Martedì 31 gennaio 2017
Seminari di Probabilità e Statistica Matematica
Aula G, Dipartimento di Matematica, Sapienza Università di Roma
Ore 14
Valentina Cammarota (Sapienza Università di Roma)
On the critical values of random spherical harmonics.
Abstract: We study the limiting distribution, in the high energy limit, of critical points and extrema of random spherical harmonics. In particular, we first derive the density functions of extrema and saddles and then we provide analytic expressions for the variances. Our arguments require a careful investigation of the validity of the Kac-Rice formula in nonstandard circumstances, entailing degeneracies of covariance matrices for first and second derivatives of the processes being analyzed. It is well known that after proper rescaling random spherical harmonics converge to Berry's random plane waves; in the second part of the talk we focus on the spatial distribution of critical points of random plane waves. Based on joint works with Dmitry Beliaev, Domenico Marinucci and Igor Wigman.
Ore 15
Sokol Ndreca (University Center of Belo Horizonte)
Abstract: In this talk we consider a stochastic point process $i + \xi_i$, where $i\in \mathbb{N}$ and the $\xi_i's$ are i.i.d exponential random
variables with standard deviation $\sigma$. Some properties of this
process are investigated. We then study a discrete time single server
queueing system with this process as arrival process and
deterministic unit service time. We obtain a functional equation of
the bivariate probability generating function of the stationary
distribution for the system. The functional equation is quite
singular, does not admit simple solution. We find the solution of
such equation on a subset of its set of definition. Finally we prove
that the stationary distribution of the system decays
super-exponentially fast in the quarter plane. The queueing model,
motivated by air and railway traffic, has been proposed by Kendall and
others some five decades ago, but no solution of it has been found so
far. This is a joint work with Gianluca Guadagni, Carlo Lancia and
Benedetto Scoppola.