Partial Sum Process Research Articles

Limit Theorems for Partial Sum Processes of Moving Averages Based on Heterogeneous Processes

Limit Theorems for Partial Sum Processes of Moving Averages Based on Heterogeneous Processes

Simultaneous Model Change Detection in Multivariate Linear Regression With Application to Indonesian Economic Growth Data

In this paper, we study asymptotic model change detection in multivariate linear regression by using the Kolmogorov–Smirnov function of the partial sum process of recursive residuals. We approximate the rejection region and also the power function of the test by establishing a functional central limit theorem for the sequence of the partial sum processes of the recursive residuals of the observations. When the assumed model is true, the limit process is given by the standard multivariate Brownian motion which does not depend on the regression functions. However, when the assumed model is not true (some models change), the limit process is represented by a vector of deterministic trend plus the standard multivariate Brownian motion. The finite sample size rejection region and the power of the test are investigated by means of Monte Carlo simulation. The simulation study shows evidence that the proposed test is consistent in the sense that it attains the power larger than the size of the test when the hypothesis is not true. We also demonstrate the application of the proposed test method to Indonesian economic growth data in which we test the adequacy of three‐variate low‐order polynomial model. The test result shows that the growth of the Indonesian economy is neither simultaneously constant nor linear. The test has successfully detect the appearance of a change in the model which is mainly caused by the COVID‐19 pandemic in 2020.

Counterbalancing steps at random in a random walk

A random walk with counterbalanced steps is a process of partial sums \check S(n)=\check X_{1}+ \cdots + \check X_{n} whose steps \check X_{n} are given recursively as follows. For each n\geq 2 , with a fixed probability p , \check X_{n} is a new independent sample from some fixed law \mu , and with complementary probability 1-p , \check X_{n}= -\check X_{v(n)} counterbalances a previous step, with v(n) a uniform random pick from \{1, \ldots, n-\nobreak 1\} . We determine the asymptotic behavior of \check S(n) in terms of p and the first two moments of \mu . Our approach relies on a coupling with a reinforcement algorithm due to H. A. Simon, and on properties of random recursive trees and Eulerian numbers, which may be of independent interest. The method can be adapted to the situation where the step distribution \mu belongs to the domain of attraction of a stable law.

Two Monotone Processes of a Replacement Policy for a Repairable System with its Repairman Having Multiple Vacations.

For the maintenance of repairable system with a repairman having multiple vacations using two monotone processes. If the system fails and the repairman is on vacation, it will wait for repair until the repairman is available. Assume the damaged system cannot be repaired “as good as new” and the repairman can be repaired immediately with a probability of p, we optimize replacement policy using increasing alpha series and decreasing partial sum processes. The explicit expression for the expected profit under replacement policy N is derived analytically. And we also a numerical example is given to illustrate the theoretical results of the model mentioned in this paper.

Limit theorems for functionals of long memory linear processes with infinite variance

Let X={Xn:n∈N} be a long memory linear process in which the coefficients are regularly varying and innovations are independent and identically distributed and belong to the domain of attraction of an α-stable law with α∈(0,2). Then, for any integrable and square integrable function K on R, under certain mild conditions, we establish the asymptotic behavior of the partial sum process ∑n=1[Nt][K(Xn)−EK(Xn)]:t≥0as N tends to infinity, where [Nt] is the integer part of Nt for t≥0.

Convergence of partial sum processes to stable processes with application for aggregation of branching processes

We provide a generalization of Theorem 1 in Bartkiewicz et al. (2011) in the sense that we give sufficient conditions for weak convergence of finite dimensional distributions of the partial sum processes of a strongly stationary sequence to the corresponding finite dimensional distributions of a non-Gaussian stable process instead of weak convergence of the partial sums themselves to a non-Gaussian stable distribution. As an application, we describe the asymptotic behaviour of finite dimensional distributions of aggregation of independent copies of a strongly stationary subcritical Galton–Watson branching process with regularly varying immigration having index in (0,1)∪(1,4/3) in a so-called iterated case, namely when first taking the limit as the time scale and then the number of copies tend to infinity.

Impact of economic policy uncertainty on the stock markets of the G7 Countries:A nonlinear ARDL approach

This paper employs the linear and nonlinear ARDL models to examine the short-run and long-run effects of Economic Policy Uncertainty (EPU) on the stock prices of the G7 countries using monthly data up to May of 2021. Using the linear ARDL model, the results indicate that EPU has a significant negative short-run effect on the stock prices of all the G7 countries. These negative short-run effects last into the long-run in only Canada and Japan, with no long-run effect in the remaining countries. However, using the nonlinear ARDL model shows that at least one of the partial sum processes of positive and negative changes in EPU has a significant short-run effect on the stock prices of all the G7 countries, and that these short-run effects last into the long-run in all the G7 countries, except in the UK. Moreover, we find evidence of asymmetries in both the short-run and long-run in all the countries, except in the UK. Our findings have important policy implications.

