Thursday 22.09.2022, 1pm, MNO 1.020
Nikolai Leonenko (Cardiff University)
Sojourn functionals for spatiotemporal Gaussian random fields with long-memory
Abstract: The paper  addresses the asymptotic analysis of sojourn functionals of spatiotemporal Gaussian random fields with long-range dependence (LRD) in time also known as long memory. Specifically, reduction theorems are derived for local functionals of nonlinear transformation of such fields, with Hermite rank m ≥ 1, under general covariance structures. These results are proven to hold, in particular, for a family of non–separable covariance structures belonging to Gneiting class. For m = 2, under separability of the spatiotemporal covariance function in space and time, the properly normalized Minkowski functional, involving the modulus of a Gaussian random field, converges in distribution to the Rosenblatt type limiting distribution for a suitable range of the long memory parameter. For spatiotemporal isotropic stationary fields on sphere similar results obtained in Marinucci et al. . Some other related results can be found in Makogin and Spodarev . For short-memory random fields the asymptotic analysis of sojourn functionals can be done using the Mallivin-Stein technique, fourth-moment limit theorems, Breuer-Major type theorems (see [1,2,6,7,8] and the references therein).This is joint results with M.D.Ruiz-Medina (Granada University, Spain).References:
 Bourguin, S.,Campese, S., Leonenko, N. and Taqqu, M.S. (2019) Four moments theorems on Markov chaos. Ann. Probab. 47 (2019), no. 3, 1417–1446
 Ivanov A.V., Leonenko N.N, Ruiz-Medina, M.D. and Savich, I.N. (2013) Limit theorems for weighted non-linear transformations of Gaussian processes with singular spectra, Ann. of Probab., vol. 41, No 2, 1088-1114
 Leonenko, N.N. and Ruiz-Medina, M.D. (2022) Sojourn functionals for spatiotemporal Gaussian random fields with long-memory, Journal of Applied Probability, in press.
 Makogin, V. and Spodarev, E. (2022). Limit theorems for excursion sets of subordinated Gaussian random fields with long-range dependence, Stochastics, 94, 111–142
 Marinucci, D., Rossi, M. and Vidotto, A. (2020). Non-universal fluctuations of the empirical measure for isotropic stationary fields on S2 × R, Annals of Applied Probability, 31, 2311–2349
[6 ] Nourdin, I. and Peccati, G. (2015) The optimal fourth moment theorem. Proc. Amer. Math. Soc. 143 (2015), no. 7, 3123–3133.
 Nourdin, I. and Peccati, G.and Podolskij, M. (2011) Quantitative Breuer-Major theorems. Stochastic Process. Appl. 121 (2011), no. 4, 793–812.