16th Mexico-Japan Probability Seminar

Date: 18:00--19:00+QA, 15th(TUE) October (Mexico, CST) / 9:00--10:00+QA, 16th(WED) October (Japan, JST), 2024 on zoom

Speaker: Ryuya Namba (Kyoto Sangyo University)

Title: Diffusion processes in population genetics and iterates of some positive linear operators

Abstract: The Bernstein operator is a positive linear operator acting on the Banach space of continuous functions on the unit interval, which is used to show the celebrated Weierstrass approximation theorem. In this talk, we introduce an extension of the Bernstein operator to the multidimensional cases and discuss some limit theorems for the iterates of the operator. As the limit, we capture the multidimensional Wright-Fisher diffusion with mutation which is well-studied in population genetics. Some related limit theorems including infinite-dimensional cases are discussed as well. Based on a joint work with Takatoshi Hirano.

16th Mexico-Japan Probability Seminar

Date: 18:00--19:00+QA, 15th(TUE) October (Mexico, CST) / 9:00--10:00+QA, 16th(WED) October (Japan, JST), 2024 on zoom

Speaker: Ryuya Namba (Kyoto Sangyo University)

Title: Diffusion processes in population genetics and iterates of some positive linear operators

Abstract: The Bernstein operator is a positive linear operator acting on the Banach space of continuous functions on the unit interval, which is used to show the celebrated Weierstrass approximation theorem. In this talk, we introduce an extension of the Bernstein operator to the multidimensional cases and discuss some limit theorems for the iterates of the operator. As the limit, we capture the multidimensional Wright-Fisher diffusion with mutation which is well-studied in population genetics. Some related limit theorems including infinite-dimensional cases are discussed as well. Based on a joint work with Takatoshi Hirano.

16th Mexico-Japan Probability Seminar

Date: 18:00--19:00+QA, 15th(TUE) October (Mexico, CST) / 9:00--10:00+QA, 16th(WED) October (Japan, JST), 2024 on zoom

Speaker: Ryuya Namba (Kyoto Sangyo University)

Title: Diffusion processes in population genetics and iterates of some positive linear operators

Abstract: The Bernstein operator is a positive linear operator acting on the Banach space of continuous functions on the unit interval, which is used to show the celebrated Weierstrass approximation theorem. In this talk, we introduce an extension of the Bernstein operator to the multidimensional cases and discuss some limit theorems for the iterates of the operator. As the limit, we capture the multidimensional Wright-Fisher diffusion with mutation which is well-studied in population genetics. Some related limit theorems including infinite-dimensional cases are discussed as well. Based on a joint work with Takatoshi Hirano.

Date: 18:00--19:00+QA, 17th(TUE) September (Mexico, CST) / 9:00--10:00+QA, 18th(WED) September (Japan, JST), 2024 on zoom

Speaker: María Clara Fittipaldi (Facultad de Ciencias, Universidad Nacional Autónoma de México)
Joint work with Adrián González Casanova (Instituto de Matemáticas, Universidad Nacional Autónoma de México, México; Department of Statistics, University of California Berkeley) and Julio Ernesto Nava Trejo (Humboldt-Universität zu Berlin)

Title: Lookdown construction for a Moran seed-bank model

Abstract: In this talk, a lookdown construction for a Moran seed-bank model with variable active and inactive population sizes will be introduced. Furthermore, the lookdown construction will be used to show that the time to the most recent common ancestor, starting from N individuals with stationary distribution over its state (active or inactive), has the same asymptotic order as the longest inactivity period. Additionally, an explicit approximation of its distribution under extra assumption on the inactivity and activity switching rates will be provided.

Speaker: March Boedihardjo

Title: Max-sliced Wasserstein distances

Abstract: I will give essentially matching upper and lower bounds for the expected max-sliced 1-Wasserstein distance between a probability measure on a separable Hilbert space and its empirical distribution from n samples. A version of this result for Banach spaces will also be presented. From this, we will derive an upper bound for the expected max-sliced 2-Wasserstein distance between a symmetric probability measure on a Euclidean space and its symmetrized empirical distribution.

