scholarly journals The impact of the diagonals of polynomial forms on limit theorems with long memory

Bernoulli ◽  
2017 ◽  
Vol 23 (1) ◽  
pp. 710-742 ◽  
Author(s):  
Shuyang Bai ◽  
Murad S. Taqqu
Keyword(s):  
Long Memory ◽  
Limit Theorems ◽  
The Impact

Limit theorems for long-memory flows on Wiener chaos

Bernoulli ◽  
2020 ◽  
Vol 26 (2) ◽  
pp. 1473-1503 ◽  
Author(s):  
Shuyang Bai ◽  
Murad S. Taqqu
Keyword(s):  
Long Memory ◽  
Limit Theorems ◽  
Wiener Chaos

Testing for long memory in the presence of a general trend

2001 ◽  
Vol 38 (04) ◽  
pp. 1033-1054 ◽  
Author(s):  
Liudas Giraitis ◽  
Piotr Kokoszka ◽  
Remigijus Leipus
Keyword(s):  
Long Memory ◽  
Change Point ◽  
General Trend ◽  
Quantitative Description ◽  
Stationary Sequence ◽  
Change Points ◽  
Asymptotic Size ◽  
Short Memory ◽  
The Impact ◽  
Do So

The paper studies the impact of a broadly understood trend, which includes a change point in mean and monotonic trends studied by Bhattacharyaet al.(1983), on the asymptotic behaviour of a class of tests designed to detect long memory in a stationary sequence. Our results pertain to a family of tests which are similar to Lo's (1991) modifiedR/Stest. We show that both long memory and nonstationarity (presence of trend or change points) can lead to rejection of the null hypothesis of short memory, so that further testing is needed to discriminate between long memory and some forms of nonstationarity. We provide quantitative description of trends which do or do not fool theR/S-type long memory tests. We show, in particular, that a shift in mean of a magnitude larger thanN-½, whereNis the sample size, affects the asymptotic size of the tests, whereas smaller shifts do not do so.

Testing for long memory in the presence of a general trend

2001 ◽  
Vol 38 (4) ◽  
pp. 1033-1054 ◽  
Author(s):  
Liudas Giraitis ◽  
Piotr Kokoszka ◽  
Remigijus Leipus
Keyword(s):  
Long Memory ◽  
Change Point ◽  
General Trend ◽  
Quantitative Description ◽  
Stationary Sequence ◽  
Change Points ◽  
Asymptotic Size ◽  
Short Memory ◽  
The Impact ◽  
Do So

The paper studies the impact of a broadly understood trend, which includes a change point in mean and monotonic trends studied by Bhattacharyaet al.(1983), on the asymptotic behaviour of a class of tests designed to detect long memory in a stationary sequence. Our results pertain to a family of tests which are similar to Lo's (1991) modifiedR/Stest. We show that both long memory and nonstationarity (presence of trend or change points) can lead to rejection of the null hypothesis of short memory, so that further testing is needed to discriminate between long memory and some forms of nonstationarity. We provide quantitative description of trends which do or do not fool theR/S-type long memory tests. We show, in particular, that a shift in mean of a magnitude larger thanN-½, whereNis the sample size, affects the asymptotic size of the tests, whereas smaller shifts do not do so.

scholarly journals Covariances Estimation for Long-Memory Processes

2010 ◽  
Vol 42 (1) ◽  
pp. 137-157 ◽  
Author(s):  
Wei Biao Wu ◽  
Yinxiao Huang ◽  
Wei Zheng
Keyword(s):  
Time Series ◽  
Asymptotic Behavior ◽  
Long Memory ◽  
Limit Theorems ◽  
Limiting Distribution ◽  
Linear Processes ◽  
Long Memory Processes ◽  
Noncentral Limit Theorems ◽  
Dependence Properties

For a time series, a plot of sample covariances is a popular way to assess its dependence properties. In this paper we give a systematic characterization of the asymptotic behavior of sample covariances of long-memory linear processes. Central and noncentral limit theorems are obtained for sample covariances with bounded as well as unbounded lags. It is shown that the limiting distribution depends in a very interesting way on the strength of dependence, the heavy-tailedness of the innovations, and the magnitude of the lags.

scholarly journals Fluid limit theorems for stochastic hybrid systems with application to neuron models

2010 ◽  
Vol 42 (3) ◽  
pp. 761-794 ◽  
Author(s):  
K. Pakdaman ◽  
M. Thieullen ◽  
G. Wainrib
Keyword(s):  
Hybrid Systems ◽  
Central Limit ◽  
Markov Processes ◽  
Limit Theorems ◽  
Channel Noise ◽  
Stochastic Hybrid Systems ◽  
Neuron Models ◽  
Fluid Limit ◽  
The Impact ◽  
Jump Markov Processes

In this paper we establish limit theorems for a class of stochastic hybrid systems (continuous deterministic dynamics coupled with jump Markov processes) in the fluid limit (small jumps at high frequency), thus extending known results for jump Markov processes. We prove a functional law of large numbers with exponential convergence speed, derive a diffusion approximation, and establish a functional central limit theorem. We apply these results to neuron models with stochastic ion channels, as the number of channels goes to infinity, estimating the convergence to the deterministic model. In terms of neural coding, we apply our central limit theorems to numerically estimate the impact of channel noise both on frequency and spike timing coding.

