Arno Siri (UNAM)

Title: Random genealogies: models, matricial methods and inference

Abstract: In this series of talks I will give a short introduction on the theory of exchangeable coalescents and their connection with population evolution models.: this class of processes appears naturally as the limit genealogies, when the size of the population increases, of a wide class of evolution models.

In a second part, I will show how coalescent functionals can be observed from genomic material, and introduce the classical statistics used for genealogical model selection. The law of these statistics is in general hard to obtain explicitly, hence I will describe an easy computational method based on phase-type theory to study them. Finally, I will show a non-parametric method for the estimation of the coalescent measure.

Takahiro Hasebe(Hokkaido University)

Title: Proving free infinite divisibility of probability measures with explicit densities

Abstract: In general, it is not easy to see the free infinite　divisibility of very explicit probability measures. Almost all methods　are based on constructing some analytic continuation of the Cauchy　transform, originally defined on the upper half-plane, to some larger　domain which is mapped bijectively onto the lower half-plane. Typical methods for constructing and analyzing such analytic continuation are:

(1) to use a differential equation or a functional equation satisfied by the Cauchy transform, if exists.

(2) to use Cauchy's integral formula to obtain a useful formula for the Cauchy transform.

Method (1) was applied for proving e.g. the normal distribution, some beta distributions and some gamma distributions are freely ID.

In this talk I will explain method (2). The talk will be based on the paper:

T. Hasebe, Free infinite divisibility for powers of random variables, ALEA Lat. Am. J. Probab. Math. Stat. 13 (2016), no. 1, 309-336. arXiv:1509.08614.

Daniel Hernández (CIMAT)

Title: Path-dependent zero-sum deterministic games with intermediate Hamiltonians

Abstract: In this talk we consider path-dependent Isaacs partial differential equations (PDEs) of first order with intermediate Hamiltonians given by convex combinations of lower and upper Hamiltonians. We propose discrete-time approximations which converge to a unique viscosity solutions of the intermediate Isaacs PDEs. Furthermore, we give discrete-time stochastic dynamic game representations for the approximations.

This is a joint work with Hidehiro Kaise (Kumamoto University).

Ryoichi Suzuki(Ritsumeikan University)

Title: A modified $\Phi$-Sobolev inequality for canonical $L^2$-L\'{e}vy processes and related inequalities

Abstract: In this talk, we consider a new modified $\Phi$-Sobolev inequality for canonical $L^{2}$-L{\'e}vy processes, which are mixed cases of Brownian motion processes and pure jump-L{\'e}vy processes. Existing results included only a part of Brownian motion processes and pure jump processes. Therefore, in this talk, we develop a generalized version of the $\Phi$-Sobolev inequality for Poisson and Wiener spaces. Furthermore, the theorem can be applied to obtain concentration inequalities for canonical L{\'e}vy processes. Unlike the case of measure concentration inequalities for Brownian motion alone or for pure jump-L{\'e}vy processes alone, the measure concentration inequalities for canonical L{\'e}vy processes involve Lambert's $W$-function. Examples of inequalities such as the supremum of L{\'e}vy processes in the case of mixed Brownian motion and Poisson processes will also be presented. This talk is based on joint work with Noriyoshi Sakuma.

José Luis Pérez (CIMAT)

Title.- The $\Lambda$-asymmetric ancestral graph

Abstract.- In this talk, we intend to explain the meaning of having a selective disadvantage in the general context of populations with skewed reproduction mechanisms. For instance, if the reproduction events of type $1$ individuals typically are less common or have a smaller size compared to the reproduction events involving type $2$ individuals. To this end, we consider pair of populations such that the ancestry of population $i\in\{1,2\}$ lies in the universality class of a $\Lambda_i$ coalescent. Our approach consists in constructing a Moran model in which individuals of the two different populations compete, which allows us to define a frequency process of one of the types in the population. The frequency process converges to the solution an SDE, which typically is a process with discontinuous paths. We show that we can introduce a partial order in $\mathcal{M}[0,1]$ such that if two reproduction mechanisms satisfy $\Lambda_1<\Lambda_2$ then the population of type 2 individuals will have a selective advantage. This result is the consequence of a pathwise duality result that extends the well-known duality for $\Lambda$-coalescents with classic selection, which relies on the construction of the $\Lambda$-ancestral selection graph, to $\Lambda$-coalescents associated with skewed reproduction mechanisms. Finally, we derive a Griffiths representation for the generator of the SDE and use it to find a semi-explicit formula for the probability of fixation of type individuals depending on general reproduction mechanisms $\Lambda_1$, $\Lambda_2$ and the starting frequency of each of the types.