Speaker: Joshué Ricalde Guerrero

Title: Pontryagin’s Maximum Principle for Optimization in Random Environments

Abstract: One of the primary methods for solving optimization problems involves determining a set of necessary conditions for any optimal solution. These conditions can become sufficient under additional convexity assumptions on the objective/constraint functions. Pontryagin’s Maximum Principle (PMP) expands this idea to optimal control problems, i.e., optimization problems in infinite dimensional spaces. Namely, PMP states that for an optimally controlled dynamical system, there exists an adjoint set of equations; together, they both solve a two-point boundary value problem plus a maximum condition for a function called the Hamiltonian. In this talk, we present a stochastic PMP for dynamical systems affected by a measure-valued random environment, such that the state process is assumed to be driven by a diffusion with self-exciting jumps. Then, we apply our results to the problem of Mean-Variance Portfolio Selection with Regime Switching. This is a joint work with Daniel Hernández Hernández (CIMAT).

Speaker: David Croydon (RIMS, Kyoto University)

Title: Random walk on a critical percolation cluster on a random hyperbolic half-planar triangulation

Abstract: Capturing the behaviour expected for critical percolation clusters on high-dimensional integer lattices, we show that the Gromov-Hausdorff-Prohorov scaling limit of a critical percolation cluster on a random hyperbolic triangulation of the half-plane is the Brownian continuum random tree. As a corollary, we obtain that the simple random walk on the critical cluster rescales to Brownian motion on the continuum random tree. This is a joint work with Eleanor Archer (Université Paris Nanterre).

Speaker: Mario Diaz (UNAM)

Title: Stein Method in Free Probability Theory

In classical probability theory, the Stein Method is a powerful technique that could be used to derive quantitative versions of the central limit theorem. In this talk, we revisit the classical Stein method and present a natural analogue in the context of free probability theory. This talk is based on work in progress with Arturo Jaramillo (CIMAT).

Speaker: Juan Carlos Pardo Millan

Title: Limit theorems for occupation times of  symmetric Markov processes

Abstract: Limit theorems of occupation times for Markov process has attracted a lot of interest since the seminal work of Darling and Kac in the late fifties of the last century. In, the authors  generalised and unified previous results by showing that, under suitable conditions, such limit distribution must be Mittag-Leffler. The approach used by Darling and Kac  relies on analytic tools which allow them to apply the celebrated Tauberian theorem.  In this talk, we provide a probabilistic approach to such limit theorems but for  symmetric L\'evy process. Our approach provides further information of the appearance of such limiting distributions.   Our methodology relies on excursion theory and on a  generalisation of the second Ray-Knight theorem of such families of processes due to Eisenbaum et al. We believe that our methodology can be applied for strongly symmetric Markov processes. This is an ongoing joint work with Joseph Najnudel (University of Bristol) and Ju-Yi Yen (University of Cincinnati)

Speaker: Airam Blancas(ITAM) and Sandra Palau (IIMAS-UNAM)

Title: The fixation times for the Lambda-Wright-Fisher

Abstract: The Wright-Fisher and Moran model, have been used to describe the dynamics of a population with two types. The first model is basically, a discrete time Markov chain, whereas the second one is a continuous time Markov chain. It is know, that an appropriate scaling make them converge in distribution, to the Wright-Fisher diffusion. 

Based on the lookdown model developed by Donnelly and Kurtz. In this talk, we present a construction of a d-types Lambda-Wright-Fisher diffusion. This process describe the evolution of a very large population, where the individuals reproduce at random and carry one of d types. Moreover, we address the question of absorption time and extinction of some types in the population.

Speaker: Adrian Gonzalez

Title: Sampling Duality

Abstract: Heuristically, two stochastic processes are dual if one can study one using the other. More formally, let (X_t) and (Y_t) be two real-valued processes and let H be a measurable, R^2 \mapsto R function. We say that (X_t) and (Y_t) are H-dual if E[H(X_t,y)|X_0=x]=E[H(x,Y_t)|Y_0=y].
Sampling Duality is stochastic duality using a duality function S(n,x) of the form ¨what is the probability that all the members of a sample of size n are of a particular type, given that the number (or frequency) of that type of individuals is x¨. Implicitly, this technique can be traced back to the work of Pascal. Explicitly, it was studied in a paper by Martin Möhle in 1999 in the context of population genetics. We will discuss several examples in which this technique is useful, including Haldane's formula for the fixation probability of a beneficial mutation and the long-standing open question in the theoretical evolution of the rate of the Muller Ratchet.