Symmetric or asymmetric: how is Nigeria's bilateral trade balance responding to oil price and exchange rate changes?

PurposeThe experience of rising trade imbalance between Nigeria and its key trading partners in recent years motivated this study. Previous studies on this issue either ignored bilateral level or assumed that the effect of crude oil price and/or exchange rate changes on trade balance is symmetric. Consequently, this study investigates whether Nigeria's bilateral trade balance with Belgium, China, United Kingdom (UK) and USA is responding symmetrically or asymmetrically to changes in oil price and exchange rate.Design/methodology/approachThe authors used nonlinear autoregressive-distributed lag (NARDL) model that decomposed oil price and exchange rate into partial sum processes of positive and negative changes over the period 1999Q1–2019Q4.FindingsThe study finds that the effects of oil price hike and plunge asymmetrically influence Nigeria's trade balance with the UK and USA. Further evidence indicated that oil price plunge exerts greater influence than price hike in all the cases, except the UK in the long run. Furthermore, Nigeria's trade balance responds asymmetrically and significantly to changes in exchange rate with China in the long run and with China and the UK in the short run. Specifically, the depreciation effect is more prominent than appreciation.Originality/valueSignificant contributions to the existing literature in Nigeria include the recognition that the effects of oil price and exchange rate changes on trade are asymmetric and the disaggregation of trade into bilateral level to identify country-specific effect.

Likelihood ratio test for the mean of asymptotic spatial regression with the Brownian sheet noise

Likelihood ratio test (LR-test) is frequently applied in regression analysis when the observations are normally distributed. However, when the normality assumption is violated, such a test can not be directly adopted in practice. In this work, we propose an approach by firstly transforming the observations to a set-indexed partial sums process. The limit model under this transformation is presented as a deterministic trend plus a random noise given by the set-indexed Brownian sheet. We show that the problem of testing the appropriateness of a model can be handled step-wise by testing the validity of the trend in the limit model by defining an LR-test based on the ratio between the likelihood function under H 0 and under H 0 ∪ H 1. The rejection region as well as the power of the size α LR-test are obtained based on the Cameron-Martin-Girsanov formula of the Radon-Nikodym derivative of the set-indexed partial sums limit process of the observation with respect to the set-indexed Brownian sheet. Simulation study shows that the proposed test behaves as a consistent test in that it maximizes the power when H 1 is true. Application of the method to real data can help in detecting valid regression model describing the variability of the maximum height of corn plants over the experimental region.

Limit Theorems for Conservative Flows on Multiple Stochastic Integrals

We consider a stationary sequence \((X_n)\) constructed by a multiple stochastic integral and an infinite-measure conservative dynamical system. The random measure defining the multiple integral is non-Gaussian and infinitely divisible and has a finite variance. Some additional assumptions on the dynamical system give rise to a parameter \(\beta \in (0,1)\) quantifying the conservativity of the system. This parameter \(\beta \) together with the order of the integral determines the decay rate of the covariance of \((X_n)\). The goal of the paper is to establish limit theorems for the partial sum process of \((X_n)\). We obtain a central limit theorem with Brownian motion as limit when the covariance decays fast enough, as well as a non-central limit theorem with fractional Brownian motion or Rosenblatt process as limit when the covariance decays slowly enough.

An adjusted-range based self-normalization test for correlation change

This paper proposes an adjusted-range based self-normalized test for change in correlation. Unlike the self-normalization approach proposed by Lobato (2001) and Shao (2010), which relies on the variance of the partial sum process as a self-normalizer, and used by Choi and Shin (2020) to formulate the self-normalized Q test statistic, here we propose the use of an adjusted-range of the partial sum process instead. Similar to the self-normalized Q test statistic in Choi and Shin (2020), our adjusted-range based correlation change test is also nuisance parameter and distribution free, which helps to circumvent the kernel-based long-run variance (LRV) estimation in Wied et al. (2012) and the computationally expensive bootstrapping in Wied (2017). Moreover, the proposed adjusted-range based correlation change test is a portmanteau test with a neater formulation than Q; it is found to have adequate sizes and powers in many cases and it has much better powers than Q under serial dependence with either homoscedasticity or heteroscedasticity. The empirical analysis also suggests the adequacy of the proposed test.