Henry Pantí (UADY)

Title.- Gerber-Shiu function for a class of Markov-modulated Lévy risk processes with two-sided jumps

Abstract.- In this work we investigate the Gerber-Shiu discounted penalty function for Markov-modulated Lévy risk processes with random incomes. Firstly, we consider the case when the downward and upward jumps (respectively, claims and random gains) are given by independent compound Poisson processes, with claim sizes with a general distribution function and gains in such a way that their distribution has a rational Laplace transform. Afterwards, we use the above results and weak convergence techniques to study the case when the claims are given by a subordinator and, subsequently, we establish results when the claims are governed by a pure jump spectrally positive Lévy process.

Joint work with Ehyter Martín-González and Antonio Murillo-Salas

Sergio López (UNAM)

Title: Two-point function of the KPZ equation

Abstract: We derive the two-point function of the slope of the one-dimensional KPZ equation starting from an arbitrary two-sided Brownian motion initial condition. Malliavin calculus basic tools allow us to compute this function in terms of the polymer end-point annealed distribution associated with the stochastic heat equation. We prove that this distribution can be computed using the derivative of the variance of the solution of the KPZ equation. We also show upper bounds for asymptotic independence. (Joint work with Leandro Pimentel, Universidade Federal do Rio de Janeiro).

Luis Iván Hernández (Kyoto University)

Title.-Law of Large Numbers and Functional Central Limit Theorem for the Renewal Hawkes Process.

Abstract.-The Renewal Hawkes Process (RHP) was proposed by Wheatley (2016) as an extension to the self-exciting Hawkes Process. This generalization allows the immigration process to be a Renewal Process instead of an homogeneous Poisson.

In this work we established an existence result for the RHP exploiting its cluster structure. We then proved a Law of Large Numbers and Functional Central Limit Theorem using a martingale theory approach; similar to what was done by Bacry (2013) in the case of Multivariate Hawkes Processes.

Juan Carlos Pardo Millán (CIMAT)

Title: Growth-fragmentation embedded in Brownian excursions from hyperplanes

Abstract: In this talk, we present a self-similar growth-fragmentation process linked to a Brownian excursion from hyperplanes, obtained by cutting the excursion at heights along horizontal hyperplanes. More precisely by slicing these excursions, we obtain a collection of excursions which exhibit a branching structure. We define the size of such an excursion as the difference between the endpoint and the starting point. We show that considering the collection of these sizes at varying heights constructs a special growth-fragmentation in $\mathbb{R}^{d-1}$. This is a joint work with William Da Silva (University of Vienna).

Kosuke Yamato (Kyoto Univerisity)

Title: Existence of quasi-stationary distributions for spectrally positive Lévy processes on the half-line

Abstract: For spectrally positive Lévy processes killed on exiting the half-line, we characterize the existence of a quasi-stationary distribution by the exponential integrability of the exit time, the Laplace exponent and the non-negativity of scale functions. We also show if there is a quasi-stationary distribution, there are necessarily infinitely many ones and identify the set of quasi-stationary distributions.

Tulio Gaxiola (UAS)

Title:Spectral Analysis of Graph and Growing Graphs

Abstract: In this talk we introduce some results about the relation of spectral distributions of some products of graphs with notions of non-commutative independence. We present the Quantum Decomposition Method and we show how this method works for the Spectral Analysis of Graphs and Growing Graphs. Finally, we will make a brief compilation of some interesting results in the area.

Kouji Yano (Kyoto University) [online]

Title: Local time penalizations with various clocks for Lévy processes

Abstract: For one-dimensional Levy processes, we discuss local time penalizations, a generalization of conditioning to avoid zero. The limits are taken via certain families of random times, called clocks. The limit processes may differ according to the choice of the clocks when the original Levy process is recurrent and of finite variance.

This talk is based on a joint work with Shosei Takeda.

arXiv:2203.08428.

