Page 559 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 559
DELING ROUGHNESS AND LONG-RANGE DEPENDENCE WITH FRACTIONAL
PROCESSES (MS-18)
The multifaceted behaviour of supOU processes: intermittency,
multiscaling in limit theorems
Nikolai Leonenko, LeonenkoN@Cardiff.ac.uk
Cardiff University, United Kingdom
Superpositions of Ornstein-Uhlenbeck type (supOU) processes provide a rich class of stationary
stochastic processes for which the marginal distribution and the dependence structure may be
modeled independently [1]. In this paper we investigate the limiting behavior of integrated
supOU processes with finite variance [2,3] (see also [5] for multifaceted behavior of supOU
processes in the case the infinite variance).
We show that after suitable normalization four different limiting processes may arise. The
type of limit depends on the decay of the correlation function as well as on the characteristic
triplet of the marginal distribution. supOU processes, moreover, may exhibit intermittency, a
phenomenon affecting the rate of growth of moments [2,3,4]. We establish this rate for each of
the four limiting scenarios. The rate changes at some point indicating that there is a change-
point in the asymptotic behavior of absolute moments. For such a behavior to be possible,
the moments in the limit theorem do not converge beyond some critical point. We show that
this point is related to the dependence structure of the supOU process. The intermittency phe-
nomenon appears also in some other models, for example, in the subclass of ambit processes
known as trawl processes [3]. We also discuss the contrasts between convergence in distribu-
tions and almost sure convergence using intermittency and multiscaling.
Joint work with D. Grahovac (Osijek University) and M. Taqqu (Boston University).
References
[1] Barndorff-Nielsen, O.E. and Leonenko, N.N., (2005), Spectral properties of superpositions
of Ornstein-Uhlenbeck type processes, Meth. Computing in Applied Probability, 7, 335-
352
[2] Grahovac, D., Leonenko, N., Sikorskii, A. and Taqqu.M.S. (2019) The unusual properties
of aggregated superpositions of Ornstein-Uhlenback type processes, Bernoulli 25 (2019),
2029-2050
[3] Grahovac, D., Leonenko, N., Taqqu, M.S. (2017) Intermittency of trawl processes, Statistic
and Probability Letters, 137, 235-242
[4] Grahovac, D., Leonenko, N., Taqqu, M.S. (2019) Limit theorems, scaling of moments and
intermittency for integrated finite variance supOU processes, Stoch. Proc. Appl., 129, 5113-
5150
[5] Grahovac, D., Leonenko, N., Taqqu, M.S. (2020) The multifaceted behaviour of supOU
processes: the infinite variance case, Journal of Theoretical Probability, in press.
557
PROCESSES (MS-18)
The multifaceted behaviour of supOU processes: intermittency,
multiscaling in limit theorems
Nikolai Leonenko, LeonenkoN@Cardiff.ac.uk
Cardiff University, United Kingdom
Superpositions of Ornstein-Uhlenbeck type (supOU) processes provide a rich class of stationary
stochastic processes for which the marginal distribution and the dependence structure may be
modeled independently [1]. In this paper we investigate the limiting behavior of integrated
supOU processes with finite variance [2,3] (see also [5] for multifaceted behavior of supOU
processes in the case the infinite variance).
We show that after suitable normalization four different limiting processes may arise. The
type of limit depends on the decay of the correlation function as well as on the characteristic
triplet of the marginal distribution. supOU processes, moreover, may exhibit intermittency, a
phenomenon affecting the rate of growth of moments [2,3,4]. We establish this rate for each of
the four limiting scenarios. The rate changes at some point indicating that there is a change-
point in the asymptotic behavior of absolute moments. For such a behavior to be possible,
the moments in the limit theorem do not converge beyond some critical point. We show that
this point is related to the dependence structure of the supOU process. The intermittency phe-
nomenon appears also in some other models, for example, in the subclass of ambit processes
known as trawl processes [3]. We also discuss the contrasts between convergence in distribu-
tions and almost sure convergence using intermittency and multiscaling.
Joint work with D. Grahovac (Osijek University) and M. Taqqu (Boston University).
References
[1] Barndorff-Nielsen, O.E. and Leonenko, N.N., (2005), Spectral properties of superpositions
of Ornstein-Uhlenbeck type processes, Meth. Computing in Applied Probability, 7, 335-
352
[2] Grahovac, D., Leonenko, N., Sikorskii, A. and Taqqu.M.S. (2019) The unusual properties
of aggregated superpositions of Ornstein-Uhlenback type processes, Bernoulli 25 (2019),
2029-2050
[3] Grahovac, D., Leonenko, N., Taqqu, M.S. (2017) Intermittency of trawl processes, Statistic
and Probability Letters, 137, 235-242
[4] Grahovac, D., Leonenko, N., Taqqu, M.S. (2019) Limit theorems, scaling of moments and
intermittency for integrated finite variance supOU processes, Stoch. Proc. Appl., 129, 5113-
5150
[5] Grahovac, D., Leonenko, N., Taqqu, M.S. (2020) The multifaceted behaviour of supOU
processes: the infinite variance case, Journal of Theoretical Probability, in press.
557