Asymmetric oil price and Asian economies: A nonlinear ARDL approach

We study the asymmetric effects of oil price changes on the domestic output of the ASEAN-5 countries (Indonesia, Malaysia. Singapore, Philippines, and Thailand) plus Japan and Korea. Asymmetries are introduced by accumulating oil price increases separately from decreases using partial sum processes in a nonlinear ARDL framework. Utilizing annual data for the period 1973–2018, the results from the linear ARDL model suggest that oil price changes do not affect the domestic output of Indonesia, Korea, Singapore, and Thailand. However, the nonlinear ARDL model reveals that oil price changes asymmetrically affect the domestic output of all seven Asian countries in both the short-run and in the long-run. We observe an asymmetrically larger effect on output from rising oil prices than from falling prices, but effects vary across countries. Moreover, nonlinear causality tests confirm causality from oil price to output in all the countries.

On Aggregation of Subcritical Galton–Watson Branching Processes with Regularly Varying Immigration

. We study an iterated temporal and contemporaneous aggregation of N independent copies of a strongly stationary subcritical Galton–Watson branching process with regularly varying immigration having index α ∈ (0, 2). We show that limits of finite-dimensional distributions of appropriately centered and scaled aggregated partial-sum processes exist when first taking the limit as N → ∞and then the time scale n→ ∞. The limit process is an α-stable process if α ∈ (0, 1) ∪ (1, 2) and a deterministic line with slope 1 if α = 1.

Random discretization of stationary continuous time processes

This paper investigates second order properties of a stationary continuous time process after random sampling. While a short memory process always gives rise to a short memory one, we prove that long-memory can disappear when the sampling law has very heavy tails. Despite the fact that the normality of the process is not maintained by random sampling, the normalized partial sum process converges to the fractional Brownian motion, at least when the long memory parameter is preserved.

Bootstrapping covariance operators of functional time series

For testing hypothesis on the covariance operator of functional time series, we suggest to use the full functional information and to avoid dimension reduction techniques. The limit distribution follows from the central limit theorem of the weak convergence of the partial sum process in general Hilbert space applied to the product space. In order to obtain critical values for tests, we generalise bootstrap results from the independent to the dependent case. This results can be applied to covariance operators, autocovariance operators and cross covariance operators. We discuss one sample and changepoint tests and give some simulation results.

A functional non-central limit theorem for multiple-stable processes with long-range dependence

A functional limit theorem is established for the partial-sum process of a class of stationary sequences which exhibit both heavy tails and long-range dependence. The stationary sequence is constructed using multiple stochastic integrals with heavy-tailed marginal distribution. Furthermore, the multiple stochastic integrals are built upon a large family of dynamical systems that are ergodic and conservative, leading to the long-range dependence phenomenon of the model. The limits constitute a new class of self-similar processes with stationary increments. They are represented by multiple stable integrals, where the integrands involve the local times of intersections of independent stationary stable regenerative sets.

Asymmetric effect of exchange rate changes on cross-border trade in Nigeria

This paper investigates the asymmetric effect of exchange rate changes on cross-border trade in Nigeria. The investigation becomes necessary because several studies have reported insignificant results in attempting to establish a link between these two variables using symmetric specification. Whereas, there are strong evidence of nonlinear mean-reverting association because some exchange rate changes of the same magnitude exhibit different effects on other variables of interest. Having separated the real effective exchange rate into both depreciation and appreciation regimes using the partial sum processes based on logistic smooth-transition and exponential smooth-transition, results from the nonlinear autoregressive distributed lags show that exchange rate appreciation had a statistically significant negative relationship with cross-border trade in Nigeria. The study concludes that the relationship between real effective exchange rate and cross-border trade is asymmetric. (Depreciation and appreciation of equal magnitude do not have the same effect on cross-border trade in Nigeria.) The study recommends that policy makers should consider models that allow a nonlinear adjustment of exchange rates which may produce outcomes supporting an effective devaluation or appreciation policy, at least against some trading partners.

On the rate of Poisson approximation to Bernoulli partial sum processes

We investigate approximation of a Bernoulli partial sum process to the accompanying Poisson process in the non-i.i.d. case. The rate of closeness is studied in terms of the minimal distance in probability. In particular, a new lower bound for the total variation distance between a Bernoulli partial sum process and the accompanying Poisson process is obtained.

On The Distribution of Partial Sums of Randomly Weighted Powers of Uniform Spacings

Objectives:To study the asymptotic theory of the randomly wieghted partial sum process of powers of k-spacings from the uniform distribution.Methods:Earlier results on the distribution of the uniform incremental randomly weighted sums.Methods:Based on theorems on weak and strong approximations of partial sum processes.Results and conculsions:Our main contribution is to prove the weak convergence of weighted sum of powers of uniform spacings.

Partial Sum Processes of Residual-Based and Wald-type Break-Point Statistics in Time Series Regression Models

Partial Sum Processes of Residual-Based and Wald-type Break-Point Statistics in Time Series Regression Models