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Семінар "Стохастика та її застосування"

Lah distributions, Stirling numbers, and phase transitions for convex hulls of random walks

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21 квітня 2021 року о 14 год 15 хв

Let \(X_1,X_2,\ldots\) be independent random vectors in \(\mathbb{R}^d\) having an absolutely continuous distribution. Consider the random walk \(S_k:=X_1+\ldots+X_k\), and let \(P_n:={\rm conv}\{0,S_1,S_2,\ldots,S_n\}\) be the convex hull of its first \(n\) steps. We shall be interested in the number of the \(k\)-dimensional faces of the polytope \(P_n\) and in particular, whether this number is equal to the maximal possible number \(\binom {n+1}{k+1}\) with high probability, as \(n\), \(d\), and possibly also \(k\) diverge to \(\infty\). There is an explicit formula for the expected number of \(k\)-dimensional faces which involves Stirling numbers of both kinds. Motivated by this formula, we introduce a distribution, called the Lah distribution, and study its properties.

Доповідач: Zakhar Kabluchko (University of Münster, Germany)

Дата проведення: 21 квітня 2021 року о 14 год 15 хв.


 

Функціональні граничні теореми для збурених випадкових блукань

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14 квітня 2021 року о 14 год 15 хв (Увага! Змінено дату!)

Розглянуто ланцюги Маркова, стрибки яких є незалежними однаково розподіленими центрованими випадковими  величинами ззовні певної обмеженої множини А. За певних припущень щодо стрибків із А встановлено функціональну граничну теорему для  донскєровських шкалювань  відповідного блукання.

Доповідач: Андрій Пилипенко (Інститут математики НАН України)

Дата проведення: 14 квітня 2021 року о 14 год 15 хв.


 

Weak convergence of random geometric objects in non-Euclidean spaces

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8 квітня 2021 року о 13 год 15 хв

The analysis of conventional convex hulls of random points sampled from the uniform distribution on a compact convex subset of the d-dimensional Euclidean space is the classical topic in stochastic geometry. In the past years there has been a splash of activity around various generalized concepts of convexity both in geometric and probabilistic communities. The talk is devoted to the discussion of two particular models of this kind. In the first model we consider a sample picked from the uniform distribution on the upper semi-sphere and analyze its spherical convex hull. In the second model the sample is taken from the uniform distribution in a convex body K in the d-dimensional Euclidean space and we focus on the analysis of its K-convex hull, that is, intersection of all affine translates of K which contain the sample. Considered from an appropriate viewpoint, these two models incorporating generalized notion of convexity exhibit a similar behavior which turns out to be very different from such in the classical setting

Доповідач: Alexander Marynych

Дата проведення: 8 квітня 2021 року о 13 год 15 хв.


 

 

Reflected Brownian motion in a wedge and q-difference equations

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17 березня 2021 року о 14 год 15 хв

 

We consider a semimartingale reflected Brownian motion in a two-dimensional wedge. Under standard assumptions on the parameters of the model (opening of the wedge, angles of the reflections on the axes, drift), we study the algebraic and differential nature of the Laplace transform of its stationary distribution. We derive necessary and sufficient conditions for this Laplace transform to be rational, algebraic, differentially finite or more generally differentially algebraic. These conditions are explicit linear dependencies among the angles involved in the definition of the model. To prove these results, we start from a functional equation that the Laplace transform satisfies, to which we apply tools from diverse horizons. To establish differential algebraicity, a key ingredient is Tutte's invariant approach, which originates in enumerative combinatorics. To establish differential transcendence, we turn the functional equation into a difference equation and apply Galoisian results on the nature of the solutions to such equations.

This is a joint work with M. Bousquet-Mélou, A. Elvey Price, S. Franceschi and C. Hardouin (see https://arxiv.org/abs/2101.01562).

 

Доповідач: Kilian Raschel (CNRS, Institut Denis Poisson, Université de Tours, France)

Дата проведення: 17 березня 2021 року о 14 год 15 хв.


 

 

On nested occupancy scheme in random environment

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11 березня 2021 року о 13 год 15 хв

A nested occupancy scheme in random environment is a generalization of the Karlin infinite balls-in-boxes occupancy scheme in random environment (with random probabilities). Unlike the Karlin scheme in which the collection of boxes is unique, there is a nested hierarchy of boxes, and the hitting probabilities of boxes are defined in terms of iterated fragmentation of a unit mass. We say that the boxes belong to the j-th level provided that their hitting probabilities are given by the j-fold fragmentation. Assuming that the number of balls is n, we shall present functional limit theorems for the number of occupied boxes in the j-th level in two different settings: 1) j is fixed; 2) j=j(n) diverges to infinity and j(n)=o((\log n)^{1/2}) as n tends to infinity.

Доповідач: Alexander Iksanov

Дата проведення: 11 березня 2021 року о 13 год 15 хв.


 

 


Сторінка 1 з 18

 

Joan Miró (Жоан Миро), Personnage devant le soleil (Характер перед сонцем), 1968

Joan Miró (Жуан Міро)

Personnage devant le soleil (Характер перед сонцем), 1968

Помічено Д. Загребельною в одному з музеїв Барселони