# Past Probability Seminars Fall 2008

## Probability Seminar Fall 2008

The Probability Seminar meets Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. Organized by Tom Kurtz

### Schedule

• September 4  *
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• Large gaps between random eigenvalues*

One of the first problems of random matrix theory was to identify the asymptotic probability of a large gap between random eigenvalues. In the 1950s Wigner proposed a formula for the large gap probabilities of the Gaussian Unitary Ensemble which was later improved and generalized by Dyson for a large class of models called beta ensembles.  By characterizing the point process limits of the beta ensembles as a functional of Brownian motion in the hyperbolic plane we are able to analyze the large gap probabilities using stochastic calculus. This leads to a proof of a modified version of Dyson's predictions. (Joint with Balint Virag, University of Toronto)

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• September 11  *
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• Yong Zeng, *University of Missouri at Kansas City
• Econometric analysis via filtering for financial ultra-high frequency (UHF) data  (slides)*

We propose a general nonlinear filtering framework with marked point process observations for financial UHF data. The signal contains the intrinsic value and the related parameters and is modeled as a general Markov process. Trading times are driven by a generic point process, and the noise is described by a random transformation from the intrinsic value to trading price. Other observable variables (such as initiators of trade, and economic news) are allowed to affect the intrinsic value, the trading intensity and the noise. The proposed model encompasses many important existing models.  We derive the SPDEs such as filtering equations to characterize the likelihoods, the posterior, the likelihood ratios and the Bayes factors of the proposed model. We further study the Bayesian inference via filtering.  Especially, we employ the Markov chain approximation method to construct easily-parallelizable, recursive efficient algorithms to compute the posteriors and others, and we prove the convergence of such algorithms.  The general theory is applied to a specific model built for UHF Treasury notes data from GovPX. We find that the buyer-seller initiation dummy and the economic news dummy in volatility are statistically significant, which has important implications in financial market microstructure theory.

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• September 18  *
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• No Seminar***
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• September 25  *
• Properties of the averaged large deviation rate function for random walks in random environments*

An averaged (annealed) large deviation principle for multi-dimensional random walks in random environments (RWRE) was proved by Varadhan in 2003. However, Varadhan's proof is quite complicated and thus it is difficult to derive much qualitative information about the large deviation rate function from the formulation given in his proof. In this talk I will describe a different, much simpler, approach to large deviations of multidimensional RWRE using regeneration times. I will use this approach to show that when the distribution on environments is “non-nestling,” the averaged large deviation rate function is analytic in a neighborhood of the limiting velocity of the RWRE.

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• October 2 *
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• Improved mixing time bounds for the Thorp shuffle and L-reversal chain*

We prove a theorem that reduces bounding the mixing time of a card shuffle to verifying a condition that involves only pairs of cards, then  we use it to obtain improved bounds for two previously studied models.

E. Thorp introduced the following card shuffling model in 1973. Suppose the number of cards n is even. Cut the deck into two equal piles. Drop the first card from the left pile or from the right pile according to the outcome of a fair coin flip. Then drop from the other pile. Continue this way until both piles are empty. We obtain a mixing time bound of O(\log^4 n). Previously, the best known bound was O(\log^{29} n) and previous proofs were only valid for n a power of 2.

We also analyze the following model, called the L-reversal chain, introduced by Durrett. There are n cards arrayed in a circle. Each step, an interval of cards of length at most L is chosen uniformly at random and its order is reversed.

Durrett has conjectured that the mixing time is O(\max(n, {n^3 \over L^3})\log n). We obtain a bound that is within a factor

O(\log^2 n) of this, the first bound within a poly log factor of the conjecture.

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• October 9 *
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• Fluctuations of the empirical quantiles of independent Brownian motions (slides)*

We consider _n_ iid, one-dimensional Brownian motions, _B,,j,,__(__t)_, where _B,,j,,__(0)_ has a rapidly decreasing, smooth density function _f_. The empirical quantiles, or pointwise order statistics, are denoted by _B,,j:n,,__(t)_, and we are interested in a sequence of quantiles _Q,,n,,__(t) = B,,j,,,,(n):n,,(t)_, where _j(n)/n → α __∈__ (0,1)_. This sequence converges in probability in _C[__0,∞)_ to _q(t)_, the α-quantile of the law of _B,,j,,__(t)_. Our main result establishes the convergence in law in _C[__0,∞)_ of the fluctuation processes _F,,n,, = n^1/2^(Q,,n,, - q)_. The limit process _F_ is a centered Gaussian process and we derive an explicit formula for its covariance function. We also show that _F_ has many of the same local properties as _B^1/4^_, the fractional Brownian motion with Hurst parameter _H = 1/4_. For example, it is a quartic variation process, it has Hölder continuous paths with any exponent γ < 1/4, and (at least locally) it has increments whose correlation is negative and of the same order of magnitude as those of _B^1/4^_.  See http://arxiv.org/abs/0812.4102

