6th Cincinnati Symposium on Probability
University of Cincinnati
May 1–4, 2024

Titles and abstracts of invited talks• An invariance principle for de-randomized conductance models
Marek Biskup, University of California, Los Angeles• On some possible combinations of mixing rates for strictly stationary, reversible Markov chains
Rick Bradley, Indiana University, Bloomington• On the optimality of McLeish’s conditions for the central limit theoremJérôme Dedecker, Université Paris Descartes, France• Scaling limits for growth driven by reflecting Brownian motion
Amir Dembo, Stanford University• Some recent advances on limit theorems for stationary random fields
Davide Giraudo, Université de Strasbourg, France• Elephant random walks
Allan Gut, Uppsala University, Sweden• Darboux transformation of diffusion processes
Alexey Kuznetsov, York University, Canada• Global central limit theorems for stationary Markov chains
Michael Lin, Ben-Gurion University, Israel• On Markov chains based on some new copula families
Martial Longla, University of Mississippi• Around Rosenthal’s type inequalities for dependent structures
Florence Merlevède, Université Gustave Eiffel, France• Secular coefficients and the holomorphic multiplicative chaos
Joseph Najnudel, University of Bristol, UK• Revisiting random matrices
Tamer Oraby, University of Texas, Rio Grande Valley• Some recent developments on linear processes and linear random fields
Hailin Sang, University of Mississippi• Approximating the stationary distribution of the open ASEP
Dominik Schmid, University de Bonn, Germany• Busemann processes in positive and zero temperature, on the lattice, and in the continuum
Timo Seppäläinen, University of Wisconsin, Madison• On the motion of a tagged particle in simple exclusion
Sunder Sethuraman, University of Arizona• Wandering around the self-normalized limit theory
Qi-Man Shao, Southern University of Science and Technology, China• Central limit theorems in deterministic dynamical systems
Dalibor Volný, Université de Rouen, France• Several short stories on quadratic harnesses
Jacek Wesolowski, Warsaw University of Technology, Poland• Fast algorithms for estimating covariance matrices of stochastic gradient descent solutions
Wei Biao Wu, University of Chicago• Convergence of the KPZ equation
Xuan Wu, University of Illinois, Urbana-Champaign• Askey–Wilson signed measures and their applications in open ASEP
Zongrui Yang, Columbia University
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An invariance principle for de-randomized conductance models
Marek Biskup
University of California, Los Angeles
Abstract. The theory of random walks in disordered reversible environments, a.k.a. conductance models,
has advanced remarkably under the assumption that the environment is drawn at random from a law thatis stationary and ergodic under translates. Indeed, the concept of the “point of view of the particle” thenenables ergodic theorems that conveniently produce limits that would be difficult to establish otherwise andthe corrector method from stochastic homogenization yields a proof of diffusive scaling to Brownian motion.
Unfortunately, the fact that both the input and the statement are stochastic make it all but impossible todecide whether the stated conclusion applies to a given (non-periodic) conductance configuration. I will show
how to overcome this by de-randomizing both the mixing theory for the “point of view of the particle” andthe corrector method. An invariance principle will then hold for *every* conductance configuration (modulo
certain growth/decay restrictions) whose block averages converge and define an ergodic law on the space ofenvironments.

On some possible combinations of mixing rates for strictly stationary, reversible Markov
chainsRick Bradley
Indiana University, Bloomington
Abstract. The papers of G.O. Roberts and J.S. Rosenthal [Electronic Comm. Probab. 2 (1997) 13-25] andG.O. Roberts and R.L. Tweedie [J. Appl. Probab. 38A (2001) 37-41] together developed for Markov chainssome theory that included (in greater generality) the following “key result”: For strictly stationary Markov
chains that are reversible and satisfy a certain “irreducibility” condition, “geometric ergodicity” is equivalent
to a certain “spectral gap” condition. It is well known that that “key result” can be formulated in termsof the dependence coefficients associated with the absolute regularity (β-mixing) and ρ-mixing conditions.
In an exposition of that “key result” that was provided by the author [J. Time Series Anal. 42 (2021)
499-533], heavy use was made of a certain rather tight connection, for strictly stationary, reversible Markov
chains, between the dependence coefficients associated with the strong mixing (α-mixing) and ρ-mixing
conditions. This talk will describe a class of examples constructed in a “follow-up” paper by the author[Rocky Mountain J. Math. (accepted for publication)], which shows that for strictly stationary Markov
chains that are reversible, the simultaneous mixing rates for the strong mixing and ρ-mixing conditions can(at least within a certain “log convexity” condition) be fairly arbitrary, subject to certain unavoidable tightrestrictions. Those examples are countable-state and have the added property that the mixing rate for theabsolute regularity condition is within a constant factor of that for strong mixing.

