Limit theorems for trawl processes

Mikko S. Pakkanen; Riccardo Passeggeri; Orimar Sauri; Almut E.D. Veraart

doi:10.1214/21-EJP652

Limit theorems for trawl processes

Mikko S. Pakkanen, Riccardo Passeggeri, Orimar Sauri, Almut E.D. Veraart

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Abstract

In this work we derive limit theorems for trawl processes. First, we study the asymptotic behavior of the partial sums of the discretized trawl process (X_i∆n )^⌊nt^⌋−1 i=0, under the assumption that as n ↑ ∞, ∆_n ↓ 0 and n∆_n → µ ∈ [0, +∞]. Second, we prove a general result on functional convergence in distribution of trawl processes. As an application of this result, we show that a trawl process whose Lévy measure tends to infinity converges in distribution, under suitable rescaling, to a Gaussian moving average process.

Original language	English
Article number	116
Journal	Electronic Journal of Probability
Volume	26
DOIs	https://doi.org/10.1214/21-EJP652
Publication status	Published - 2021

Bibliographical note

Funding Information:
*R. P. acknowledges the support provided by the Fondation Sciences Mathématiques de Paris (FSMP) fellowship, held at LPSM (Sorbonne University). O. S. would like to thank the Villum Fonden for providing partial funding for this research as part of the project number 11745 titled “Ambit Fields: Probabilistic Properties and Statistical Inference”. †Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK. E-mail: m.pakkanen@imperial.ac.uk ‡Department of Statistical Sciences, University of Toronto, 700 University Ave., Toronto, ON, M5G 1X6, Canada. E-mail: r.passeggeri@utoronto.ca §Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220, Aalborg, Denmark. E-mail: osauri@math.aau.dk ¶Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK. E-mail: a.veraart@imperial.ac.uk

Publisher Copyright:
© 2021, Institute of Mathematical Statistics. All rights reserved.

Keywords

Functional limit theorem
Moving average
Partial sum
Stable convergence
Trawl process

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title = "Limit theorems for trawl processes",

abstract = "In this work we derive limit theorems for trawl processes. First, we study the asymptotic behavior of the partial sums of the discretized trawl process (Xi∆n )⌊nt⌋−1 i=0, under the assumption that as n ↑ ∞, ∆n ↓ 0 and n∆n → µ ∈ [0, +∞]. Second, we prove a general result on functional convergence in distribution of trawl processes. As an application of this result, we show that a trawl process whose L{\'e}vy measure tends to infinity converges in distribution, under suitable rescaling, to a Gaussian moving average process.",

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author = "Pakkanen, {Mikko S.} and Riccardo Passeggeri and Orimar Sauri and Veraart, {Almut E.D.}",

note = "Funding Information: *R. P. acknowledges the support provided by the Fondation Sciences Math{\'e}matiques de Paris (FSMP) fellowship, held at LPSM (Sorbonne University). O. S. would like to thank the Villum Fonden for providing partial funding for this research as part of the project number 11745 titled “Ambit Fields: Probabilistic Properties and Statistical Inference”. †Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK. E-mail: m.pakkanen@imperial.ac.uk ‡Department of Statistical Sciences, University of Toronto, 700 University Ave., Toronto, ON, M5G 1X6, Canada. E-mail: r.passeggeri@utoronto.ca §Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220, Aalborg, Denmark. E-mail: osauri@math.aau.dk ¶Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK. E-mail: a.veraart@imperial.ac.uk Publisher Copyright: {\textcopyright} 2021, Institute of Mathematical Statistics. All rights reserved.",

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doi = "10.1214/21-EJP652",

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volume = "26",

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N2 - In this work we derive limit theorems for trawl processes. First, we study the asymptotic behavior of the partial sums of the discretized trawl process (Xi∆n )⌊nt⌋−1 i=0, under the assumption that as n ↑ ∞, ∆n ↓ 0 and n∆n → µ ∈ [0, +∞]. Second, we prove a general result on functional convergence in distribution of trawl processes. As an application of this result, we show that a trawl process whose Lévy measure tends to infinity converges in distribution, under suitable rescaling, to a Gaussian moving average process.

AB - In this work we derive limit theorems for trawl processes. First, we study the asymptotic behavior of the partial sums of the discretized trawl process (Xi∆n )⌊nt⌋−1 i=0, under the assumption that as n ↑ ∞, ∆n ↓ 0 and n∆n → µ ∈ [0, +∞]. Second, we prove a general result on functional convergence in distribution of trawl processes. As an application of this result, we show that a trawl process whose Lévy measure tends to infinity converges in distribution, under suitable rescaling, to a Gaussian moving average process.

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Limit theorems for trawl processes

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