Asymptotic Behavior of Partial Sums: A More Robust Approach Via Trimming and Self-Normalization

1991 ◽  
pp. 1-53 ◽  
Author(s):  
Marjorie G. Hahn ◽  
Jim Kuelbs ◽  
Daniel C. Weiner
Keyword(s):  
Asymptotic Behavior ◽  
Partial Sums ◽  
Robust Approach

Linear Processes

2019 ◽  
pp. 345-381
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Asymptotic Behavior ◽  
Fourier Transforms ◽  
Gaussian Approximation ◽  
Related Result ◽  
Weak Limit ◽  
Linear Process ◽  
Partial Sums ◽  
Linear Processes ◽  
Discrete Fourier Transforms ◽  
Linear Statistics

Here we apply different methods to establish the Gaussian approximation to linear statistics of a stationary sequence, including stationary linear processes, near-stationary processes, and discrete Fourier transforms of a strictly stationary process. More precisely, we analyze the asymptotic behavior of the partial sums associated with a short-memory linear process and prove, in particular, that if a weak limit theorem holds for the partial sums of the innovations then a related result holds for the partial sums of the linear process itself. We then move to linear processes with long memory and obtain the CLT under various dependence structures for the innovations by analyzing the asymptotic behavior of linear statistics. We also deal with the invariance principle for causal linear processes or for linear statistics with weakly associated innovations. The last section deals with discrete Fourier transforms, proving, via martingale approximation, central limit behavior at almost all frequencies under almost no condition except a regularity assumption.

scholarly journals Converging factors for some asymptotic moment series that arise in numerical quadrature

1982 ◽  
Vol 24 (2) ◽  
pp. 223-233 ◽  
Author(s):  
Avram Sidi
Keyword(s):  
Asymptotic Behavior ◽  
Numerical Quadrature ◽  
Partial Sums ◽  
Weight Functions ◽  
Infinite Range

AbstractIn this work the asymptotic behavior of the partial sums of the divergent asymptotic moment series , where μi are the moments of the weight functions w(x) = xαe−x, α > −1, and w(x) = xαEm (x), α > −1, m + α > 0, on the interval [0, ∞), is analyzed. Expressions for the converging factors are derived by the author for the infinite range integras with w(x) as given above.

scholarly journals On Bernstein–Kantorovich invariance principle in Hölder spaces and weighted scan statistics

2020 ◽  
Vol 24 ◽  
pp. 186-206
Author(s):  
Alfredas Račkauskas ◽  
Charles Suquet
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Weak Convergence ◽  
Invariance Principle ◽  
Brownian Bridge ◽  
Polygonal Line ◽  
Scan Statistics ◽  
Partial Sums ◽  
Kantorovich Theorem ◽  
The One

Let ξn be the polygonal line partial sums process built on i.i.d. centered random variables Xi, i ≥ 1. The Bernstein-Kantorovich theorem states the equivalence between the finiteness of E|X1|max(2,r) and the joint weak convergence in C[0, 1] of n−1∕2ξn to a Brownian motion W with the moments convergence of E∥n−1/2ξn∥∞r to E∥W∥∞r. For 0 < α < 1∕2 and p (α) = (1 ∕ 2 - α) -1, we prove that the joint convergence in the separable Hölder space Hαo of n−1∕2ξn to W jointly with the one of E∥n−1∕2ξn∥αr to E∥W∥αr holds if and only if P(|X1| > t) = o(t−p(α)) when r < p(α) or E|X1|r < ∞ when r ≥ p(α). As an application we show that for every α < 1∕2, all the α-Hölderian moments of the polygonal uniform quantile process converge to the corresponding ones of a Brownian bridge. We also obtain the asymptotic behavior of the rth moments of some α-Hölderian weighted scan statistics where the natural border for α is 1∕2 − 1∕p when E|X1|p < ∞. In the case where the Xi’s are p regularly varying, we can complete these results for α > 1∕2 − 1∕p with an appropriate normalization.

