Past sessions

Thursday 29.09.2022, 1pm, MNO 1.020
Sébastien Darses (AixMarseille University), On probabilistic generalizations of the NymanBeurling criterion and new closedform identities involving the Riemann Zeta function
Abstract: One of the seemingly innocent reformulations of the terrifying Riemann Hypothesis (RH) is the NymanBeurling criterion: The indicator function of (0,1) can be linearly approximated in L^2 by dilations of the fractional part function. Randomizing these dilations generates new structures and criteria for RH, regularizing very intricate ones. One other possible nice feature is to consider polynomials instead of Dirichlet polynomials for the approximations. The Zeta function is then encoded in this last framework as a weighted density measure on the critical line. We prove closed form identities for the involved moments (determinate Hamburger problem, and thus a full characterization).
The talk will be very accessible, especially for graduate students and a quick review on the Zeta function will be given.
Joint work with F. Alouges and E. Hillion.
 Thursday 22.09.2022, 1pm, MNO 1.020
Nikolai Leonenko (Cardiff University), Sojourn functionals for spatiotemporal Gaussian random fields with longmemory
Abstract: The paper [3] addresses the asymptotic analysis of sojourn functionals of spatiotemporal Gaussian random fields with longrange 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 shortmemory random fields the asymptotic analysis of sojourn functionals can be done using the MallivinStein technique, fourthmoment limit theorems, BreuerMajor type theorems (see [1,2,6,7,8] and the references therein).This is joint results with M.D.RuizMedina (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, RuizMedina, M.D. and Savich, I.N. (2013) Limit theorems for weighted nonlinear transformations of Gaussian processes with singular spectra, Ann. of Probab., vol. 41, No 2, 10881114
[3] Leonenko, N.N. and RuizMedina, M.D. (2022) Sojourn functionals for spatiotemporal Gaussian random fields with longmemory, Journal of Applied Probability, in press.
[4] Makogin, V. and Spodarev, E. (2022). Limit theorems for excursion sets of subordinated Gaussian random fields with longrange dependence, Stochastics, 94, 111–142
[5] Marinucci, D., Rossi, M. and Vidotto, A. (2020). Nonuniversal 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 BreuerMajor theorems. Stochastic Process. Appl. 121 (2011), no. 4, 793–812.
 Monday 05.09.2022, 4pm, MNO 1.050
ZhenQing Chen (University of Washington – Seattle), Long range random walk on infinite groups
Abstract: Given the original discrete group and a random walk on it driven by a certain type of symmetric probability measure, there exists a homogeneous nilpotent Lie group which carries an adapted dilation structure and a stablelike process which appears in a Donskertype functional limit theorem as the limit of a rescaled version of the random walk. Both the limit group and the limit process on that group depend on the driving probability measure. In addition to the functional limit theorem, a local limit theorem is also established.
Based on joint work with Takashi Kumagai, Laurent SaloffCoste, Jian Wang and Tianyi Zheng.