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 [3] 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. [5]. Some other related results can be found in Makogin and Spodarev [4]. 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:**

[1] 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

[2] 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

[3] 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.

[4] Makogin, V. and Spodarev, E. (2022). Limit theorems for excursion sets of subordinated Gaussian random fields with long-range dependence, *Stochastics*, 94, 111–142

[5] 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.

[7] Nourdin, I. and Peccati, G.and Podolskij, M. (2011) Quantitative Breuer-Major theorems. *Stochastic Process. Appl.* 121 (2011), no. 4, 793–812.