scholarly journals On simultaneous limits for aggregation of stationary randomized INAR(1) processes with poisson innovations

2021 ◽  
Vol 71 (5) ◽  
pp. 1241-1268
Author(s):  
Mátyás Barczy ◽  
Fanni K. Nedényi ◽  
Gyula Pap
Keyword(s):  
Density Function ◽  
Explicit Formula ◽  
Time Scale ◽  
Random Coefficient ◽  
Partial Sums ◽  
Characteristic Functions ◽  
Autoregressive Processes ◽  
Finite Dimensional ◽  
Joint Characteristic

Abstract We investigate joint temporal and contemporaneous aggregation of N independent copies of strictly stationary INteger-valued AutoRegressive processes of order 1 (INAR(1)) with random coefficient α ∈ (0, 1) and with idiosyncratic Poisson innovations. Assuming that α has a density function of the form ψ(x) (1 − x) β , x ∈ (0, 1), with β ∈ (−1, ∞) and lim x ↑ 1 ψ ( x ) = ψ 1 ∈ ( 0 , ∞ ) $\lim\limits_{x\uparrow 1} \psi(x) = \psi_1 \in (0, \infty)$ , different limits of appropriately centered and scaled aggregated partial sums are shown to exist for β ∈ (−1, 0] in the so-called simultaneous case, i.e., when both N and the time scale n increase to infinity at a given rate. The case β ∈ (0, ∞) remains open. We also give a new explicit formula for the joint characteristic functions of finite dimensional distributions of the appropriately centered aggregated process in question.

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).

Invariant measures for Markov chains with no irreducibility assumptions

1988 ◽  
Vol 25 (A) ◽  
pp. 275-285 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Probability Theory ◽  
Invariant Measure ◽  
Markov Chains ◽  
Invariant Measures ◽  
Random Coefficient ◽  
General Function ◽  
Autoregressive Processes ◽  
Detailed Consideration ◽  
Positive Recurrence ◽  
Countable Space

Foster's criterion for positive recurrence of irreducible countable space Markov chains is one of the oldest tools in applied probability theory. In various papers in JAP and AAP it has been shown that, under extensions of irreducibility such as ϕ -irreducibility, analogues of and generalizations of Foster's criterion give conditions for the existence of an invariant measure π for general space chains, and for π to have a finite f-moment ∫π (dy)f(y), where f is a general function. In the case f ≡ 1 these cover the question of finiteness of π itself. In this paper we show that the same conditions imply the same conclusions without any irreducibility assumptions; Foster's criterion forces sufficient and appropriate regularity on the space automatically. The proofs involve detailed consideration of the structure of the minimal subinvariant measures of the chain. The results are applied to random coefficient autoregressive processes in order to illustrate the need to remove irreducibility conditions if possible.

Invariant measures for Markov chains with no irreducibility assumptions

1988 ◽  
Vol 25 (A) ◽  
pp. 275-285 ◽  
Author(s):  
R. L. Tweedie
Keyword(s):  
Probability Theory ◽  
Invariant Measure ◽  
Markov Chains ◽  
Invariant Measures ◽  
Random Coefficient ◽  
General Function ◽  
Autoregressive Processes ◽  
Detailed Consideration ◽  
Positive Recurrence ◽  
Countable Space

Foster's criterion for positive recurrence of irreducible countable space Markov chains is one of the oldest tools in applied probability theory. In various papers in JAP and AAP it has been shown that, under extensions of irreducibility such as ϕ -irreducibility, analogues of and generalizations of Foster's criterion give conditions for the existence of an invariant measure π for general space chains, and for π to have a finite f-moment ∫π (dy)f(y), where f is a general function. In the case f ≡ 1 these cover the question of finiteness of π itself.In this paper we show that the same conditions imply the same conclusions without any irreducibility assumptions; Foster's criterion forces sufficient and appropriate regularity on the space automatically. The proofs involve detailed consideration of the structure of the minimal subinvariant measures of the chain.The results are applied to random coefficient autoregressive processes in order to illustrate the need to remove irreducibility conditions if possible.

Random coefficient autoregression, regime switching and long memory

2003 ◽  
Vol 35 (03) ◽  
pp. 737-754 ◽  
Author(s):  
Remigijus Leipus ◽  
Donatas Surgailis
Keyword(s):  
Probability Density ◽  
Power Law ◽  
Renewal Process ◽  
Long Memory ◽  
Regime Switching ◽  
Covariance Function ◽  
Random Coefficient ◽  
Levy Process ◽  
Partial Sums ◽  
Stable Lévy Process

We discuss long-memory properties and the partial sums process of the AR(1) process {X t , t ∈ 𝕫} with random coefficient {a t , t ∈ 𝕫} taking independent values A j ∈ [0,1] on consecutive intervals of a stationary renewal process with a power-law interrenewal distribution. In the case when the distribution of generic A j has either an atom at the point a=1 or a beta-type probability density in a neighborhood of a=1, we show that the covariance function of {X t } decays hyperbolically with exponent between 0 and 1, and that a suitably normalized partial sums process of {X t } weakly converges to a stable Lévy process.

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