scholarly journals Aggregation of a random-coefficient ar(1) process with infinite variance and idiosyncratic innovations

2010 ◽  
Vol 42 (2) ◽  
pp. 509-527 ◽  
Author(s):  
Donata Puplinskaitė ◽  
Donatas Surgailis
Keyword(s):  
Long Memory ◽  
Moving Average ◽  
Domain Of Attraction ◽  
Random Coefficient ◽  
Partial Sums ◽  
Long Range Dependence ◽  
Infinite Variance ◽  
Stable Law ◽  
Slope Coefficient ◽  
Finite Dimensional

Contemporaneous aggregation ofNindependent copies of a random-coefficient AR(1) process with random coefficienta∈ (−1, 1) and independent and identically distributed innovations belonging to the domain of attraction of an α-stable law (0 < α < 2) is discussed. We show that, under the normalizationN1/α, the limit aggregate exists, in the sense of weak convergence of finite-dimensional distributions, and is a mixed stable moving average as studied in Surgailis, Rosiński, Mandrekar and Cambanis (1993). We focus on the case where the slope coefficientahas probability density vanishing regularly ata= 1 with exponentb∈ (0, α − 1) for α ∈ (1, 2). We show that in this case, the limit aggregate {X̅t} exhibits long memory. In particular, for {X̅t}, we investigate the decay of the codifference, the limit of partial sums, and the long-range dependence (sample Allen variance) property of Heyde and Yang (1997).

scholarly journals Aggregation of a random-coefficient ar(1) process with infinite variance and idiosyncratic innovations

2010 ◽  
Vol 42 (02) ◽  
pp. 509-527 ◽  
Author(s):  
Donata Puplinskaitė ◽  
Donatas Surgailis
Keyword(s):  
Long Memory ◽  
Moving Average ◽  
Domain Of Attraction ◽  
Random Coefficient ◽  
Partial Sums ◽  
Long Range Dependence ◽  
Infinite Variance ◽  
Stable Law ◽  
Slope Coefficient ◽  
Finite Dimensional

Contemporaneous aggregation of N independent copies of a random-coefficient AR(1) process with random coefficient a ∈ (−1, 1) and independent and identically distributed innovations belonging to the domain of attraction of an α-stable law (0 &lt; α &lt; 2) is discussed. We show that, under the normalization N 1/α, the limit aggregate exists, in the sense of weak convergence of finite-dimensional distributions, and is a mixed stable moving average as studied in Surgailis, Rosiński, Mandrekar and Cambanis (1993). We focus on the case where the slope coefficient a has probability density vanishing regularly at a = 1 with exponent b ∈ (0, α − 1) for α ∈ (1, 2). We show that in this case, the limit aggregate {X̅ t } exhibits long memory. In particular, for {X̅ t }, we investigate the decay of the codifference, the limit of partial sums, and the long-range dependence (sample Allen variance) property of Heyde and Yang (1997).

scholarly journals On a random-coefficient AR(1) process with heavy-tailed renewal switching coefficient and heavy-tailed noise

2006 ◽  
Vol 43 (02) ◽  
pp. 421-440
Author(s):  
Remigijus Leipus ◽  
Vygantas Paulauskas ◽  
Donatas Surgailis
Keyword(s):  
Unit Root ◽  
Domain Of Attraction ◽  
Stationary Solutions ◽  
Partial Sums ◽  
Infinite Variance ◽  
Finite Variance ◽  
Stable Law ◽  
Stable Lévy Process ◽  
Heavy Tailed ◽  
Independent Identically Distributed

We discuss the limit behavior of the partial sums process of stationary solutions to the (autoregressive) AR(1) equation X t = a t X t−1 + ε t with random (renewal-reward) coefficient, a t , taking independent, identically distributed values A j ∈ [0,1] on consecutive intervals of a stationary renewal process with heavy-tailed interrenewal distribution, and independent, identically distributed innovations, ε t , belonging to the domain of attraction of an α-stable law (0 &lt; α ≤ 2, α ≠ 1). Under suitable conditions on the tail parameter of the interrenewal distribution and the singularity parameter of the distribution of A j near the unit root a = 1, we show that the partial sums process of X t converges to a λ-stable Lévy process with index λ &lt; α. The paper extends the result of Leipus and Surgailis (2003) from the case of finite-variance X t to that of infinite-variance X t .

