsilon ^{ - 2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^b }} {{n^{3/2} \log n}}E\left\{ {\sigma \varepsilon \sqrt {\frac{{\pi ^2 n}} {{8\log \log n}}} - M_n } \right\}} _ + = \frac{{\Gamma (b + 1/2)}} {{\sqrt 2 (b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2b + 2} }}} ,$$ where x + = max{x, 0}, N is a standard normal random variable, and Γ(·) is a Gamma function.

scholarly journals Asymptotic Conditional Distribution of Exceedance Counts

2012 ◽  
Vol 44 (1) ◽  
pp. 270-291 ◽  
Author(s):  
Michael Falk ◽  
Diana Tichy
Keyword(s):  
Asymptotic Distribution ◽  
Conditional Distribution ◽  
Random Variables ◽  
Extremal Index ◽  
Stationary Sequence ◽  
Independent Random Variables ◽  
Fragility Index ◽  
Expected Number

We investigate the asymptotic distribution of the number of exceedances amongdidentically distributed but not necessarily independent random variables (RVs) above a sequence of increasing thresholds, conditional on the assumption that there is at least one exceedance. Our results enable the computation of thefragility index, which represents the expected number of exceedances, given that there is at least one exceedance. Computed from the firstdRVs of a strictly stationary sequence, we show that, under appropriate conditions, the reciprocal of the fragility index converges to the extremal index corresponding to the stationary sequence asdincreases.

scholarly journals Chaoticity for Multiclass Systems and Exchangeability Within Classes

2008 ◽  
Vol 45 (04) ◽  
pp. 1196-1203 ◽  
Author(s):  
Carl Graham
Keyword(s):  
Distributed System ◽  
Random Variables ◽  
Asymptotic Independence ◽  
Empirical Measure ◽  
Empirical Measures ◽  
Partial Exchangeability ◽  
Independent And Identically Distributed

Classical results for exchangeable systems of random variables are extended to multiclass systems satisfying a natural partial exchangeability assumption. It is proved that the conditional law of a finite multiclass system, given the value of the vector of the empirical measures of its classes, corresponds to independent uniform orderings of the samples within {each} class, and that a family of such systems converges in law {if and only if} the corresponding empirical measure vectors converge in law. As a corollary, convergence within {each} class to an infinite independent and identically distributed system implies asymptotic independence between {different} classes. A result implying the Hewitt-Savage 0-1 law is also extended.

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