Katsunori Fujie (Hokkaido University)

Title: Law of large numbers for finite free multiplicative convolution

Abstract: Law of large numbers (LLN) is a well-known theorem in probability theory.

It describes that the sample average of independent identically distributed random variables with finite mean concentrates on the theoretical mean when the sample size is sufficiently large. Finite free probability is a discrete approximation theory for free probability.

Some of the typical limit theorems have already known in finite free probability, for example, LLN for finite free additive convolution, the central limit theorem, the law of small numbers, etc. In this talk, we provide the Law of large numbers for finite free multiplicative convolution.

This talk is based on the joint work with Yuki Ueda (Hokkaido University of Education).

Mauricio Salazar (UASLP)

Title: Speed of convergence for measures of zero third moment in some non-commutative central limit theorems.

Abstract: The classical Berry-Esseen theorem gives a general bound for the speed of convergence in the central limit theorem for measures of finite absolute third moment. One may expect a better bound on the rate of convergence for measures of zero third moment, however it is not true. In this talk, we will see that in the free and Boolean central limit theorems we have indeed an improvement on the rate of convergence for such class of measures.

Victor Rivero (CIMAT)

Title: Towards a Ray-Knight theorem for spectrally negative Lévy processes

Abstract: It has been established by Kaspi and Eisenbaum that the local time process $(L^{x}_{\zeta}, x\in \mathbb{R})$, as a process in space, associated to an $\mathbb{R}$-valued Markov process $(X_t, 0\leq t\leq \zeta)$ bears itself the Markov property, if and only if, the process $X$ has continuous paths. In that case, the local time process can be described using branching processes. Besides, one can think of spectrally negative Lévy processes (SNLP) as those Lévy processes that bear properties closer to diffusions. Also, an analysis of the paths of SNLP allow to intuit that the local time process, as a process in space, associated to a SNLP killed at suitable stopping times, also bears a branching property. Even though the Markov property is not preserved. We will make this precise and describe the law of the local time process. This is based in a work in process with José Contreras, PhD student in CIMAT.

Noriyoshi Sakuma (Nagoya City University)

Title: Selfsimilar free additive processes and freely selfdecomposable distributions

Abstract:In the paper by Fan(2006), he introduced the marginal selfsimilarity of non-commutative stochastic processes and proved the marginal distributions of selfsimilar processes with freely independent increments are freely selfdecomposable. In this talk, we, first, introduce a new definition of selfsimilarity via linear combinations of non-commutative stochastic processes and prove the converse of Fan's result, to complete the relationship between selfsimilar free additive processes and freely selfdecomposable distributions. Furthermore, we construct stochastic integrals with respect to free additive processes for constructing the background driving free Lévy processes of freely selfdecomposable distributions. A relation in terms of their free cumulant transforms is also given and several examples are also discussed.

Josué Vasquez Becerra (UAM)

Title: The lack of infinitesimal freeness in Wishart real matrices

Abstract.- Infinitesimal freeness is an extension of free independence. For random matrices, infinitesimal laws can help detect the outliers in limiting eigenvalue distributions. In this talk, we will see a combinatorial proof for the joint infinitesimal law of real Wishart matrices. Our formula implies the lack of infinitesimal freeness of independent real Wishart matrices, a result which is opposite to the complex case. This talk is based on joint work with James A. Mingo.

Mario Diaz (UNAM)

Title: On the analytic structure of second-order non-commutative probability spaces

Abstract: In this talk we present a general approach to the central limit theorem for the continuously differentiable linear statistics of random matrix ensembles. This approach, which is based on a weak large deviation principle for the operator norm, a Poincaré-type inequality for the linear statistics, and the existence of a second-order limit distribution, allows us to recover known central limit theorems and establish new ones. Furthermore, we define an analytic version of a second-order non-commutative probability space that allows us to recover in an abstract form some of the results obtained for the random matrix ensembles considered by our approach. Joint work with James Mingo (Queen's University).

Andrés García (CIMAT)

Title.- Rotationally invariant estimators: an application on the dynamics of financial risk and some simulations of new proposals.