On the optimality of McLeish’s conditions for the central limit theoremJérôme Dedecker
Université Paris Descartes, France
Abstract. We begin with a historical review of the sufficient conditions for the CLT expressed in terms ofconditional expectation, starting with the seminal paper by Gordin (1969). We then show that McLeish’s
conditions (1975) for the CLT (which can be deduced from Hannan’s criterion (1973)) are optimal in a very
precise sense. We will also compare our counterexample with other counterexamples in the literature.

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Scaling limits for growth driven by reflecting Brownian motion
Amir Dembo
Stanford University
Abstract. In joint works with Kevin Yang, we consider a stochastic Laplacian growth model, that canbe viewed as a continuum version of origin-excited random walks. Here, we grow the (d + 1)-dimensional
manifold M (t) according to a reflecting Brownian motion (RBM) on M (t), stopped at level sets of its
boundary local time. An averaging principle for the RBM characterizes the scaling limit for the leading
order behavior of the interface (namely, the boundary of M (t)). This limit is given by a locally well-posed,
geometric flow-type PDE, whose blow-up times correspond to changes in the diffeomorphism class of thegrowing set. Smoothing the interface as we inflate M (t), yields an SPDE for the large-scale fluctuations ofan associated height function. This SPDE is a regularized KPZ-type equation, modulated by a Dirichletto-Neumann operator. For d = 1 we can further remove the regularization, so the fluctuations of M (t) now
have a double-scaling limit given by a singular KPZ-type equation.

Some recent advances on limit theorems for stationary random fields
Davide Giraudo
Université de Strasbourg, France
Abstract. In this talk, we will present some recent results obtained by us and various authors on limit
theorem for stationary random fields. We will consider partials sums on rectangle of random fields indexed
by Zd the functional central limit theorem and its quenched version, as well as the law of large numbers
and the law of the iterated logarithms. The main tools for these limit theorems are approximation by amulti-indexed martingale or approximation by m-dependent random fields and will be introduced.

Elephant random walks
Allan Gut
Uppsala University, Sweden
Abstract. In the classical simple symmetric random walk the steps are equal to ±1 and independent.
In contrast, in the so called elephant random walk (ERW), introduced by Schütz and Trimper [3] in
2004, ”the next step” depends on the whole process so far. This has suggested its name via the fact thatelephants have a very long memory.
Our point of departure was Bercu [1], who establishes asymptotics for ERWs using martingale methods.
It turns out that the asymptotics depends crucially on the success probability p. More precisely, there isa diffusive regime (0 ≤ p < 3/4), a superdiffusive regime (3/4 < p ≤ 1), and a critical one (p = 3/4).
Technically, Bercu defines the ERW as a random walk in which the first step, X1 , equals 1 with probabilityPhs ∈ [0, 1] and −1 with probability 1 − s. After n steps, at position Sn = k=1 Xk , the next step is definedas(+XK , with probability p ∈ [0, 1],
Xn+1 =−XK , with probability 1 − p,where K has a uniform distribution on the integers 1, 2, . . . , n.
The aim of our work has been to investigate variations and extensions of ERWs. Main focus has beenon cases when the elephant has a restricted memory, such as models in which the elephant remembers only
some distant past, only a recent past or a mixture of both. In those settings there is, contrary to the generalcase, no phase transition. Another idea was to allow for stops, or pauses, at every step with some probabilityr = 1 − p − q ∈ (0, 1). In some instances it turns out that the behavior is dramatically different comparedto the general case. For more, check our paper [2] and references given there.
Joint work with Ulrich Stadtmüller.

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References[1] Bercu, B., A martingale approach for the elephant random walk. J. Phys. A: Math. Theor. 51 (2018), 015201.
[2] Gut, A., and Stadtmüller, U., Elephant random walks; a review. Ann. Univ. Sci. Budapest Sect. Comp. 54 (2023),
171-198.
[3] Schütz, G.M., and Trimper, S., Elephants can always remember: Exact long-range memory effects in a non-Markovian
random walk. Phys. Rev. E 70 (2004), 045101.

Darboux transformation of diffusion processes
Alexey KuznetsovYork University, Canada
Abstract. Darboux transformation is a well-known technique in the study of second order linear differential
operators. Darboux transform of a second order linear differential operator gives another such operator,
which is ‘almost’ isospectral to the first one – the spectrum may be different at most at one point. In otherwords, this transform allows us to insert or remove eigenvalues in the spectrum of an operator. We willshow how to use Darboux transformation to construct new families of diffusion processes whose transition
probability densities (and their spectral expansions) are given in closed form. We will also discuss connections
to exceptional orthogonal polynomials. This talk is based on joint work with Minjian Yuan.