On Asymptotics of the Sample Distribution for a Class of Linear Process Models in Economics

1992 ◽  
Vol 8 (3) ◽  
pp. 330-342
Author(s):  
C. H. Hesse
Keyword(s):  
Empirical Distribution ◽  
Moving Average ◽  
Domain Of Attraction ◽  
Rates Of Convergence ◽  
Linear Process ◽  
Process Models ◽  
Infinite Variance ◽  
Stable Law ◽  
Sample Distribution ◽  
Parameter Sequence

Let … be a moving average process of infinite order where the innovations ε(k) are in the domain of attraction of a stable law with index α ε (0, 2) and the parameter sequence decreases at a polynomial or exponential rate. These and similar processes have recently received increased attention both in the econometrics and statistics/probability literature. The present paper studies almost sure uniform rates of convergence of the empirical distribution function. Applications of these infinite variance processesin econometrics are mentioned.

scholarly journals On a random-coefficient AR(1) process with heavy-tailed renewal switching coefficient and heavy-tailed noise

2006 ◽  
Vol 43 (2) ◽  
pp. 421-440
Author(s):  
Remigijus Leipus ◽  
Vygantas Paulauskas ◽  
Donatas Surgailis
Keyword(s):  
Unit Root ◽  
Domain Of Attraction ◽  
Stationary Solutions ◽  
Partial Sums ◽  
Infinite Variance ◽  
Finite Variance ◽  
Stable Law ◽  
Stable Lévy Process ◽  
Heavy Tailed ◽  
Independent Identically Distributed

We discuss the limit behavior of the partial sums process of stationary solutions to the (autoregressive) AR(1) equation Xt = atXt−1 + εt with random (renewal-reward) coefficient, at, taking independent, identically distributed values Aj ∈ [0,1] on consecutive intervals of a stationary renewal process with heavy-tailed interrenewal distribution, and independent, identically distributed innovations, εt, belonging to the domain of attraction of an α-stable law (0 < α ≤ 2, α ≠ 1). Under suitable conditions on the tail parameter of the interrenewal distribution and the singularity parameter of the distribution of Aj near the unit root a = 1, we show that the partial sums process of Xt converges to a λ-stable Lévy process with index λ < α. The paper extends the result of Leipus and Surgailis (2003) from the case of finite-variance Xt to that of infinite-variance Xt.

Weak convergence and first passage times

1975 ◽  
Vol 12 (02) ◽  
pp. 324-332
Author(s):  
Allan Gut
Keyword(s):  
First Passage Time ◽  
Domain Of Attraction ◽  
Passage Time ◽  
Partial Sums ◽  
First Passage ◽  
Stable Law ◽  
Common Distribution Function ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit ◽  
Passage Times

Let Sn, n = 1, 2, ‥, denote the partial sums of i.i.d. random variables with the common distribution function F and positive, finite mean. Let N(c) = min [k; Sk &gt; c‥kp ], c ≥ 0, 0 ≤ p &lt; 1. Under the assumption that F belongs to the domain of attraction of a stable law with index α, 1 &lt; α ≤ 2, functional central limit theorems for the first passage time process N(nt), 0 ≤ t ≤ 1, when n → ∞, are derived in the function space D[0,1].

Law of the iterated logarithm for self-normalized sums and their increments

2006 ◽  
Vol 43 (1) ◽  
pp. 79-114
Author(s):  
Han-Ying Liang ◽  
Jong-Il Baek ◽  
Josef Steinebach
Keyword(s):  
Random Variables ◽  
Domain Of Attraction ◽  
Weighted Sums ◽  
Partial Sums ◽  
Stable Law ◽  
Iterated Logarithm ◽  
Independent And Identically Distributed

Let X1, X2,… be independent, but not necessarily identically distributed random variables in the domain of attraction of a stable law with index 0<a<2. This paper uses Mn=max 1?i?n|Xi| to establish a self-normalized law of the iterated logarithm (LIL) for partial sums. Similarly self-normalized increments of partial sums are studied as well. In particular, the results of self-normalized sums of Horváth and Shao[9]under independent and identically distributed random variables are extended and complemented. As applications, some corresponding results for self-normalized weighted sums of iid random variables are also concluded.

