Convergence of long-memory discrete <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math>th order Volterra processes

Bai, Shuyang; Taqqu, Murad S.

doi:10.1016/j.spa.2014.12.006

Cited by 8 publications

(34 citation statements)

References 21 publications

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“…Remark 2.4. Since by (5), the self-similarity parameter H equals γ 1 + γ 2 + 2, we get that H tends to 1 as (γ 1 , γ 2 ) → (−1/2, −1/2). It is known (see e.g., Theorem 3.1.1 of Embrechts and Maejima [12]) that the only self-similar finite-variance processes with stationary increments having H = 1 are degenerate processes.…”

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confidence: 87%

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Behavior of the generalized Rosenblatt process at extreme critical exponent values

Bai¹,

Taqqu²

2017

Ann. Probab.

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The generalized Rosenblatt process is obtained by replacing the single critical exponent characterizing the Rosenblatt process by two different exponents living in the interior of a triangular region. What happens to that generalized Rosenblatt process as these critical exponents approach the boundaries of the triangle? We show by two different methods that on each of the two symmetric boundaries, the limit is non-Gaussian. On the third boundary, the limit is Brownian motion. The rates of convergence to these boundaries are also given. The situation is particularly delicate as one approaches the corners of the triangle, because the limit process will depend on how these corners are approached. All limits are in the sense of weak convergence in C[0, 1]. These limits cannot be strengthened to convergence in L 2 (Ω).

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confidence: 87%

“…One has γ 1 + γ 2 = −3/2 all through the diagonal d. The corners of the triangle are excluded by the requirement γ 1 , γ 2 > −1 + ǫ. Convergence to Brownian motion in (7) is expected heuristically since the self-similarity parameter H = γ 1 + γ 2 + 2 → 1/2 (see (5)), and 1/2 is the self-similarity parameter of Brownian motion.…”

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confidence: 99%

Behavior of the generalized Rosenblatt process at extreme critical exponent values

Bai¹,

Taqqu²

2017

Ann. Probab.

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“…These processes Z(t) include the Hermite process considered in [6,16] and [15]. In [2], a noncentral limit theorem is established for a polynomial-form process called kth order discrete Volterra process:…”

Section: Introductionmentioning

confidence: 99%

“…which differs from Y ′ (n) in (2) by including the diagonals, and where a(·) is asymptotically g(·), some special type of generalized Hermite kernel called generalized Hermite kernel of Class (B) (GHK(B)). The limit Z(t) can be heuristically thought as (3) with diagonals included, and is precisely expressed as a k-fold centered Wiener-Stratonovich integral, which is a linear combination of certain Wiener-Itô integrals of orders lower than or equal to k (see [2]).…”

Section: Introductionmentioning

confidence: 99%

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

Bai¹,

Taqqu²

2017

Bernoulli

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We start with an i.i.d. sequence and consider the product of two polynomial-forms moving averages based on that sequence. The coefficients of the polynomial forms are asymptotically slowly decaying homogeneous functions so that these processes have long memory. The product of these two polynomial forms is a stationary nonlinear process. Our goal is to obtain limit theorems for the normalized sums of this product process in three cases: exclusion of the diagonal terms of the polynomial form, inclusion, or the mixed case (one polynomial form excludes the diagonals while the other one includes them). In any one of these cases, if the product has long memory, then the limits are given by Wiener chaos. But the limits in each of the cases are quite different. If the diagonals are excluded, then the limit is expressed as in the product formula of two Wiener-Itô integrals. When the diagonals are included, the limit stochastic integrals are typically due to a single factor of the product, namely the one with the strongest memory. In the mixed case, the limit stochastic integral is due to the polynomial form without the diagonals irrespective of the strength of the memory.

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“…The process Z g and other related processes appear as limits in various types of non-central limit theorems involving Voterra-type nonlinear process. See Bai and Taqqu (2014c) and Bai and Taqqu (2014a) for details. The following is a natural question:…”

Section: Introductionmentioning

confidence: 99%

Structure of the third moment of the generalized Rosenblatt distribution

Bai¹,

Taqqu²

2014

Statistics & Probability Letters

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The Rosenblatt distribution appears as limit in non-central limit theorems. The generalized Rosenblatt distribution is obtained by allowing different power exponents in the kernel that defines the usual Rosenblatt distribution. We derive an explicit formula for its third moment, correcting the one in Maejima and Tudor (2012) and Tudor (2013). Evaluating this formula numerically, we are able to confirm that the class of generalized Hermite processes is strictly richer than the class of Hermite processes.

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Convergence of long-memory discrete kth order Volterra processes

Cited by 8 publications

References 21 publications

Behavior of the generalized Rosenblatt process at extreme critical exponent values

Behavior of the generalized Rosenblatt process at extreme critical exponent values

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

Structure of the third moment of the generalized Rosenblatt distribution

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