Global central limit theorems for stationary Markov chains
Michael Lin
Ben-Gurion University, Israel
Abstract. Let P = P (x, A) be a Markov transition probability on a general state space (S, Σ), withinvariant probability m. Let Ω := S N be the space of trajectories with σ-algebra A := Σ⊗N , and let Pm be
the probability on A of the chain with transition probability P and initial distribution m. By invarianceof m, Pm is shift-invariant on (Ω, A). Let Xn be the projection of Ω on the nth coordinate. Then (Xn )on (Ω, A, Pm ) is a stationary Markov chain with state space S. We assume m ergodic for P , so the chain is
ergodic too, i.e. the shift on (Ω, Pm ) is ergodic.
We say that a real centered f ∈ L2 (m) satisfies the annealed CLT if in (Ω, Pm ) we have
n

1 X
D
√
f (Xk ) → N (0, σ 2 ),
n

where

N (0, 0) := δ0 .

k=1

We say that a real centered 0 6= f ∈ L2 (m) satisfies the L2 -normalized CLT ifn

1 X
Df (Xk ) → N (0, 1),
σn (f )
k=1

where σn (f ) := kPn

k=1 f (Xk )k2 > 0 for large n.

We study conditions which yield that for every centered 0 6= f ∈ L2 (m) a non-degenerate (σ 2 > 0) annealed
CLT and an L2 -normalized CLT hold. Joint work with Christophe Cuny.

On Markov chains based on some new copula families
Martial Longla
University of Mississippi

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Abstract. This talk considers Markov chains generated by a new class of bivariate copulas. These copulas
depend on some orthogonal functions and their form allows to obtain closed forms of joint distributions andlimit theorems for sample averages of functions of the Markov chain. The limiting variance can be computedin closed form, which allows better confidence intervals. Maximum likelihood theory is also developed forsome examples. Several examples are constructed to explain the methods and a simulation study is provided.

Around Rosenthal’s type inequalities for dependent structures
Florence MerlevèdeUniversité Gustave Eiffel, France
Abstract. Rosenthal inequalities relate moments of order higher than 2 of partial sums of random variables
to the variance of partial sums. Besides of being useful to compare the norms Lp and L2 of partial sums,
these inequalities are important tools for obtaining a variety of results, including tightness of the empirical
process, convergence rates with respect to the strong law of large numbers, almost sure invariance principles
or Lp -integrated risk for kernel estimators of the density.
Since the 70’s, there has been a great amount of works which extended the Rosenthal inequality todependent sequences. Magda Peligrad has been a leader in this type of extension for a wide range of
dependence structures and has led many of her collaborators (including myself) to take an interest in thisproblem. Among the key results we can cite the following works: Peligrad (1985) and Shao (1995) for thecase of ρ-mixing sequences; Shao (1988), Peligrad (1989) and Utev (1991) for the case of φ-mixing sequences;Peligrad and Gut (1999) and Utev and Peligrad (2003) for interlaced mixing; Rio (2000) for the strongly
mixing case; Rio (2009), Dedecker (2010) and Merlevède and Peligrad (2013) for projective criteria.
In the talk I would like to survey some of these deep results and in particular the Rosenthal inequalitythat Magda and I obtained in 2013, which reads as follows: Suppose that (Xk ) is a real stationary sequenceof real-valued random variables that are in Lp , p > 2, thennnX
1/(2δ) 
X
1
1
1/p
2 δ
k max |Sj | kp ≤ Cn
kX1 kp +
kE
(S
)k
+
kE
(S
)k
,
0
k
p
0
k
p/2
1≤j≤n
k 1+1/p
k 1+2δ/pk=1
k=1
where δ = min(1, 1/(p − 2)) and E0 (·) = E(·|σ(Xi , i ≤ 0)).
This Rosenthal inequality is an important refinement of the Burkholder-like inequality in Peligrad-UtevWu (2007). In addition, it is a starting point for many other results including an extension of Rio’s inequality(2000) for strong mixing sequences to the larger class of weak α-dependent sequences which leads to new
insights in case of intermittent maps of the interval.

Secular coefficients and the holomorphic multiplicative chaos
Joseph Najnudel
University of Bristol, UK
Abstract. We study the coefficients of the characteristic polynomial of unitary matrices drawn from theCircular Beta Ensemble. When the inverse temperature parameter beta is strictly larger than 4, we obtaina new class of limiting distributions that arise when both the order of the coefficient and the dimension of
the matrix goes to infinity. For beta equal to 2, we solve an open problem of Diaconis and Gamburd by
showing that the middle coefficient tends to zero in probability when the dimension goes to infinity. We
introduce a new stochastic object associated to the coefficients of the characteristic polynomial, which we call
Holomorphic Multiplicative Chaos (HMC). Viewing the HMC as a random distribution, we prove a sharp
result about its regularity in an appropriate Sobolev space. Our proofs expose and exploit several novel
connections with other areas, including random permutations, Tauberian theorems and combinatorics.

