Contributed 09

Topics Related to RMT

Conference
9:30 PM — 10:00 PM KST
Local
Jul 22 Thu, 8:30 AM — 9:00 AM EDT

On eigenvalue distributions of large auto-covariance matrices

Wangjun Yuan (The University of Hong Kong)

4
In comparison to Hermitian random matrices, the study of random matrices without Hermitian or Unitary structure is more limited. The famous circular law, that the eigenvalue empirical measures of random matrices whose entries are i.i.d. centered complex random variables with unit variance converges almost surely to the uniform distribution on the unit disk, was posed in 1950s. Important breakthroughs were made by Bai, Girko, Tao and Vu, and the conjecture was finally proved by Tao et al. Auto-covariance matrices are important in statistics, especially in high-dimensional time series analysis. Let (X_j)_{1<=j<=n} be the time series observed at time 1<=j<=n. We study the non-Hermitian matrix Y(k)=1/n (X_{k+1}X_1^* +...+ X_nX_{n-k}^*) is known as lag-k auto-covariance matrix of the time series. Recently, a similar model was studied in , where the matrix Z=Y(1)+1/n X_1X_n^* was considered. To obtained the limit of the sequence of eigenvalue empirical measures,  used the linearization technique, the small ball probability as well as the logarithmic potential. Although the two matrices Y(1) and Z differ only by a rank-one matrix, the limit eigenvalue empirical measure of Z does not imply anything a priori on the asymptotic properties of Y(1) since rank-one perturbations may destroy the limit completely. The linearization technique in  fails in our case. We design another auxiliary matrix to obtain the lower bound for the least singular value. The lower bound together with the small rank perturbations on the limit of singular values empirical measure leads to the limit of the sequence of eigenvalue empirical measures of Y(k). The results can be found in .

 Arup Bose and Walid Hachem, Smallest singular value and limit eigenvalue distribution of a class of non-Hermitian random matrices with statistical application, J. Multivariate Anal. 2020.
 Jianfeng Yao and Wangjun Yuan, On eigenvalue distributions of large auto-covariance, arXiv:2011.09165.

Linear spectral statistics of sequential sample covariance matrices

Nina Dörnemann (Ruhr University Bochum)

4
Estimation and testing of a high-dimensional covariance matrix is a fundamental problem of statistical inference with numerous applications in biostatistics, wireless communications and finance. Linear spectral statistics are frequently used to construct tests for various hypotheses. In this work, we consider linear spectral statistics from a sequential point of view. To be precisely, we prove that the stochastic process corresponding to a linear spectral statistic of the sequential empirical covariance estimator converges weakly to a non-standard Gaussian process. As an application we use these results to develop a novel approach for monitoring the sphericity assumption in a high-dimensional framework, even if the dimension of the underlying data is larger than the sample size. Compared to previous contributions in this field, the results of the present work are conceptually different, because the sequential parameter used in the definition of the process also appears in the eigenvalues. This “non-linearity” results in a substantially more complicated structure of the problem. In particular, the limiting processes are non-standard Gaussian processes, and the proofs of our results (in particular the proof of tightness) require an extended machinery, which has so far not been considered in the literature on linear spectral statistics. As a consequence, we provide a substantial generalization of the classical CLT for linear spectral statistics proven by Bai and Silverstein.

Couplings for Andersen dynamics and related piecewise deterministic Markov processes

Nawaf Bou-Rabee (Rutgers University Camden)

3
Piecewise Deterministic Markov Processes (e.g. bouncy particle and zigzag samplers) have recently garnered increased research interest for their potential to rapidly sample high-dimensional probability distributions. However, there remain many open questions and challenges to understanding their mixing time and convergence properties. In this talk, I will highlight recent progress on Andersen dynamics: a PDMP that iterates between Hamiltonian flows and velocity randomizations of randomly selected particles. Various couplings of Andersen dynamics will be used to obtain explicit convergence bounds in a Wasserstein sense. The bounds are dimension free for not necessarily convex potentials with weakly interacting components on a high dimensional torus, and for strongly convex and gradient Lipschitz potentials on a Euclidean product space.

