This semester, we are meeting on Mondays 3:45-4:40pm in the Seminar Room 4206 French Hall. Please email yizao.wang@uc.edu
if you want to be added to our email list or have any questions.
Cincinnati Symposium on Probability Theory and Applications, November 9-11, 2018.
- Aug 31, organizational meeting.
- Sep 3, Labor Day, no talk.
- Sep 10, no talk.
- Sep 17, Yizao Wang, On a positive and self-similar Markov process.
In this talk, we go over a few recent developments related to a positive and self-similar Markov process with explicit transition density function. We explain how the process arises as the tangent process at the boundary of a large family of Markov processes, including in particular the $q$-Brownian motions. We then show how the convergence to the tangent process plays a role in the limit fluctuations for height functions of random Motzkin paths and for cumulative density functions of open asymmetric simple exclusion processes in the steady state. It is known that the positive self-similar Markov process has also intrinsic connection to free probability; the talk however does not require any knowledge on free probability. The talk is based on joint works with Wlodek Bryc.
- Sep 24, Magda Peligrad, On the local limit theorem for lower psi-mixing Markov chains.
Local limit theorem for a sequence of centered random variables is about the
rate of convergence of $P(x - h < S_n< x + h)$. This problem was intensively
studied in the i.i.d. case by Shepp (1964), Stone (1965) and Feller (1967). When
the variable are independent but not necessarily identically distributed, in the
so called non lattice case, we mention the papers by Mineka and Silverman
(1970), Maller (1978), Shore (1978). Hafouta and Kifer (2016) proved a local limit theorem for nonconventional sums of stationary psi-mixing Markov chains. These results raise the natural question if a local limit theorem can be found for certain
nonstationary Markov chains. We treat here a class larger than psi-mixing
Markov chains, defined by using the one sided lower psi-mixing coefficient.
- Oct 1, Ju-Yi Yen, On excursions inside an excursion.
The distribution of ranked heights of excursions of a Brownian bridge is given in a paper by Pitman and Yor. In this talk, we consider excursions of a Brownian excursion above a certain level. We study the maximum heights of these excursions as Pitman and Yor did for excursions of a Brownian bridge.
- Oct 8, Zuopeng Fu, Fractional stable processes parameterized by metric spaces.
In this talk, we first review a few examples of Brownian motion and fractional Brownian motions indexed by metric spaces, in particular the Euclidean space and the Euclidean sphere, and stable processes with Chentsov representation. We then introduce a new family of stable processes indexed by metric spaces with Chentsov representation. We shall describe the integral representation and series representation of these processes. At the core of our argument is a result on measure definite kernels which is of independent interest. We end this talk by showing some simulation results. This is a joint work with Yizao Wang.
- Oct 15, Na Zhang, On the quenched CLT for stationary random fields under projective criteria.
This talk is motivated by random evolutions which do not start from equilibrium, in a recent work, Peligrad and Voln\'{y} (2018) showed that the quenched CLT (central limit theorem) holds for ortho-martingale random fields. In this paper, we study the quenched CLT for a class of random fields larger than the ortho-martingales. To get the results, we impose sufficient conditions in terms of projective criteria under which the partial sums of a stationary random field admit an ortho-martingale approximation. More precisely, the sufficient conditions are of the Hannan's projective type. As applications, we establish quenched CLT's for linear and nonlinear random fields with independent innovations. This is a joint work with Magda Peligrad and Lucas Reding.
- Oct 22, Marcin Swieca, Multidimensional quantum Bessel processes.
In 1996 Biane constructed a semigroup of a non-commutative Brownian motion on the group $C_*$-algebra of the Heisenberg group $H$, and then, considering a restriction of the semigroup to the commutative sub-$C_∗$-algebra of radial functions on H, he obtained a classical Markov process on the real plane, which he called quantum Bessel process.
In my talk I will describe Biane's construction and show how can this construction be generalized to obtain multidimesional versions of quantum Bessel Process.
- Oct 29, no talk.
- Nov 4, Joseph Najnudel, Spiking and collapsing in large noise limits of stochastic differential equations.
In relation with measurements of some quantum systems, we study the behavior of some stochastic differential equations, when the intensity of the noise goes to infinity. For some particular equations, we show that the solutions tend, in a sense which can be made precise, to a jump process, which somehow surprisingly is decorated with spikes. Our result is proven for one-dimensional processes, it may be possible to extend it for some equations in higher dimension.
- Nov 9-Nov 11, Cincinnati Symposium on Probability Theory and Applications.
- Nov 12, Veterans Day, no talk.
- Nov 14, Swift Hall 608, 3:35pm, Hailin Sang, University of Mississippi, Fourier transform and project methods in kernel entropy estimation for linear processes.
Entropy is widely applied in the fields of information theory, statistical classification, pattern recognition and so on since it is a measure of uncertainty in a probability distribution. The quadratic functional plays an important role in the study of quadratic Renyi entropy and the Shannon entropy
It is a challenging problem to study the estimation of the quadratic functional and the corresponding entropies for dependent case. In this talk, we consider the estimation of the quadratic functional for linear processes. With a Fourier transform on the kernel function and the projection method, it is shown that, the kernel estimator has similar asymptotical properties as the i.i.d. case studied in Gine and Nickl (2008) if the linear process $\{X_n : n \in\mathbb N\}$ has the defined short range dependence. We also provide an application to $L^2$ divergence and the extension to multivariate linear processes. The simulation study for linear processes with Gaussian and $\alpha$-stable innovations confirms the theoretical results. As an illustration, we estimate the $L^2$ divergences among the density functions of average annual river flows for four rivers and obtain promising results.
This is a joint work with Yongli Sang and Fangjun Xu.
- Nov 19, Joseph Najundel, Orthogonal Polynomials on the Unit Circle and Gaussian multiplicative chaos.
For any probability measure on the unit circle, the corresponding orthogonal polynomials satisfy a recurrence relation called Szego recursion. The corresponding coefficients are called the Verblunsky coefficients. If we take these coefficients as random, they correspond to a random measure on the unit circle. On the other hand, from a Gaussian logarithmically correlated field on the unit circle, one can construct another random measure, called Gaussian multiplicative chaos. In a join work in progress with Reda Chhaibi, we conjecture that these two measure coincide if we take a particular probability distribution for the Verblunsky coefficients, related to the distribution of eigenvalues of some ensembles of random unitary matrices.
- Nov 26, Joseph Najnudel, On the maximum of two log-correlated fields: the logarithms of the characteristic polynomial of the Circular Beta Ensemble and the Riemann zeta function.
For different random fields whose correlation is logarithmic with respect to the distance between the points, we observe similar behavior for their extreme values, either proven or conjectured, depending on the model. In this talk, we present two different examples of such fields: the logarithm of the characteristic polynomial of the Circular Beta Ensemble (an ensemble of random unitary matrices generalizing the Circular Unitary Ensemble), and the logarithm of the Riemann zeta function, on a random interval of the critical line.
- Dec 3, Rafal Kulik, University of Ottawa. Limit theorems for empirical cluster functionals with applications to statistical inference.
Limit theorems for empirical cluster functionals are discussed. Conditions for weak convergence are provided in terms of tail and spectral tail processes and can be verified for a large class of multivariate time series, including geometrically ergodic Markov chains. Applications include asymptotic normality of blocks and runs estimators for the extremal index and other cluster indices. Results for multiplier bootstrap
processes are also provided.
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