Partial Sum Process Research Articles

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

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.

On partial-sum processes of ARMAX residuals

We establish general and versatile results regarding the limit behavior of the partial-sum process of ARMAX residuals. Illustrations include ARMA with seasonal dummies, misspecified ARMAX models with autocorrelated errors, nonlinear ARMAX models, ARMA with a structural break, a wide range of ARMAX models with infinite-variance errors, weak GARCH models and the consistency of kernel estimation of the density of ARMAX errors. Our results identify the limit distributions, and provide a general algorithm to obtain pivot statistics for CUSUM tests.

Universal kriging of corn plant data under isotropic power type variogram model

Kriging is a well known notion in geostatistics which aim at the construction of optimal prediction for unobserved quantities of interest. In this paper we derive optimal prediction of a second-order stationary spatial process by minimizing the quadratic risk or mean square error of the predictor subject to some additional conditions. The coefficients of the linear predictor are calculated by a formula which is obtained by applying Lagrange method. The spatial process under study is assumed to be isotropic with a variogram model belongs to quadratic family. Least squares estimation to the parameters of the variogram model is calculated by graphical method. The validity of the mean function is investigated by utilizing asymptotic test based on the partial sums process of weighted least squares residuals. The application of the method to spatial process of rate of growth of corn plants results in the kriging map of the process. The maps are generated under two different variogram models: power and linear (Tent) models.

Uniform asymptotic normality of self-normalized weighted sums of random variables*

Let X, X1, X2, . . . be a sequence of nondegenerate i.i.d. random variables, let μ = {μni : n ∈ ℕ+, i = 1, …, n} be a triangular array of possibly random probabilities on the interval [0, 1], and let $$ \mathcal{F} $$ be a class of functions with bounded q-variation on [0, 1] for some q ∈ [1, 2). We prove the asymptotic normality uniformly over $$ \mathcal{F} $$ of self-normalized weighted sums $$ {\sum}_{i=1}^n{X}_i{\mu}_{ni} $$ when μ is the array of point measures, uniform probabilities, and their random versions. Also, we prove a weak invariance principle in the Banach space of functions of bounded p-variation with p > 2 for partial-sum processes, polygonal processes, and their adaptive versions.

The principle of invariance in the Donsker form to the partial sum processes of finite order moving averages

We consider the process of partial sums of moving averages of finite order with a regular varying memory function, constructed from a stationary sequence, variance of the sum of which is a regularly varying function. We study the Gaussian approximation of this process of partial sums with the aid of a certain class of Gaussian processes, and obtain sufficient conditions for the $C$-convergence in the invariance principle in the Donsker form

Efficient computation of the stochastic behavior of partial sum processes

In this paper the computational aspects of probability calculations for dynamical partial sum expressions are discussed. Such dynamical partial sum expressions have many important applications, and examples are provided in the fields of reliability, product quality assessment, and stochastic control. While these probability calculations are ostensibly of a high dimension, and consequently intractable in general, it is shown how a recursive integration methodology can be implemented to obtain exact calculations as a series of two-dimensional calculations. The computational aspects of the implementation of this methodology, with the adoption of Fast Fourier Transforms, are discussed.

A note on linear processes with tapered innovations

In the paper, we consider the partial-sum process \( {\sum}_{k=1}^{\left[ nt\right]}{X}_k^{(n)}, \) where \( \left\{{X}_n^{(n)},k\in \mathbb{Z}\right\},n\ge 1, \) is a series of linear processes with innovations having heavy-tailed tapered distributions with tapering parameter bn depending on n. We show that, depending on the properties of a filter of a linear process under consideration and on the parameter bn defining if the tapering is hard or soft, the limit process for such partial-sum process can be a fractional Brownian motion or linear fractional stable motion.

Infinite Paths on a Random Environment of $${\mathbb {Z}}^2$$ with Bounded and Recurrent Sums

This paper considers a random structure on the lattice $\mathbb{Z}^2$ of the following kind. To each edge $e$ a random variable $X_e$ is assigned, together with a random sign $Y_e \in \{-1,+1\}$. For an infinite self-avoiding path on $\mathbb{Z}^2$ starting at the origin consider the sequence of partial sums along the path. These are computed by summing the $X_e$'s for the edges $e$ crossed by the path, with a sign depending on the direction of the crossing. If the edge is crossed rightward or upward the sign is given by $Y_e$, otherwise by $-Y_e$. We assume that the sequence of $X_e$'s is i.i.d., drawn from an arbitrary common law and that the sequence of signs $Y_e$ is independent, with independent components drawn from a law which is allowed to change from horizontal to vertical edges. First we show that, with positive probability, there exists an infinite self-avoiding path starting from the origin with bounded partial sums. Moreover the process of partial sums either returns to zero or at least it returns to any neighborhood of zero infinitely often. These results are somewhat surprising at the light of the fact that, under rather mild conditions, there exists with probability $1$ two sites with all the paths joining them having the partial sums exceeding in absolute value any prescribed constant.

Exponential inequalities for the distributions of canonical multiple partial sum processes

В работе получены экспоненциальные оценки для хвоста распределения равномерной нормы процесса частичных кратных сумм с каноническим ограниченным ядром, построенных как по независимым, так и по слабо зависимым наблюдениям. При этом показатель экспоненты имеет правильный порядок.

Exponential Inequalities for the Distributions of Canonical Multiple Partial Sum Processes

Exponential inequalities are obtained for the distribution tail of the sup-norm of a multiple partial sum process with canonical bounded kernel based on independent and weakly dependent observation...