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Revisiting random matrices
Tamer Oraby
University of Texas, Rio Grande Valley
Abstract. In this talk, I will revisit results about random block matrices and products of random matrices
and discuss new directions. The work started in my PhD under supervision of Dr. Wlodek Bryc.

Some recent developments on linear processes and linear random fields
Hailin Sang
University of Mississippi
Abstract. In this talk we review some recent developments for the limit theorems and nonparametricestimations of linear random fields with linear processes as their one-dimensional special cases. The linear
random fields may have short or long memory. We establish the limit theorems including the local version for
linear random fields when the i.i.d. innovations have finite second moment, or the innovations have infinite
second moment and belong to the domain of attraction of a stable law with index 0 < α ≤ 2. When thecoefficients are absolutely summable we do not have restriction on the regions of summation. However, when
the coefficients are not absolutely summable we add the variables on unions of rectangles with some regularity
conditions. Entropy is widely applied in the fields of information theory, statistical classification, patternrecognition and so on since it is a measure of uncertainty in a probability distribution. We review somerecent work on the estimation of density and quadratic entropy for linear processes with kernel and wavelet
methods. We study bias, the integrated square error and central limit theorems for density and entropy
estimators with these two methods. We finally propose some open questions on parametric estimations forlong memory linear random fields.

Approximating the stationary distribution of the open ASEP
Dominik Schmid
University de Bonn, Germany
Abstract. The exclusion process is one of the best-studied examples of an interacting particle system.
In this talk, we consider the stationary distribution of asymmetric simple exclusion processes with open
boundaries. We project the stationary distribution onto a subinterval, whose size is allowed to grow withthe length of the underlying segment. Depending on the boundary parameters for the exclusion process,we provide sufficient conditions such that the projected stationary distribution is close in total variation
distance to a product measure. This talk is based on joint work with Evita Nestoridi.

Busemann processes in positive and zero temperature, on the lattice, and in thecontinuum
Timo Seppäläinen
University of Wisconsin, Madison
Abstract. This talk is an overview of the construction and properties of multicomponent stationary measures and Busemann functions of random growth models and directed polymer models, and of the utility ofthese measures in the study of the properties of the models. Potential examples include the corner growthmodel, the planar directed polymer model, the Kardar-Parisi-Zhang equation, and the directed landscape.

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On the motion of a tagged particle in simple exclusion
Sunder Sethuraman
University of Arizona
Abstract. Informally, the simple exclusion process follows a collection of continuous time random walks
on Zd interacting as follows: When a clock rings, the particle jumps to a nearest-neighbor vertex, if thatlocation is unoccupied. If occupied, the jump is suppressed and clocks start again.
In this model, the motion of a distinguished or ‘tagged’ particle is of interest. Part of the difficulty in its
analysis is that it is not Markovian by itself, due to interaction with other particles.
We will review some of the previous results for the asymptotics of the tagged particle, before coming to recent research on Gumbel limits and large deviations in 1D models, starting from different initial distributions.
Based on works with Michael Conroy and SRS Varadhan.

Wandering around the self-normalized limit theory
Qi-Man Shao
Southern University of Science and Technology, China
Abstract. Last two decades has witnessed significant progress on self-normalized limit theory in probability
and statistics. In contrast with the classical limit theorems, self-normalized limit theorems, especially Cramértype moderate deviation theorems, require much less moment assumptions. In this talk, we shall review recent
developments of limit theory for self-normalized processes as well as applications to statistical inference.

Central limit theorems in deterministic dynamical systems
Dalibor Volný
Université de Rouen, France
Abstract. Any strictly stationary process can be represented in a dynamical system with probability
preserving transformation T . If the dynamical system is deterministic (T is of zero entropy) then it seems
that none of the known methods of proving the CLT can apply. A CLT can hold true however. We will also
show convergence to stable laws. Joint work with Zemer Kosloff.

Several short stories on quadratic harnesses
Jacek Wesolowski
Warsaw University of Technology, Poland
Abstract. In early 1980s several people in Warsaw University of Technology were excited with the question
to what extent conditional moments of the first and second order of the Gaussian process uniquely determine
the process. Of course, one of the answers gives the celebrated Lévy characterization of the Brownian motion.
The idea was to avoid conditions imposed on trajectories though while conditioning with respect to the pastfuture filtration (Fs,u )s<u of the process. It lead Wlodek Bryc (and collaborators) to the definition ofquadratic harnesses (QH) - the basic object of these short stories: QH is a real-valued stochastic process(Xt )t≥0 which satisfies two fundamental conditions: for i = 1, 2E(Xti | Fs,u ) is a polynomial of degree i in variables Xs and Xu , s < t < u.
In the talk I would like to report on several highlights of the mathematical journey of Wlodek and his crew (Iwas lucky to be a member of) in (and around) fascinating land of QHs. These highlights include: infinite integrability; reaching, first, beyond Gaussian and then beyond the Lévy–Meixner processes; non-commutative
probability connections; Cauchy kernel families; free Laha–Lukacs theorem; orthogonal martingale polynomials; Askey–Wilson (AW) processes; a near-algebra connection; algebraic structure of polynomial processes;

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QH/AW representations of open ASEPs with a view towards KPZ related asymptotics; Motzkin paths
asymptotics etc, etc.
And the journey is far from being over...