Stationary Sequences in a Random Time Scenery

2019 ◽  
pp. 319-344
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Asymptotic Behavior ◽  
Markov Chain ◽  
Random Walk ◽  
Random Time ◽  
Partial Sums ◽  
Stationary Sequences ◽  
Invariance Principles ◽  
Weak Invariance ◽  
Functional Clt ◽  
Random Scenery

In this chapter, we analyze the asymptotic behavior of the partial sums process associated with examples of stationary sequences in a random time scenery. The examples considered are stationary sequences sampled by shifted renewal Markov chains and random walks in a strictly stationary scenery. The asymptotic behavior of the partial sums process is essentially investigated with the help of the weak invariance principles stated in Chapter 4. More precisely, for the partial sums process associated with a stationary process sampled by a renewal Markov chain stated at zero, due to the non-stationarity of the underlying sequence, the functional CLT is obtained as an application of the functional CLT for non-stationary sequences developed in Section 4.4. In the case where we are sampling a strictly stationary random scenery by a random walk, stationarity is preserved, and the invariance principle is then derived by using the functional CLT under Maxwell–Woodroofe condition.

scholarly journals Limit Theorems for Long-Memory Stochastic Volatility Models with Infinite Variance: Partial Sums and Sample Covariances

2012 ◽  
Vol 44 (4) ◽  
pp. 1113-1141 ◽  
Author(s):  
Rafał Kulik ◽  
Philippe Soulier
Keyword(s):  
Asymptotic Behavior ◽  
Stochastic Volatility ◽  
Long Memory ◽  
Convergence Rates ◽  
Partial Sums ◽  
Infinite Variance ◽  
Tail Behavior ◽  
Stochastic Volatility Models ◽  
Crucial Difference ◽  
Volatility Models

In this paper we extend the existing literature on the asymptotic behavior of the partial sums and the sample covariances of long-memory stochastic volatility models in the case of infinite variance. We also consider models with leverage, for which our results are entirely new in the infinite-variance case. Depending on the interplay between the tail behavior and the intensity of dependence, two types of convergence rates and limiting distributions can arise. In particular, we show that the asymptotic behavior of partial sums is the same for both long memory in stochastic volatility and models with leverage, whereas there is a crucial difference when sample covariances are considered.

ASYMPTOTIC BEHAVIOR OF TRIMMED SUMS

2012 ◽  
Vol 12 (01) ◽  
pp. 1150002 ◽  
Author(s):  
ISTVÁN BERKES ◽  
LAJOS HORVÁTH ◽  
JOHANNES SCHAUER
Keyword(s):  
Asymptotic Behavior ◽  
Central Limit ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Stable Law ◽  
Limit Theory ◽  
Trimmed Sums ◽  
The Central Limit Theorem

Trimming is a standard method to decrease the effect of large sample elements in statistical procedures, used, e.g., for constructing robust estimators. It is also a powerful tool in understanding deeper properties of partial sums of independent random variables. In this paper we review some basic results of the theory and discuss new results in the central limit theory of trimmed sums. In particular, we show that for random variables in the domain of attraction of a stable law with parameter 0 < α < 2, the asymptotic behavior of modulus trimmed sums depends sensitively on the number of elements eliminated from the sample. We also show that under moderate trimming, the central limit theorem always holds if we allow random centering factors. Finally, we give an application to change point problems.

scholarly journals Limit Theorems for Long-Memory Stochastic Volatility Models with Infinite Variance: Partial Sums and Sample Covariances

2012 ◽  
Vol 44 (04) ◽  
pp. 1113-1141
Author(s):  
Rafał Kulik ◽  
Philippe Soulier
Keyword(s):  
Asymptotic Behavior ◽  
Stochastic Volatility ◽  
Long Memory ◽  
Convergence Rates ◽  
Partial Sums ◽  
Infinite Variance ◽  
Tail Behavior ◽  
Stochastic Volatility Models ◽  
Crucial Difference ◽  
Volatility Models

In this paper we extend the existing literature on the asymptotic behavior of the partial sums and the sample covariances of long-memory stochastic volatility models in the case of infinite variance. We also consider models with leverage, for which our results are entirely new in the infinite-variance case. Depending on the interplay between the tail behavior and the intensity of dependence, two types of convergence rates and limiting distributions can arise. In particular, we show that the asymptotic behavior of partial sums is the same for both long memory in stochastic volatility and models with leverage, whereas there is a crucial difference when sample covariances are considered.

Weakly Associated Random Variables

2019 ◽  
pp. 233-250
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Asymptotic Behavior ◽  
Gaussian Processes ◽  
Random Variables ◽  
Empirical Processes ◽  
Partial Sums ◽  
Asymptotic Results ◽  
Associated Random Variables ◽  
Maximal Inequalities ◽  
Non Gaussian ◽  
Shed Light

The purposes of this chapter are to introduce the notion of weakly associated (negatively or positively) random variables and to develop tools that will allow us, in the chapters to follow, to give estimations of moments of partial sums, maximal inequalities, and asymptotic results with both Gaussian and non-gaussian limits. As we shall see, these results shed light on the asymptotic behavior of numerous examples such as exchangeable variables, certain Gaussian processes, empirical processes, various classes of Markov chains, and determinantal processes. They are also useful to study stochastic processes that are functionals of the two independent processes mentioned above.

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