Joint functional convergence of partial sums and maxima for linear processes*

For linear processes with independent identically distributed innovations that are regularly varying with tail index α ∈ (0, 2), we study the functional convergence of the joint partial-sum and partial-maxima processes. We derive a functional limit theorem under certain assumptions on the coefficients of the linear processes, which enable the functional convergence in the space of ℝ2-valued cadlag functions on [0, 1] with the Skorokhod weak M2 topology.We also obtain a joint convergence in the M2 topology on the first coordinate and in theM1 topology on the second coordinate.

A note on joint functional convergence of partial sum and maxima for linear processes

Recently, for the joint partial sum and partial maxima processes constructed from linear processes with independent identically distributed innovations that are regularly varying with tail index α∈(0,2), a functional limit theorem with the Skorohod weak M2 topology has been obtained. In this paper we show that, if all the coefficients of the linear processes are of the same sign, the functional convergence holds in the stronger topology, i.e. in the Skorohod weak M1 topology on the space of R2-valued càdlàg functions on [0,1].

Almost sure central limit theorem for the hybrid process

ABSTRACTIn this paper, we investigate some problems related to the almost sure central limit theorem for the hybrids of empirical and partial sum processes. More precisely, under mild technical conditions, we study the almost sure (a.s.) convergence of where denote the hybrids of empirical and partial sum processes, is a bounded Lipschitz function, denotes a sequence of positive weights, and

Functional law of the iterated logarithm for partial sum processes of non-negative valued iid random variables belonging to the domain of partial attraction of a semi-stable law

Let (Xn) be a sequence of independent and identically distributed non-negative valued random variables defined over a common probability space and let F denote the common distribution function. Set Sn=X1+X2+⋯+Xn and denote inf{x;n(1−F(x))≤1} by Bn,n≥1. Assuming that F belongs to the domain of partial attraction of a positive semi-stable law, we obtain the set of almost sure limit functions of Bn−1S[nt]1loglogn,t∈[0,1], under M1-topology.

Tests for structural breaks in memory parameters of long-memory heterogeneous autoregressive models

ABSTRACTTests for structural breaks in the coefficients of the long-memory heterogeneous autoregressive (HAR) models are developed. The tests are based on the partial sum process of the normalized efficient score vector. The tests have the nice property of identifying the parameters of the daily, weekly, and monthly regressors in which breaks occur. Limiting null distributions of the proposed tests are proven to be derived from standard Brownian bridges. A finite sample Monte-Carlo experiment shows reasonable size and power properties of the proposed tests. The proposed method is illustrated by a real data analysis.

Strong approximations for the p-fold integrated empirical process with applications to statistical tests

The main purpose of this paper is to investigate the strong approximation of the $p$-fold integrated empirical process, $p$ being a fixed positive integer. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on results of Koml\'os, Major and Tusn\'ady (1975). We also obtain an exponential bound for the tail probability of the weighted approximation to the $p$-fold integrated empirical process. Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of integrated empirical processes when the parameters are estimated. We study the behavior of the self-intersection local time of the partial sum process representation of integrated empirical processes. Finally, simulation results are provided to illustrate the finite sample performance of the proposed statistical tests based on the integrated empirical processes.

Invariance principles for tempered fractionally integrated processes

We discuss invariance principles for autoregressive tempered fractionally integrated moving averages in α-stable (1<α≤2) i.i.d. innovations and related tempered linear processes with vanishing tempering parameter limN→∞λ∕N=λ∗. We show that the limit of the partial sums process takes a different form in the weakly tempered (λ∗=0), strongly tempered (λ∗=∞), and moderately tempered (0<λ∗<∞) cases. These results are used to derive the limit distribution of the ordinary least squares estimate of AR(1) unit root with weakly, strongly, and moderately tempered moving average errors.

A note on the normalizing sequences for sums of linear processes in the case of negative memory

We consider the partial-sum process $$ {S}_n(t)={\sum}_{k=0}^{\left\lfloor nt\right\rfloor }{X}_k $$ of linear processes $$ {X}_n={\sum}_{i=0}^{\infty }{c}_i{\upxi}_{n-i} $$ with independent identically distributed innovations {ξ i } belonging to the domain of attraction of α-stable law (0 < α ≤ 2). If |c k | = k −γ , k ∈ ℕ , γ > max(1, 1/α), and $$ {\sum}_{k=0}^{\infty}\kern0.5em ck=0 $$ (the case of negative memory for the stationary sequence {X n }), then it is known that the normalizing sequence of S n (1) can grow as n 1/α−γ+1 or remain bounded if the signs of the coefficients are constant or alternate, respectively. It is of interest to know whether it is possible, given ⋋ ∈ (0, 1/α − γ + 1), to change the signs of c k so that the rate of growth of the normalizing sequence would be n ⋋ . In this paper, we give the positive answer: we propose a way of choosing the signs and investigate the finite-dimensional convergence of appropriately normalized S n (t) to linear fractional Levy motion.

Partial sum processes and continued fractions

Given{Xi}i=1∞, a sequence of real valued random variables, we define S0=0, Sn=∑i=1nXi, and define the normalized partial sum process{Yn(t):0≤t≤1} by linear interpolation of Ynin=SiSn (assuming P(Sn=0)=0 for all n≥1). In this note the convergence of Yn(⋅) in [0,1] is investigated under various assumptions on {Xi}i=1∞. Of particular interest is the special case where the Xi are the coefficients in the continued fraction expansion of a point x∈[0,1] chosen according to Gauss measure.

Markov Property in Discrete Schur-constant Models

This paper is concerned with Schur-constant survival models for discrete random variables. Our main purpose is to prove that the associated partial sum process is a non-homogeneous Markov chain. This is shown in three different situations where the random variables considered take values in the sets 0, {0,1} or {0,…,m}, m ≥ 2. The property of Schur-constancy is also compared for these three cases.