Fast algorithms for estimating covariance matrices of stochastic gradient descentsolutions
Wei Biao Wu
University of Chicago
Abstract. Stochastic gradient descent (SGD), an important optimization method in machine learning, iswidely used for parameter estimation especially in online setting where data comes in stream. While thisrecursive algorithm is popular for the computation and memory efficiency, it suffers from randomness of thesolutions. In this talk we shall estimate the asymptotic covariance matrices of the averaged SGD iterates(ASGD) in a fully online fashion. Based on the recursive estimator and classic asymptotic normality resultsof ASGD, we can conduct online statistical inference of SGD estimators and construct asymptotically valid
confidence intervals for model parameters. The algorithm for the recursive estimator is efficient and onlyuses SGD iterates: upon receiving new observations, we update the confidence intervals at the same time
as updating the ASGD solutions without extra computational or memory cost. This approach fits in online
setting even if the total number of data is unknown and takes the full advantage of SGD: computation andmemory efficiency. This work is joint with Wanrong Zhu and Xi Chen.

Convergence of the KPZ equation
Xuan Wu
University of Illinois, Urbana-Champaign
Abstract. The KPZ universality is a wide class of models for growing interfaces, including the longest
increasing subsequence, the Dyson Brownian motion, interacting particle systems and directed polymers.
Among these models, the KPZ equation is a canonical member and its asymptotic behaviors have beenmajor topic in this field. In this talk we will give a complete description of the large time limits of the KPZ
equation.

Askey–Wilson signed measures and their applications in open ASEP
Zongrui Yang
Columbia University
Abstract. We introduce the Askey–Wilson signed measures as a new tool for studying the stationary
measure of the open asymmetric simple exclusion process (ASEP). As applications of this technique, we
access several asymptotics of open ASEP: the density profile, limit fluctuations, open KPZ fixed-point limit,
half-line ASEP limit, and open ASEP with a light particle. Based on joint works with Dominik Schmid,
Yizao Wang and Jacek Wesolowski.

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Titles and abstracts of poster presentations• On a learning scheme for Value-at-Risk and expected shortfall
Juan David Barrera Cano, Universidad de los Andes, Colombia• Empirical limit theorems for Wiener chaos
Jiemiao Chen, University of Georgia, Athens• Cutoff for random walks on contingency tables
Zihao Fang and Andrew Heeszel, The Ohio State University• Random permutation matrices induced by the Chinese restaurant process
Jaime Garza, University of Cincinnati• Scaling limit of a stochastic Hegselmann–Krause model
Aditya Suresh Gopalan, University of Illinois, Urbana-Champaign• Estimation problems for some perturbations of the independence copula
Mous-Abou Hamadou, University of Mississippi• Non-equilibrium fluctuations for a tagged particle in a one-dimensional zero range processes withregularized random environment
Marcel Hudiani, University of Arizona• An insight into economic scenario generation for central Africa through time series analysis andcopulasRégine Constella Imandi, University of Mississippi• Large deviations for the volume of k-nearest neighbor balls
Taegyu Kang, Purdue University• Asymptotic joint distributions of tree shapes for random phylogenetic trees
Gursharn Kaur, University of Virginia• Free energy of the elastic random manifold
Pax Kivimae, New York University• Phase transition of the consistent maximal displacement of branching Brownian motion
Jiaqi Liu, University of Pennsylvania• On planar Brownian motion singularly tilted through a point potential
Barkat Mian, University of Mississippi• A point on discrete versus continuous state-space Markov chains
Mattias Muia, University of Mississippi• Conditional coalescence of a diploid Moran model with selfing
Maximillian Newman, Indiana University, Bloomington• Weighted average iterated filtering using p-generalized Gaussian smoothing
Zamzam Qazi, University of Mississippi• Two-steps estimation problem for a new family of copula
Sahifa Siddiqua, University of Mississippi• Large deviations for empirical measures of self-interacting Markov chains
Adam Waterbury, Denison University• On the local limit theorems for linear sequences of lower psi-mixing Markov chains
Na Zhang, Towson University

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On a learning scheme for Value-at-Risk and expected shortfall
Juan David Barrera Cano
Universidad de los Andes, Colombia
Abstract. In this poster we exhibit a regression scheme recently proposed with collaborators to achieve
the empirical learning of conditional quantiles and superquantiles from a given random sample. Our results
provide convergence guarantees and quantitative upper bounds on the speed of convergence which seemto be realized by the implemented examples and are consistent with the widespread observation that the‘generalization power’ of neural networks is preserved under LASSO regularizations.