Speaker: Yuki Ueda (Hokkaido University of Education Asahikawa Campus)

Title: New combinatorial formulas and limit theorems for finite free convolution

Abstract: Finite free probability theory was introduced by Marcus, Spielman and Srivastava as an approximate theory of free probability. The following results are important to understand a connection between finite free probability and free probability; (i) the empirical root distributions of Hermite polynomials converge weakly to semicircle law as degree tends to infinity; (ii) the free convolution of probability distributions is approximated by the finite free convolution. At the first of this talk, we introduce the central limit theorem for finite free multiplicative convolution. Our motivation in this talk is to investigate convergence of the finite free cumulants of the limit polynomial which appears from CLT, as its degree tends to infinity. An important key to compute the finite free cumulants is a new combinatorial identities over partitions of a finite poset. In this talk, we show the combinatorial formula and the limit of finite free cumulants. This is a joint work with Octavio Arizmendi (CIMAT) and Katsunori Fujie (Hokkaido University).

  Speaker: Luis Iván Hernández Ruíz (Kyoto University)

Title: Distribution of the maximum and decay rates for Renewal Processes with application to Hawkes Processes.Abstract: In this work we study the distribution of the maximum value for a process that evolves between the epochs of a renewal process under the assumption of the Regenerative Property and apply the results to the Intensity Process and the Backward Recurrence Time of a Renewal Point Process.
Additionally, we use a coupling argument to find power decay rates for the Key Renewal Theorem in the spread out case and use the result to prove a law of large numbers for a Renewal Hawkes process under more relaxed assumptions than our previous work.

 Speaker 1: Takahiro Hasebe(Hokkaido University)
Title: Multi-faced independences arising from ``universal lifts'' of operators
Abstract: We attack the classification problem of multi-faced independences i.e. independences for non-commutative random vectors.  Although we do not achieve a complete classification, we formalize a key concept of a ``universal lift'' which underlies many examples; such a lift allows to extend (or lift) an operator on a pre-Hilbert space to a larger pre-Hilbert space obtained by taking a monoidal product with another pre-Hilbert space. It will be shown how universal lifts can be used to construct very well-behaved (multi-faced) independences. Furthermore, these lifts will be entirely classified in the cases where the monoidal product is taken to be the tensor product and where it is taken to be the free product. Our work also brings to light surprising new examples. Most noteworthy, for many known 2-faced independences, we find that they admit continuous deformations within the class of 2-faced independences, showing in particular that, in contrast with the single-faced case, this class is infinite (and even uncountable).

Speaker 2: Henry Pantí (Universidad Autónoma de Yucatán)
Title: Gerber-Shiu function for a class of Markov-modulated  Lévy risk processes with two-sided jumps
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

"18:30--20:00", 23rd May(Mexico, CST) /" 8:30--10:00", 24th May(Japan, JST) on zoom

Speaker 1: Hiroshi Tsukada(Kagoshima University)

Title: Pathwise uniqueness of SDEs driven by Brownian motions and finite variation Lévy processes

Abstract: We study the pathwise uniqueness of the solutions to one-dimensional stochastic differential equations (SDEs) driven by Brownian motions and Lévy processes with finite variation paths. The driving Lévy processes are not necessarily one-sided jump processes. In this talk, we give some non-Lipschitz conditions on the drift, diffusion and jump coefficients, under which the pathwise uniqueness of the solution to the SDEs is established.

Speaker 2: 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.

Speaker 1: Octavio Arizmendi(CIMAT)

Title: Cyclic independence: Boolean and Monotone.