Q&A for Contributed Session 09

0
This talk does not have an abstract.

Session Chair

Kyeongsik Nam (University of California at Los Angeles)

Contributed 11

Topics Related to KPZ Universality

Conference
9:30 PM — 10:00 PM KST
Local
Jul 22 Thu, 8:30 AM — 9:00 AM EDT

Upper tail decay of KPZ models with Brownian initial conditions

Balint Veto (Budapest University of Technology and Economics)

2
We consider the limiting distribution of KPZ growth models with random but not stationary initial conditions introduced by Chhita, Ferrari and Spohn. The one-point distribution of the limit is given in terms of a variational problem. We deduce the right tail asymptotic of the distribution function. This gives a rigorous proof and extends the results obtained by Meerson and Schmidt.

Bijective matching between q-Whittaker and periodic Schur measures

Matteo Mucciconi (Tokyo Institute of Technology)

4
We report on a combinatorial construction that allows to relate marginal distributions of the q-Whittaker and the periodic Schur measures. The periodic Schur measure is a generalization of the Schur measure introduced by Borodin in 2006 and that models lozenge tilings in a cylindrical domain. Its free fermionic origin yields a nice mathematical structure and its correlations are determinantal.The q-Whittaker measure is another generalization of the Schur measure, which has found application in the rigorous description of KPZ models. Since mathematical properties of q-Whittaker polynomials are much more complicated than the Schur polynomials, the q-Whittaker measure is a more difficult object to handle. Using a bijective combinatorial approach we are able to relate the theories of Schur and q-Whittaker polynomials producing a remarkable correspondence between the two measures. Our arguments pivot around a combination of various theories, which had not yet been used in integrable probability, that include Kirillov-Reschetikhin crystals, Demazure modules, the Box-Ball system or the skew RSK correspondence.

The talk is based on collaborations with Takashi Imamura and Tomohiro Sasamoto. Motivations and general ideas of our work are addressed by T. Imamura and applications to probabilistic systems are explained by T. Sasamoto.

A new approach to KPZ models by determinantal and Pfaffian measures

Tomohiro Sasamoto (Tokyo Institute of Technology)

3
Recently we have established bijectively an identity which relates certain sums of q-Whittaker polynomials and skew Schur polynomials. More precisely we have found that a marginal of the q-Whittaker measure is equivalent to that of the periodic Schur measure. This enables us to study various models in the KPZ universality class by using methods associated with determinantal point processes. The main purpose of this talk is to explain this new approach to KPZ models. By imposing a symmetry on the bijection, we have also found an identity which relates sums which include single q-Whittaker polynomials and skew Schur polynomials. This is in fact related to the KPZ models in half-space and our identity allows us to study such models using Pfaffian measures. In particular we will establish the limiting distributions for models in half-space, which have not been achieved by other approaches.

The talk is based on collaborations with Takashi Imamura and Matteo Mucciconi. Motivations and general ideas of our work are addressed by T. Imamura and the bijective proofs of identities are explained by M. Mucciconi.

Q&A for Contributed Session 11

0
This talk does not have an abstract.

Session Chair

Jaehoon Kang (Korea Advanced Institute of Science and Technology (KAIST))

Contributed 21

Dimension Reduction and Model Selection

Conference
9:30 PM — 10:00 PM KST
Local
Jul 22 Thu, 8:30 AM — 9:00 AM EDT

Probabilistic principal curves on Riemannian manifolds

Seungwoo Kang (Seoul National University)

7
This paper studies a new curve fitting approach that is useful for representation and dimension reduction of data on Riemannian manifolds. In this study, we extend the probabilistic formulation of the curve passing through the middle of data on Euclidean space by Tibshirani (1992) to Riemannian symmetric space. To this end, we define a principal curve based on a mixture model for observations and unobserved latent variables, and propose a new algorithm to estimate the principal curve for given data points on Riemannian manifolds using a series of procedures in ‘unrolling, unwrapping, and wrapping’ and EM algorithm. Some properties for justification of the estimation algorithm are further investigated. Results from numerical examples, including several simulation sets on hyperbolic space, sphere, special orthogonal group, and a real data example, demonstrate the promising empirical properties of the proposed probabilistic approach.