scholarly journals Projective Stochastic Equations and Nonlinear Long Memory

2014 ◽  
Vol 46 (4) ◽  
pp. 1084-1105
Author(s):  
Ieva Grublytė ◽  
Donatas Surgailis
Keyword(s):  
Long Memory ◽  
Volterra Series ◽  
Bernoulli Shift ◽  
Moving Average ◽  
Stochastic Equations ◽  
Partial Sums ◽  
Martingale Transform ◽  
New Class ◽  
Moving Average Processes ◽  
Nonlinear Stochastic Equations

A projective moving average {Xt, t ∈ ℤ} is a Bernoulli shift written as a backward martingale transform of the innovation sequence. We introduce a new class of nonlinear stochastic equations for projective moving averages, termed projective equations, involving a (nonlinear) kernel Q and a linear combination of projections of Xt on ‘intermediate’ lagged innovation subspaces with given coefficients αi and βi,j. The class of such equations includes usual moving average processes and the Volterra series of the LARCH model. Solvability of projective equations is obtained using a recursive equality for projections of the solution Xt. We show that, under certain conditions on Q, αi, and βi,j, this solution exhibits covariance and distributional long memory, with fractional Brownian motion as the limit of the corresponding partial sums process.

Limiting behaviour of the distributions of the maxima of partial sums of certain random walks

1972 ◽  
Vol 9 (3) ◽  
pp. 572-579 ◽  
Author(s):  
D. J. Emery
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Asymptotic Property ◽  
Regular Variation ◽  
Domain Of Attraction ◽  
Hitting Times ◽  
Partial Sums ◽  
Symmetric Distributions ◽  
Stable Law ◽  
Limiting Behaviour

It is shown that, under certain conditions, satisfied by stable distributions, symmetric distributions, distributions with zero mean and finite second moment and other distributions, the distribution function of the maxima of successive partial sums of identically distributed random variables has an asymptotic property. This property implies the regular variation of the tail of the distribution of the hitting times of the associated random walk, and hence that these hitting times belong to the domain of attraction of a stable law.

On distribution tails and expectations of maxima in critical branching processes

1996 ◽  
Vol 33 (3) ◽  
pp. 614-622 ◽  
Author(s):  
K. A. Borovkov ◽  
V A. Vatutin
Keyword(s):  
Related Problem ◽  
Branching Processes ◽  
Domain Of Attraction ◽  
Infinite Variance ◽  
Global Maximum ◽  
Stable Law ◽  
Distribution Tail ◽  
Limit Behaviour ◽  
Watson Process

We derive the limit behaviour of the distribution tail of the global maximum of a critical Galton–Watson process and also of the expectations of partial maxima of the process, when the offspring law belongs to the domain of attraction of a stable law. Thus the Lindvall (1976) and Athreya (1988) results are extended to the infinite variance case. It is shown that in the general case these two asymptotics are closely related to each other, and the latter follows readily from the former. We also discuss a related problem from the theory of general branching processes.

MULTIVARIATE AUTOREGRESSION OF ORDER ONE WITH INFINITE VARIANCE INNOVATIONS

2008 ◽  
Vol 24 (3) ◽  
pp. 677-695 ◽  
Author(s):  
M. Zarepour ◽  
S.M. Roknossadati
Keyword(s):  
Autoregressive Model ◽  
Domain Of Attraction ◽  
Ordinary Least Squares ◽  
Infinite Variance ◽  
Regularity Conditions ◽  
Vector Autoregressive Model ◽  
Stable Law ◽  
Limiting Behavior ◽  
Vector Autoregressive ◽  
Multivariate Autoregression

We consider the limiting behavior of a vector autoregressive model of order one (VAR(1)) with independent and identically distributed (i.i.d.) innovations vector with dependent components in the domain of attraction of a multivariate stable law with possibly different indices of stability. It is shown that in some cases the ordinary least squares (OLS) estimates are inconsistent. This inconsistency basically originates from the fact that each coordinate of the partial sum processes of dependent i.i.d. vectors of innovations in the domain of attraction of stable laws needs a different normalizer to converge to a limiting process. It is also revealed that certain M-estimates, with some regularity conditions, as an appropriate alternative, not only resolve inconsistency of the OLS estimates but also give higher consistency rates in all cases.

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