Abstract.- An introduction to the Markowitz portfolio theory and the effect on the estimation of the optimal frontier when working in high-dimensional scenarios is described. The family of rotationally invariant estimators (RIE) and the recent proposals from the theory of random matrices and free probability are introduced to estimate the covariance matrix when the number of variables tends to infinity. An application of these estimators is shown to determine the dynamics of the financial risk of the American and European markets. The associated risks of the minimum variance portfolio under RIE are found to increase the active information storage (AIS), suggesting that in times of financial turbulence these techniques can be of great advantage to minimize the exposure to risk[1]. Finally, work in progress in relation to simulations about new proposals for estimators of the covariance matrix is discussed[2]. The approach of the work was carried out from the paradigm of econophysics, therefore the methodology has been phenomenological and close to that followed in data science.

[1] Garcia Medina, A., & Macías Páez, R. Rotationally Invariant Estimators on Portfolio Optimization to Unveil Financial States. Available at SSRN 4126928.

[2] Joint work with Rosario Mantegna and Salvatore Micciche’ (University of Palermo).

Kazutoshi Yamazaki (U. Queensland) (online)

Title: On the CUSUM procedure for phase-type distributions: a Lévy fluctuation theory approach (joint with J. Ivanovs)

Abstract: We introduce a new method analyzing the cumulative sum (CUSUM) procedure in sequential change-point detection. When observations are phase-type distributed and the post-change distribution is given by exponential tilting of its pre-change distribution, the first passage analysis of the CUSUM statistic is reduced to that of a certain Markov additive process. By using the theory of the so-called scale matrix and further developing it, we derive exact expressions of the average run length, average detection delay, and false alarm probability under the CUSUM procedure. The proposed method is robust and applicable in a general setting with non-i.i.d. observations. Numerical results also are given.

Benoit Collins (Kyoto University) (online)

Title: Matrix models for cyclic monotone and monotone independences

Abstract: Cyclic monotone independence is an algebraic notion of noncommutative independence, introduced in the study of multi-matrix random matrix models with small rank. Its algebraic form turns out to be surprisingly close to monotone independence, which is why it was named cyclic monotone independence. We will describe a random matrix model and explain why it is the same for cyclic monotone independence and monotone independence with an appropriately chosen state, and therefore unify the two notions. This is based on joint work with Felix Leid and Noriyoshi Sakuma, arXiv 2202.11666

Arturo Jaramillo (CIMAT)

Title: Limit theorems for additive functionals of the fractional Brownian motion.

Abstract: The fractional Brownian motion is a self-similar centered Gaussian process with stationary increments. It has a strong capacity for adapting to a variety of applications due to its flexibility in exhibiting general dependence structures. In this talk, we study the asymptotic behavior of a family of continuous ergodic averages called "additive functionals", which will turn out to be closely related to local times. The presentation will converge towards a discussion of the connection of local times and the beautiful topic of functional limit theorems with mixed Gaussian limits, which will be reminiscent of classical applications of Knight's theorem but in a context of processes that lack both the Markov and martingale properties. We will address the problem with a perspective based on Malliavin calculus and an embedding of the fractional Brownian motion into a Wiener space.

Kei Noba (ISM)

Title: Optimality of classical or periodic barrier strategies for Lévy processes

Abstract: We revisit the stochastic control problem in two cases with Lévy processes that minimize running and controlling costs. Existing studies have shown the optimality of classical or periodic barrier strategies when driven by Brownian motion or Lévy processes with one-sided jumps. Under the assumption that we can be controlled at any time or only at Poissonian dividend-decision times, we show the optimality of classical or periodic barrier strategies for a general class of Lévy processes.

Ehyter Martin Gonzalez (UdeG)

Title.- Some numerical results on the distribution of first passage times for Lévy processes

Abstract.- We will present some numerical results related to the distribution of first passage time for Lévy process; in particular, the cases when the first passage is below a given level and then the case of this time conditioned to be prior to the first passage above another given level.

This is only a numerical study, whose empirical results need to be rigorously proved and it is based on a work by Doney, Kluppelberg and Maller, where the authors prove results concerning weak convergence of a rescaled version of these first passage times. The numerical results we will discuss do not seem to require a rescaling and, as we shall see, they rely on a convergence in the sense of distance of distribution functions, as in a famous theorem in Extreme Value Theory proved by Pickands, Balkema and de Haan.