Abstract: Collins, Hasebe and Sakuma introduced Cyclic Monotone Independence. Their motivation was to study the spectrum of polynomials of certain random matrices with discrete limiting distribution. In this talk we will describe three further developments of the theory.1) We generalize the results of Collins Hasebe and Sakuma to be able to calculate explicitly the precise spectral distribution for many polynomials not considered by them. 2) In joint work with Hasebe and Lehner we use this notion of independence to describe the spectral distribution of the comb product of graphs. 3) We introduce Boolean Cyclic Independence, which describes spectral distribution of the star product of graphs. This talk is based on joint works with Celestino (2019), and with Hasebe and Lehner (2022).

Speaker 2: Yuki Ueda(Hokkaido University of Education)

Title: Freely quasi-infinitely divisible distributions and extension of Bercovici-Pata bijection

Abstract:Bo\.{z}ejko raised a question whether the Bercovici-Pata bijection can be extended to an wider domain than the class of infinitely divisible distributions. We investigate the class of (classically or freely) quasi-infinitely divisible distributions to answer the question affirmatively. More strictly, this research is to find the common characteristic triplets of classically and freely quasi-infinitely divisible distributions appeared from the L\'{e}vy-Khintchine type representation. In the first of this talk, we introduce classical and free quasi-infinite divisibility and study the same or different distributional properties between classically and freely quasi-infinitely divisible distributions. Next, we give examples of freely quasi-infinitely divisible distributions that appeared from several viewpoints (for example, the subordination method for free deconvolution by Arizmendi, Tarrango and Vargas (2020) and some relatives of the Catalan sequences by Liszewska and M\l otkowski (2020), etc). Finally,  by finding the common characteristic triplets, we show that the Bercovici-Pata bijection has an extension to the subclass of quasi-infinitely divisible distributions. This is a joint-work of Ikkei Hotta, Wojciech M\l otkowski and Noriyoshi Sakuma.

Speaker1 : José Luis Pérez(CIMAT)

Title: Frequency processes associated with continuous-state branching processes, moment duality and genealogies

Abstract:When two (possibly different in distribution) continuous-state branching processes with immigration are present, we study the relative frequency of one of them when the total mass is forced to be constant at a dense set of times. This leads to a SDE whose unique strong solution will be the definition of a Lambda-asymmetric frequency process (Lambda-AFP). We prove that it is a Feller process and we calculate a large population limit when the total mass tends to infinity. This allows us to study the fluctuations of the process around its deterministic limit. Furthermore, we find conditions for the Lambda-AFP to have a moment dual. The dual can be interpreted in terms of selection, (coordinated) mutation, pairwise branching (efficiency), coalescence, and a novel component that comes from the asymmetry between the reproduction mechanisms. In the particular case of a pair of equally distributed continuous-state branching processes the associated Lambda-AFP will be the dual of a Lambda-coalescent. The map that sends each continuous-state branching process to its associated Lambda- coalescent (according to the former procedure) is a homeomorphism between metric spaces.

Speaker2: Shosei Takeda(Kyoto U.)

Title: Local time penalizations with various clocks for Lévy processes

Abstract: In this talk, we study several long-time limit theorems of one-dimensional  Lévy processes, called local time penalization, which may be seen as a generalization of  Lévy processes conditioned to avoid zero. The limit processes may differ according to the choice of the clocks (familes of random times) when the original Lévy processes is recurrent and of finite variance.

Speaker1 : Arturo Jaramillo(CIMAT)

"Limit theorems for linear statistics of matrix-valued Gaussian processes"

We study the asymptotic behavior of the second-order fluctuations of linear statistics of matrix-valued Gaussian processes by means of Malliavin calculus techniques. We discuss as well the challenges arising from the problem of proving sequential compactness for sequences of processes involving matrix-valued processes and discuss some open problems and research under development related to matrix-valued Gaussian processes.

Speaker2 : Kei Noba(ISM)

"On the optimality of the refraction--reflection strategy for Lévy processes"

In this talk, we study de Finetti's optimal dividend problem with capital injection under the assumption that the dividend strategies are absolutely continuous. In many previous studies, the process before being controlled was assumed to be a spectrally one-sided Lévy process, however in this paper we use a Lévy process that may have both positive and negative jumps. In the main theorem, we show that a refraction--reflection strategy is an optimal strategy. We also mention the existence and uniqueness of solutions of the stochastic differential equations that define refracted Lévy processes.