Empirical limit theorems for Wiener chaos
Jiemiao Chen
University of Georgia, Athens
Abstract. We consider a scheme of central limit theorem for empirical measures. In contrast to a classicalempirical central limit theorem where the distribution governing the samples is fixed, we let the distributionchange as the sample size grows in a triangular array setup. Certain asymptotic requirement is imposed on
the changing distributions in order to create asymptotic uncorrelatedness of the empirical measures evaluatedat disjoint subsets. This leads to an independently scattered Gaussian random measure as the limit. We
establish weak convergence of multiple integrals with respect to the normalized empirical measures towards
multiple Wiener-Itô integrals. This empirical limit theorem is also extended to one involving an infinite
series of multiple Wiener-Itô integrals. This is a joint work with Shuyang Bai.

Cutoff for random walks on contingency tables
Zihao Fang and Andrew Heeszel
The Ohio State University
Abstract. Contingency tables are matrices with fixed row and column sums. We studied the Diaconis–
Gangolli random walk as a Markov chain Monte Carlo model and observed the cutoff phenomena, a sharptransition from almost deterministic to almost random in the context of mixing time. We gave cutoff time
for the random walks on both 1 × n and n × n contingency tables, and we generalized our method to show
cutoff for a family of random walks on the torus (Z/qZ)n .

Random permutation matrices induced by the Chinese restaurant process
Jaime Garza
University of Cincinnati
Abstract. In this work, we are interested in random permutation matrices whose distribution is induced by
the Chinese restaurant process. The Chinese restaurant process is an algorithm depending on two parameters(α, θ) that is used to generate random permutations. For α = 0 and θ > 0, permutations generated by the
Chinese restaurant process adhere to Ewens distribution with parameter θ. Random permutation matrices
sampled using Ewens distribution have been studied extensively in the literature. We complement theseresults by investigating the linear statistics of the eigenvalues for random permutation matrices induced by
the Chinese restaurant process when α ∈ (0, 1). Joint work with Yizao Wang.

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Scaling limit of a stochastic Hegselmann–Krause model
Aditya Suresh Gopalan
University of Illinois, Urbana-Champaign
Abstract. We consider an infinite-dimensional stochastic Hegselmann-Krause model on the real line. In
discrete time, each point of a unit-intensity simple point process independently moves halfway toward eitherof its left or right neighbors, chosen uniformly at random. Co-located points are merged into a single point,
and the resulting simple point process is re-scaled to unit intensity. We show that when the point processes
are shifted such that there is a point at the origin, there is a unique weak limit of these dynamics when theinitial point process is renewal. Using a time-reversal argument, we construct a random positive measure M
on R for a dual problem, and the limiting gap between the first two points is the total mass of M assigned to
R. Finally, we discuss some ongoing research directions. Joint work with Partha Dey and Rasoul Etesami.

Estimation problems for some perturbations of the independence copula
Mous-Abou Hamadou
University of Mississippi
Abstract. In this work, we consider symmetric copulas with density of the form:C(u, v) = 1 +∞X

λk ϕk (u)ϕk (v),k=1where {ϕk (x), k ∈ N} is an orthogonal basis of L2 (0, 1) and the sequence |λk | has a finite number of values
or converges to 0. For the finite case, we propose an estimator of the vector λ = (λ1 , . . . , λs ) by considering
the Markov chains generated by the copula using uniform margins as stationary distribution. In order todetermine the confidence interval of parameters, we provide a multivariate central limit theorem for thevector λ.
For s = 2, several examples are considered such as the cosine copula, the sine-cosine copula and theLegendre copula. For each of those copulas, we determine the estimators of the parameters and the central
limit theorem as well. A simulation study is provided with a comparison to other known estimators such as
MLE and that of Longla and Peligrad (2021).

Non-equilibrium fluctuations for a tagged particle in a one-dimensional zero range
processes with regularized random environment
Marcel Hudiani
University of Arizona
Abstract. We prove a non-equilibrium functional central limit theorem for the position of a tagged particle
in a one dimensional ε-regularized Sinai random environment zero range process. This is a generalization of
the work by Jara-Landim-Sethuraman [Jara et al., 2009] and answers a question in the paper by LandimPacheco-Sethuraman-Xue [Landim et al., 2023]. Joint work with Cláudio Landim and Sunder Sethuraman.
References[Jara et al., 2009] Jara, M., Landim, C., and Sethuraman, S. (2009). Nonequilibrium fluctuations for a tagged particle inmean-zero one-dimensional zero-range processes. Probability Theory and Related Fields, 145(4):565–590.
[Landim et al., 2023] Landim, C., Pacheco, C. G., Sethuraman, S., and Xue, J. (2023). On a nonlinear SPDE derived from ahydrodynamic limit in a Sinai-type random environment. The Annals of Applied Probability, 33(1):200 – 237.

