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  • Probability & Statistics Seminar
The Probability and Statistics seminar is a meeting of the research teams of Prof. Baraud, Prof. Ley, Prof. Nourdin, Prof. Peccati, Prof. Podolskij and Prof. Thalmaier. Its aim is to present both research works and surveys of mathematical areas of common interest.
You can use this agenda to get to know when our next seminars are scheduled.
If you want to propose a talk at our seminar, please contact L. Loosveldt.

 

Upcoming sessions

  • Thursday 06.10.2022, 1pm, MNO 1.020
    Alexandre Tsybakov (CREST – ENSAE)Statistical decision for variable selection

    Abstract: For the core variable selection problem under the Hamming loss, we derive a non-asymptotic exact minimax selector over the class of all s-sparse vectors, which is also the Bayes selector with respect to the uniform prior. While this optimal selector is, in general, not realizable in polynomial time, we show that its tractable counterpart (the scan selector) attains the minimax expected Hamming risk to within factor 2 and moreover is exact minimax under the probability of wrong recovery criterion. In the monotone likelihood ratio framework, we establish explicit lower bounds on the minimax risk and provide its tight characterization in terms of the best separable selector risk. As a consequence, we obtain sharp necessary and sufficient conditions of exact and almost full recovery in the location model with light tail distributions and in the problem of group variable selection under Gaussian noise. The talk is based on a joint work with Cristina Butucea, Enno Mammen and Simo Ndaoud.​

  • Thursday 20.10.2022, 1pm, MNO 1.040
    Yassine Nachit (Cadi Ayyad University), Local times for systems of non-linear stochastic heat equations 

    Abstract: We consider u(t,x)=(u_1(t,x),\cdots,u_d(t,x)) the solution to a system of non-linear stochastic heat equations in spatial dimension one driven by a d-dimensional space-time white noise. We prove that, when d\leq 3, the local time L(\xi,t) of \{u(t,x)\,,\;t\in[0,T]\} exists and L(\bullet,t) belongs a.s. to the Sobolev space H^{\alpha}(\R^d) for \alpha<\frac{4-d}{2}, and when d\geq 4, the local time does not exist. We also show joint continuity and establish H\”{o}lder conditions for the local time of \{u(t,x)\,,\;t\in[0,T]\}. These results are then used to investigate the irregularity of the coordinate functions of {u(t,x)\,,\;t\in[0,T]\}. Comparing to similar results obtained for the linear stochastic heat equation (i.e., the solution is Gaussian), we believe that our results are sharp. Finally, we get a sharp estimate for the partial derivatives of the joint density of (u(t_1,x)-u(t_0,x),\cdots,u(t_n,x)-u(t_{n-1},x)), which is a new result and of independent interest.

  • Thursday 27.10.2022, 1pm, MNO 1.040
    Jean-Michel Poggi (Paris-Cité University and Paris-Saclay University)
    Random Forests: Introduction and industrial applications


    Abstract:
    Random forests (RF) are a statistical learning method extensively used in many fields of application, thanks to its excellent predictive performance. RF are part of the family of tree-based methods and inherit intrinsic flexibility of trees: adapted to both classification and regression problems, easily extended to many different types of data.

    We will first focus on an introduction of RF, the definition of the variable importance measures and the related variable selection capability. We then illustrate their practical power in two different applied contexts. The first one relates to physiological signal processing and addresses the functional variable selection for driver’s stress level classification. The second is about the aggregation of multi-scale experts for bottom-up electricity load forecasting.

           References

    • Genuer, Poggi, Random Forests with R, 98 p., Use’R!, Springer, 2020
    • Genuer, Poggi, Tuleau, Variable selection using Random Forests, Pattern Recognition Letters, 31(14), p. 2225-2236, 2010
    • El Haouij, Poggi, Ghozi, Sevestre Ghalila, Jaïdane, Random Forest-Based Approach for Physiological Functional Variable Selection for Driver’s Stress Level Classification, Statistical Methods & Applications, 1-29, 2018
    • Goehry, Goude, Massart, Poggi, Aggregation of Multi-scale Experts for Bottom-up Load Forecasting, IEEE Transactions on Smart Grid, vol. 11, 3, 1895-1904, 2020.
  • Thursday 10.11.2022, 1pm, TBA
    Taras Bodnar (Stockholm University)TBA
    Abstract: TBA

 

  • Thursday 17.11.2022, 1pm, TBA
    Pierre Alquier (RIKEN AIP)TBA
    Abstract: TBA

Past sessions

  • Thursday 29.09.2022, 1pm, MNO 1.020
    Sébastien Darses (Aix-Marseille University), On probabilistic generalizations of the Nyman-Beurling criterion and new closed-form identities involving the Riemann Zeta function

Abstract: One of the seemingly innocent reformulations of the terrifying Riemann Hypothesis (RH) is the Nyman-Beurling 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 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.

 

  • Monday 05.09.2022, 4pm, MNO 1.050
    Zhen-Qing 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 stable-like process which appears in a Donsker-type 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 Saloff-Coste, Jian Wang and Tianyi Zheng.

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