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

  • Mátyás Barczy EMAIL logo , Fanni K. Nedényi and Gyula Pap
Published/Copyright: October 4, 2021
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 71 Issue 5

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 limx↑1ψ(x)=ψ1∈(0,∞), 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.

Keywords: randomized INAR(1) process; temporal and contemporaneous aggregation; simultaneous limits
MSC 2010: Primary 60F05; 60J80; 60E10; Secondary 60G52; 60G15

Mátyás Barczy is supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, and by the ÚNKP-19-4 New National Excellence Program of the Ministry for Innovation and Technology. Fanni K. Nedényi is supported by the ÚNKP-19-3 New National Excellence Program of the Ministry for Innovation and Technology. Gyula Pap was supported by the Ministry for Innovation and Technology, Hungary grant NKFIH-1279-2/2020.

  1. (Communicated by Gejza Wimmer)

Appendices

Appendix A Generator function of finite-dimensional distributions of stationary INAR(1) processes with Poisson immigrations

Consider a strictly stationary (usual) INAR(1) process (Xk)k∈Z+ with thinning parameter a ∈ (0, 1) and with Poisson immigration distribution having parameter λ ∈ (0, ∞). Namely,

(A.1) P ( ξ 1 , 1 = 1 ) = a = 1 − P ( ξ 1 , 1 = 0 ) , P ( ε 1 = ℓ ) = λ ℓ ℓ ! e − λ , ℓ ∈ Z + , P ( X 0 = k ) = ( ( 1 − a ) − 1 λ ) k k ! e − ( 1 − a ) − 1 λ , k ∈ Z + .

As it was recalled in Section 1, (Xk)k∈Z+ is indeed a strictly stationary INAR(1) process.

Proposition A.1

Under the assumption (A.1), the joint generator function of (X0, X1,…,Xk), k ∈ ℤ+, takes the form

(A.2) F 0 , … , k ( z 0 , … , z k ) := E ( z 0 X 0 z 1 X 1 ⋯ z k X k ) = exp ⁡ { λ 1 − a ∑ 0 ≤ i ≤ j ≤ k a j − i ( z i − 1 ) z i + 1 ⋯ z j − 1 ( z j − 1 ) }

for all k ∈ ℕ and z0,…,zk ∈ ℂ, where, for i=j, the term in the sum above is zi − 1.

For the proof of Proposition A.1, see the proof of Proposition 2.1 in Barczy et al. [3] (see also Barczy et al. [4: Proposition 2.1].

Appendix B Infinite series representation of strictly stationary INAR(1) processes

Lemma B.1

Under the assumption (A.1), we have

(B.1) ( X k ) k ∈ Z = D ( ε k + ∑ i = 1 ∞ a k ( k − i ) ∘ ⋯ ∘ a k − i + 1 ( k − i ) ∘ ε k − i ) k ∈ Z ,

where {εk : k ∈ ℤ} are independent random variables with the same distribution as ε1(given in assumption (A.1)), andak(ℓ), k, ℓ ∈ ℤ, are given by

a k ( ℓ ) ∘ i := ∑ j = 1 i ξ k , j ( ℓ ) , i f i ∈ N , 0 , i f i = 0 ,

where ξ k , j ( ℓ ) , j ∈ ℕ, k, ℓ ∈ ℤ, have the same distribution as ξ1,1(given in assumption (A.1)), and {εk : k ∈ ℤ} andak(ℓ), k, ℓ ∈ ℤ, are independent in the sense that the families {εk : k ∈ ℤ} and{ξk,j(ℓ):j∈N}, k, ℓ ∈ ℤ, occurring inak(ℓ), k, ℓ ∈ ℤ, are independent families of independent random variables, and the series in the representation (B.1) converge with probability one.

Lemma B.1 is a special case of Lemma E.2 in Barczy et al. [2], where one can find a proof as well.

Appendix C Approximations of the exponential function and some of its integrals

In this appendix we collect some useful approximations of the exponential function and some of its integrals.

We will frequently use the following the well-known inequalities:

(C.1) 1 − e − x ≤ x ,       x ∈ R ,

(C.2) | e i u − 1 | ≤ | u | , | e i u − 1 − i u | ≤ u 2 / 2 , u ∈ R .

The next lemma is about how the inequalities in (C.2) change if we replace u ∈ ℝ by an arbitrary complex number (for a proof, see, e.g., the proof of Lemma B.1 in Barczy et al. [3]).

Lemma C.1

For any z ∈ ℂ it holds that

(C.3) | e z − 1 | ≤ | z | e | z | ,

(C.4) | e z − 1 − z | ≤ | z | 2 2 e | z | .

The next lemma is a variant of Lemma B.2 in Barczy et al. [4] (developed for proving limit theorems for iterated aggregation of randomized INAR(1) processes), and we use it in the proofs of Theorems 1.1 and 1.2.

Lemma C.2

Suppose that (0, 1) ∋ x↦ψ(x) (1 − x)βis a probability density, where ψ is a function on (0, 1) having a limitlimx↑1ψ(x)=ψ1∈(0,∞)(and then necessarily β ∈ (−1, ∞)). For all a ∈ (0, 1), let (zn(a))n∈ℕbe a sequence of complex numbers, let n0 ∈ ℕ, (εn)n≥n0be a sequence in (0, 1) withlimn→∞εn=0, and let (Nn)n∈ℕbe a sequence of positive integers such that

(C.5) sup n ≥ n 0 ε n − 1 N n sup a ∈ ( 0 , 1 − ε n ) | z n ( a ) | < ∞ ,

lim sup n → ∞ N n ∫ 1 − ε n 1 1 − e λ 1 − a z n ( a ) ( 1 − a ) β d a < ∞ , lim n → ∞ | N n ∫ 1 − ε n 1 1 − e λ 1 − a z n ( a ) ( 1 − a ) β d a − I | = 0

with some I ∈ ℂ. Then

lim n → ∞ N n ∫ 0 1 1 − e λ 1 − a z n ( a ) ψ ( a ) ( 1 − a ) β d a = ψ 1 I .