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An insight into economic scenario generation for central Africa through time series
analysis and copulasRégine Constella Imandi
University of Mississippi
Abstract. In this study, we present an Economic Scenario Generator (ESG) tailored for Central Africancountries, utilizing six variables, including macroeconomic indicators (GDP, GDP deflator inflation, anddiscount rates), along with three financial variables from BVMAC (The Central Africa Stock Exchange).
Our approach incorporates time series analysis, employing ARIMA models to assess variables. To address
non-elliptical distributions, we analyze residuals’ dependencies using copulas. Subsequently, we use copula
results and Cholesky decomposition (specific to elliptical distributions) to project macroeconomic variables.
Challenges arise in projecting non-normally distributed financial variables, posing an open problem. The
dataset is divided into training and test sets, enhancing the ESG’s robustness for economic sector risk
management. Joint work with Martial Longla and Louis Aimé Fono.

Large deviations for the volume of k-nearest neighbor balls
Taegyu Kang
Purdue University
Abstract. This paper develops the large deviations theory for the point process associated with the Euclidean volume of k-nearest neighbor balls centered around the points of a homogeneous Poisson or a binomial
point process in the unit cube. Two different types of large deviation behaviors of such point processes areinvestigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption thatthe centering terms for the volume of k-nearest neighbor balls grow to infinity more slowly than those needed
for Poisson convergence. Additionally, we also study large deviations based on the notion of M0 -topology,
which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson
convergence. As applications of our main theorems, we discuss large deviations for the number of Poissonor binomial points of degree at most k in a random geometric graph in the dense regime.
This is joint work with Christian Hirsch (Aarhus University) and Takashi Owada (Purdue University).

Asymptotic joint distributions of tree shapes for random phylogenetic trees
Gursharn Kaur
University of Virginia
Abstract. We study two subtree counting statistics, the number of cherries and the number of pitchforks,
for random phylogenetic trees generated by two widely used tree models: the proportional to distinguishable
arrangements (PDA) and the Yule–Harding–Kingman (YHK) models. We use an extended version of Polyaurn models where negative entries are permitted in the replacement matrix. We obtain the strong laws oflarge numbers and the central limit theorems for the joint distributions of two count statistics: cherries andpitchforks, for the PDA and the YHK models. This is a joint work with Kwok Pui Choi and Taoyang Wu.

Free energy of the elastic random manifold
Pax Kivimae
New York University

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Abstract. The elastic random manifold serves as a paradigmatic example of an elastic interface suspended
in a quenched disordered medium, with random polymers models being the most well-studied special case.
Work in the physics community has suggested that the behavior of these models in high dimensions should
enter a number of distinct phases, depending on the temperature, and structure of the disorder, with thefamed depinning transition being the most studied.
We confirm a number of predictions for these phases, including existence, rigorously. The central result
is a formula for the free energy in the high-dimensional limit, from which we are also able to obtain the
behavior of a number of other statistics. The key tool in our computation involves adapting the multi-species
synchronization method of Panchenko. Joint work with Gerard Ben Arous.

Phase transition of the consistent maximal displacement of branching Brownian motion
Jiaqi Liu
University of Pennsylvania
Abstract. Branching Brownian motion (BBM) is a random particle system that incorporates both thetree-like structure and the diffusion process. In BBM, each particle moves as Brownian motion and splits
into two√ at rate 1. It is well known that the right-most particle at time t will be near the linear line withslope 2. Roberts considered the so-called consistent maximal displacement to answer the question of how
close particles can stay to this critical line. We generalize√ this question and consider how close the trajectoryof a particle can stay to the linear line with slope 2 + ε where ε is small and positive. We show thatthe consistent maximal displacement undergoes a phase transition. This is based on joint work with Julien
Berestycki, Bastien Mallein and Jason Schweinsberg.

On planar Brownian motion singularly tilted through a point potential
Barkat Mian
University of Mississippi
Abstract. We will discuss a special family of two-dimensional diffusions, defined over a finite time interval [0, T ]. These diffusions have transition density functions that are given by the integral kernels of thesemigroup corresponding to the two-dimensional Schrödinger operator with a point potential at the origin.
Although, in a few ways, our processes of interest are closely related to two- dimensional Brownian motion,
they have a singular drift pointing in the direction of the origin that is strong enough to enable the possiblyof visiting there with positive probability. Our main focus is on characterizing a local time process at theorigin for these diffusions analogous to that for a one-dimensional Brownian motion. Joint work with Jeremy
Clark.