scholarly journals The Optimal Bandwidth Parameter Selection in GPH Estimation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Weijie Zhou ◽  
Huihui Tao ◽  
Feifei Wang ◽  
Weiqiang Pan
Keyword(s):  
Time Series ◽  
Long Memory ◽  
Financial Time Series ◽  
Optimal Choice ◽  
Estimation Accuracy ◽  
Optimal Bandwidth ◽  
Bandwidth Parameter ◽  
High Level ◽  
The Impact ◽  
Memory Parameter

In this paper, the optimal bandwidth parameter is investigated in the GPH algorithm. Firstly, combining with the stylized facts of financial time series, we generate long memory sequences by using the ARFIMA (1, d, 1) process. Secondly, we use the Monte Carlo method to study the impact of the GPH algorithm on existence test, persistence or antipersistence judgment of long memory, and the estimation accuracy of the long memory parameter. The results show that the accuracy of above three factors in the long memory test reached a relatively high level within the bandwidth parameter interval of 0.5 < a < 0.7. For different lengths of time series, bandwidth parameter a = 0.6 can be used as the optimal choice of the GPH estimation. Furthermore, we give the calculation accuracy of the GPH algorithm on existence, persistence or antipersistence of long memory, and long memory parameter d when a = 0.6.

scholarly journals Approximating quasi-stationary distributions with interacting reinforced random walks

2021 ◽  
Author(s):  
Amarjit Budhiraja ◽  
Nicolas Fraiman ◽  
Adam Waterbury
Keyword(s):  
Limit Theorems ◽  
Almost Sure Convergence ◽  
Time Instant ◽  
Stationary Distributions ◽  
Reinforced Random Walks ◽  
Simulation Based ◽  
Finite State ◽  
Absorbing States ◽  
The Impact ◽  
Number Of Particles

We propose two numerical schemes for approximating quasi-stationary distributions (QSD) of finite state Markov chains with absorbing states. Both schemes are described in terms of interacting chains where the interaction is given in terms of the total time occupation measure of all particles in the system and has the impact of reinforcing transitions, in an appropriate fashion, to states where the collection of particles has spent more time. The schemes can be viewed as combining the key features of the two basic simulation-based methods for approximating QSD originating from the works of Fleming and Viot (1979) and  Aldous, Flannery and Palacios (1998), respectively. The key difference between the two schemes studied here is that in the first method one starts with $a(n)$ particles at time $0$ and number of particles stays constant over time whereas in the second method we start with one particle and at most one particle is added at each time instant in such a manner that there are $a(n)$ particles at time $n$. We prove almost sure convergence to the unique QSD and establish Central Limit Theorems for the two schemes under the key assumption that $a(n)=o(n)$. Exploratory numerical results are presented to illustrate the performance.

scholarly journals Quasi-Maximum Likelihood Estimation for Long Memory Stock Transaction Data—Under Conditional Heteroskedasticity Framework

2019 ◽  
Vol 12 (2) ◽  
pp. 74
Author(s):  
A. M. M. Shahiduzzaman Quoreshi ◽  
Reaz Uddin ◽  
Naushad Mamode Khan
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Long Memory ◽  
Likelihood Estimation ◽  
Monte Carlo Experiment ◽  
Empirical Estimation ◽  
Transaction Data ◽  
Quasi Maximum Likelihood ◽  
Stock Transaction ◽  
The Impact

This paper introduces Quasi-Maximum Likelihood Estimation for Long Memory Stock Transaction Data of unknown underlying distribution. The moments with conditional heteroscedasticity have been discussed. In a Monte Carlo experiment, it was found that the QML estimator performs as well as CLS and FGLS in terms of eliminating serial correlations, but the estimator can be sensitive to start value. Hence, two-stage QML has been suggested. In empirical estimation on two stock transaction data for Ericsson and AstraZeneca, the 2SQML turns out relatively more efficient than CLS and FGLS. The empirical results suggest that both of the series have long memory properties that imply that the impact of macroeconomic news or rumors in one point of time has a persistence impact on future transactions.

Spectral Analysis for Bivariate Time Series with Long Memory

1996 ◽  
Vol 12 (5) ◽  
pp. 773-792 ◽  
Author(s):  
J. Hidalgo
Keyword(s):  
Time Series ◽  
Density Matrix ◽  
Spectral Density ◽  
Long Memory ◽  
Limit Theorems ◽  
Stationary Process ◽  
Absolute Summability ◽  
Spectral Density Matrix ◽  
Mixing Coefficients ◽  
Long Memory Processes

This paper provides limit theorems for spectral density matrix estimators and functionals of it for a bivariate covariance stationary process whose spectral density matrix has singularities not only at the origin but possibly at some other frequencies and, thus, applies to time series exhibiting long memory. In particular, we show that the consistency and asymptotic normality of the spectral density matrix estimator at a frequency, say λ, which hold for weakly dependent time series, continue to hold for long memory processes when λ lies outside any arbitrary neighborhood of the singularities. Specifically, we show that for the standard properties of spectral density matrix estimators to hold, only local smoothness of the spectral density matrix is required in a neighborhood of the frequency in which we are interested. Therefore, we are able to relax, among other conditions, the absolute summability of the autocovariance function and of the fourth-order cumulants or summability conditions on mixing coefficients, assumed in much of the literature, which imply that the spectral density matrix is globally smooth and bounded.

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