Proof

For all a ∈ (0, 1) and for sufficiently large n ∈ ℕ, we have 1 −εn > a, hence, by (C.5),

(C.6) N n | z n ( a ) | ≤ ε n ε n − 1 N n sup b ∈ ( 0 , 1 − ε n ) | z n ( b ) | → 0 as n → ∞ ,

thus we conclude limn→∞Nn|zn(a)|=0. By applying (C.3) and using (C.6), for any n ∈ ℕ and a ∈ (0, 1), we get

(C.7) | N n 1 − e λ 1 − a z n ( a ) | ≤ N n | λ 1 − a z n ( a ) | e λ 1 − a z n ( a ) → 0 as n → ∞

If n ≥ n0 and a ∈ (0, 1 −εn), then 11−a<εn−1 and

N n 1 − e λ 1 − a z n ( a ) ≤ λ ( sup n ≥ n 0 ε n − 1 N n sup a ∈ ( 0 , 1 − ε n ) | z n ( a ) | ) e λ sup n ≥ n 0 ε n − 1 sup a ∈ ( 0 , 1 − ε n ) | z n ( a ) | =: C ,

where C ∈ ℝ+ (due to (C.5)). Since ∫01ψ(a)(1−a)βda=1, we have

| N n ∫ 0 1 − ε n 1 − e λ 1 − a z n ( a ) ψ ( a ) ( 1 − a ) β d a | = | ∫ 0 1 N n 1 − e λ 1 − a z n ( a ) 1 ( 0 , 1 − ε n ) ( a ) ψ ( a ) ( 1 − a ) β d a | ≤ ∫ 0 1 C ψ ( a ) ( 1 − a ) β d a < ∞

for n = n0. Therefore, (0, 1) ∋ a ↦ Cψ(a)(1 − a)β serves as a dominating integrable function. Thus, by the dominated convergence theorem, the pointwise convergence in (C.7) results

(C.8) lim n → ∞ N n ∫ 0 1 − ε n 1 − e λ 1 − a z n ( a ) ψ ( a ) ( 1 − a ) β d a = 0 .

Moreover, for all n ≥ n0, we have

(C.9) | N n ∫ 0 1 1 − e λ 1 − a z n ( a ) ψ ( a ) ( 1 − a ) β d a − ψ 1 I | ≤ | N n ∫ 0 1 − ε n 1 − e λ 1 − a z n ( a ) ψ ( a ) ( 1 − a ) β d a | + | N n ∫ 1 − ε n 1 1 − e λ 1 − a z n ( a ) ( ψ ( a ) − ψ 1 ) ( 1 − a ) β d a | + ψ 1 | N n ∫ 1 − ε n 1 1 − e λ 1 − a z n ( a ) ( 1 − a ) β d a − I | ,

where

| N n ∫ 1 − ε n 1 1 − e λ 1 − a z n ( a ) ( ψ ( a ) − ψ 1 ) ( 1 − a ) β d a | ≤ ( sup a ∈ [ 1 − ε n , 1 ) | ψ ( a ) − ψ 1 | ) N n ∫ 1 − ε n 1 1 − e λ 1 − a z n ( a ) ( 1 − a ) β d a ,

with supa∈[1−εn,1)|ψ(a)−ψ1|→0 as n → ∞, by the assumption limx↑1ψ(x)=ψ1. Taking limsupn → ∞ of both sides of (C.9), by (C.8) and the assumptions of the lemma, we obtain the statement.

Acknowledgement

We would like to thank the referees for their comments that helped us improve the paper.

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Received: 2020-06-28
Accepted: 2020-10-20
Published Online: 2021-10-04
Published in Print: 2021-10-26

© 2021 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2021-0051
Keywords for this article
randomized INAR(1) process; temporal and contemporaneous aggregation; simultaneous limits

Articles in the same Issue

  1. Regular papers
  2. Algebraic structures for pairwise comparison matrices: Consistency, social choices and Arrow’s theorem
  3. Polynomial functions on rings of dual numbers over residue class rings of the integers
  4. Sufficient conditions for p-valent functions
  5. Upper bounds for analytic summand functions and related inequalities
  6. Global structure for a fourth-order boundary value problem with sign-changing weight
  7. On the nonexistence conditions of solution of two-point in time problem for nonhomogeneous PDE
  8. Solvability of a nonlinear three-dimensional system of difference equations with constant coefficients
  9. Properties of critical and subcritical second order self-adjoint linear equations
  10. Korovkin type approximation via statistical e-convergence on two dimensional weighted spaces
  11. Approximation by a new sequence of operators involving Apostol-Genocchi polynomials
  12. Poisson like matrix operator and its application in p-summable space
  13. On the homological and algebraical properties of some Feichtinger algebras
  14. Disjoint topological transitivity for weighted translations generated by group actions
  15. On simultaneous limits for aggregation of stationary randomized INAR(1) processes with poisson innovations
  16. Marshall-Olkin Lindley-Log-logistic distribution: Model, properties and applications
  17. The shifted Gompertz-G family of distributions: Properties and applications
  18. On the testing hypothesis in uniform family of distributions with nuisance parameter
  19. Clarkson inequalities related to convex and concave functions
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