A point on discrete versus continuous state-space Markov chains
Mattias Muia
University of Mississippi
Abstract. My poster presents an exploration of the impact of discrete marginals on copula-based Markov
chains. We analyze the mixing properties of such models to emphasize the difference between continuous anddiscrete state-space Markov chains. The Maximum likelihood approach is applied to derive estimators formodel parameters in the case of a discrete-state space Markov chain with Bernoulli marginal distribution.
A stationary case and a non-stationary case are considered. The asymptotic distributions of parameter
estimators are provided. A simulation study showcases the performance of different estimators for the
Bernoulli parameter of the marginal distribution. Some statistical tests are provided for model parameters.
Joint work with Martial Longla.

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Conditional coalescence of a diploid Moran model with selfing
Maximillian Newman
Indiana University, Bloomington
Abstract. We consider a diploid Moran model with selfing with a constant population size. This process
generates a random graph consisting of edges connecting parents to children in successive generations, called
the pedigree. Classical approaches to studying the random genealogy of a subsample of genes in a population
average over the pedigrees. Here we study these random genealogies instead conditioning on the pedigree.
We find a robustness of the conditional structure of genealogies to the random dynamics of the pedigree.

Weighted average iterated filtering using p-generalized Gaussian smoothing
Zamzam Qazi
University of Mississippi
Abstract. In recent years, simulation-based inferences have garnered significant attention due to the inherent challenges in directly computing likelihood functions for many real-world problems. Iterated filtering
Ionides et al. [2] has emerged as a method to maximize likelihood functions by perturbing models and approximating the gradient of log-likelihood through sequential Monte Carlo filtering. Using Stein’s identity, Doucet
et al. [1] devised a second-order approximation of the gradient of log-likelihood using sequential Monte Carlosmoothing. In our work, we first introduce the generalization of Stein’s identity for normal distribution to pgeneralized Gaussian distribution, enabling more flexible perturbation with different tail behaviors. Building
upon these gradient approximations, we introduce a novel weighted average algorithm for maximizing likelihood through the two-time-scale stochastic approximation. We show the integration of the algorithm into
iterated filtering framework, relaxing the requirement for a bounded variance of the two-timescale stochastic
approximation. Initially, we apply p-generalised Gaussian smoothing through Weighted Average Iterated
filtering in two toy problems: a linear ou2 model and a nonlinear Gompertz model. Subsequently, wedemonstrate the potential of this technique in fitting a more complex cholera model, incorporating a highly
nonlinear structure with discrete population dynamics, seasonality, and extra-demographic stochasticity.
References[1] Arnaud Doucet, Pierre E Jacob, and Sylvain Rubenthaler (2013). Derivative-free estimation of the score vector and observed
information matrix with application to state-space models. arXiv preprint arXiv:1304.5768.
[2] Edward L Ionides, Carles Bretó, and Aaron A King (2006). Inference for nonlinear dynamical systems. Proceedings of the
National Academy of Sciences, 103(49):18438–18443.

Two-steps estimation problem for a new family of copula
Sahifa Siddiqua
University of Mississippi
Abstract. This work delves into continuous state stationary reversible Markov chains generated by a new
family of copulas with marginals from a scale-parameter family. A two-step estimation procedure is used tofind the estimators of copula parameters. This method requires a consistent estimator of the scale parameterto establish large sample properties of estimator. A simulation study was also done in R to support thefindings. Joint work with Martial Longla.

Large deviations for empirical measures of self-interacting Markov chains
Adam Waterbury
Denison University

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Abstract. Self-interacting Markov chains arise in a range of models and applications. For example, they
can be used to approximate the quasi-stationary distributions of irreducible Markov chains and to model
random walks with edge or vertex reinforcement. The term self-interacting Markov chain is something ofa misnomer, as such processes interact with their full path history at each time instant, and therefore are
non-Markovian. Under conditions on the self-interaction mechanism, we establish a large deviation principle
for the empirical measure of self-interacting chains on finite spaces. In this setting, the rate function takes
a strikingly different form than the classical Donsker–Varadhan rate function associated with the empiricalmeasure of a Markov chain; the rate function for self-interacting chains is typically non-convex and is given
through a dynamical variational formula with an infinite horizon discounted objective function.

On the local limit theorems for linear sequences of lower psi-mixing Markov chains
Na Zhang
Towson University
Abstract. In thisP project, we investigate the local limit theorem for partial sums
P of linear sequences ofthe form Xj = i∈Z ai ξj−i . Here (ai )i∈Z is a sequence of constants satisfying i∈Z a2i < ∞ and (ξi )i∈Z
are functions of a stationary Markov chain with mean zero and finite second moment. The Markov chain isassumed to satisfy one-sided lower psi-mixing condition.