The elastic information criterion for multicollinearity detection

Kimon Ntotsis (University of the Aegean)

4
When it comes to factor interpretation, multicollinearity is among the biggest issues that must be surmounted especially in this new era, of Big Data Analytics. Since even moderate size multicollinearity can prevent a proper interpretation, special diagnostics must be recommended and implemented for identification purposes. In this work, we propose the Elastic Information Criterion which is capable of capturing multicollinearity accurately and effectively without factor over-elimination. The performance in simulated and real numerical studies is demonstrated.

A bidimensional shock model driven by the space-fractional Poisson process

Alessandra Meoli (Università degli Studi di Salerno)

4
We describe a competing risks model within the framework of bivariate random shock models, this being of great interest in reliability theory. Specifically, we assume that a system or an item fails when the sum of shocks of type 1 and of type 2 reaches a random threshold that takes values in the set of natural numbers. The two kinds of shock occur according to a bivariate space-fractional Poisson process, which is a time-changed bivariate homogeneous Poisson process where the time change is an independent stable subordinator. We obtain the failure densities, the survival function and other related quantities. In this way we generalize some results in the literature, which can be recovered when the index of stability characterizing the bivariate space-fractional Poisson process is equal to 1.

Q&A for Contributed Session 21

0
This talk does not have an abstract.

Session Chair

Jisu Kim (Inria Saclay)

Contributed 35

Financial Data Analysis

Conference
9:30 PM — 10:00 PM KST
Local
Jul 22 Thu, 8:30 AM — 9:00 AM EDT

Hedging portfolio for a degenerate market model

Ihsan Demirel (Koç University)

2
The purpose of this talk is to derive the hedging portfolio in a financial market where prices-per-shares are governed by a stochastic equation with a singular volatility matrix. The main mathematical tools of the study are the representation with respect to a minimal martingale and Malliavin calculus for the functionals of a degenerate diffusion process, which have been established in recent studies. We use those developments to prove a version of the Hausmann-Bismut-Ocone type representation formula derived for these functionals under an equivalent martingale measure. Consequently, we derive the hedging portfolio as a solution to a system of linear equations. The uniqueness of the solution is achieved by a projection idea that lies at the core of the martingale representation. We apply our result to exotic options, whose value at maturity depends on the prices over the entire time horizon. This work is supported by Tubitak Project No. 118F403.

An optimal combination of proportional - excess of loss reinsurance with random premiums

Suci Sari (Statistics Research Division, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung)

2
In reinsurance policy, the insurer needs to pay a reinsurance premium to the reinsurer. Nonetheless, the insurer will get an insurance premium from the insurance policyholder. This research addresses choosing an optimal reinsurance policy using a more realistic model, i.e., individual risk model with random insurance premiums. Here, we assume that the insurance premiums are random, and the reinsurance premiums is charged by the expected value principle. We use minimizing the risk exposure of the insurer as optimization criteria. The insurer’s risk exposure is measured by tail risk measure of the net cost of the insurer. To illustrate the applicability of our results, we consider an insurance company has two lines of business and derive the optimal reinsurance explicitly for Combination of Proportional and Excess of loss Reinsurance.

A novel inventory policy for imperfect items with stock dependent demand rate

Praveen V. P. (University of Calicut)

2
Adoption of trade credit financing policy is prevalent in inventory management as an important strategy to increase profitability with a major motive of attracting new customers and also to avoid lasting price competition. We revisit an economic order quantity model under conditionally permissible delay in payments, fix a certain period to settling the account, during this period supplier charges no interest, but beyond this period interest is being charged. On the other hand, retailer can earn interest on the revenue generated during this period. Optimal inventory policy can be managed with explicitly specifying the demand for fresh produce to be a function of its freshness expiration date and displayed volume. Shortages are allowed and it is partially backlogged. Keeping this scenario in mind, an attempt has been made to formulate an inventory policy for imperfect items with permissible delay in payments and expiration date under freshness and stock dependent demand rate. The formulated model is illustrated through numerical examples to determine the effectiveness of the proposed model. Further, the effects of changes of different inventory parameters have been studied by a sensitivity analysis.

Q&A for Contributed Session 35

0
This talk does not have an abstract.

Session Chair

Jae Youn Ahn (Ewha Womans University)    