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• October 16  *
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• Persistent cookie random walk on Z*

Let M be a natural number and assume that M cookies are put at each vertex of the one-dimensional integer lattice. When the nearest-neighbor random walk is at site n and there are still cookies at that site, the walker eats one cookie and then goes to the right with probability either p_1 or p_2, depending on whether it just came to n from n-1 or n+1. Here p_1 and p_2 are two fixed numbers between zero and one. When the random walk visits a site at which there are no more cookies left, it jumps to one of the neighbor sites with equal probabilities.

We describe recurrence criteria and criteria for non-zero speed regime for this random walk. The main technical tool that we use is an encoding of a path of cookie random walks into branching processes with migration recently introduced by Basdevant and Singh (2006) and further developed by Kosygina and Zerner (2008).

This is joint work with Giora Slutzki (ISU).

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• October 23*
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• October 30*
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• November 6*
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• Genealogy of Catalytic Populations*

For neutral branching models of two types of populations there are three universality classes of behavior: independent branching,

(one-sided) catalytic branching and mutually catalytic branching. Loss of independence in the two latter models generates many new features in the way that the populations evolve.

In this talk I will focus on describing the genealogy of a catalytic branching diffusion. This is the many individual fast branching limit of an interacting branching particle model involving two populations, in which one population, the "catalyst", evolves autonomously according to a Galton-Watson process while the other population, the "reactant", evolves according to a branching dynamics that is dependent on the number of catalyst particles.

We show that the sequence of suitably rescaled family forests for the catalyst and reactant populations converge in Gromov-Hausdorff topology to limiting real forests. We characterize their distribution via a reflecting diffusion and a collection of point-processes. We compare geometric properties and statistics of the catalytic branching forests with those of the "classical" (independent branching) forest.

This is joint work with Andreas Greven and Anita Winter.

• November 13 *
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• Uniform Laws of Large Numbers under Ergodic Sampling*

A family of functions F satisfies a uniform law of large numbers with respect to a process X if, with probability one, the sample averages of functions in F converge uniformly to their limiting expectations.  Uniform laws of large numbers have been widely used, and extensively studied, in a number of areas, including empirical process theory and machine learning.  The majority of work on uniform laws to date has considered i.i.d. processes X, though there is also a substantial literature concerned with dependent samples satisfying a variety of mixing conditions.

The talk will focus on uniform laws of large numbers under ergodic sampling, without any mixing assumptions.  We show that a combinatorial sufficient condition for uniform convergence in the i.i.d. case, namely that the family F have finite Vapnik-Chervonenkis (VC) dimension, is sufficient to ensure a uniform law for every ergodic process X. We present results for classes of sets, and for classes of functions under several related notions of combinatorial dimension. Our method of proof is new and direct: it does not rely on existing results or exponential inequalities from the i.i.d. case.

This talk should be accessible to statisticians, electrical engineers and computer scientists working on the theory of machine learning.

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• November 20*
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• Nonlinear filtering of random fields in the presence of long-memory noise*

An interesting estimation problem, arising in many dynamical systems, is that of filtering; namely, one wishes to estimate a trajectory of a “signal” process (which is not observed) from a given path of an observation process, where the latter is a nonlinear functional of the signal plus noise.

In the classical mathematical framework, the stochastic processes are parameterized by a single parameter (interpreted as time), the observation noise is a martingale (say, a Brownian motion), and the best mean-square estimate of the signal, called the optimal filter, has a number of useful representations and satisfies the well-known Kushner-FKK and Duncan-Mortensen-Zakai stochastic partial differential equations.

However, there are many applications, arising, for example, in connection with denoising and filtering of images and video-streams, where the parameter space has to be multidimensional. Another level of difficulty is added if the observation noise has a long-memory structure, which leads to “nonstandard” filtering evolution equations. Each of the two features (multidimensional parameter space and long-memory observation noise) does not permit the use of the classical theory of filtering and the combination of the two has not been previously explored in mathematical literature on stochastic filtering.

This talk focuses on nonlinear filtering of a signal in the presence of long-memory fractional Gaussian noise. We will start by introducing first the evolution equations and integral representations of the optimal filter in the one-parameter case, when the noise driving the observation is represented by a fractional Brownian motion. Next, using fractional calculus and multiparameter martingale theory, the case of spatial nonlinear filtering of a random field observed in the presence of a persistent fractional Brownian sheet will be explored.

The talk is based in part on joint work with Matt Linn.

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• November 27  Thanksgiving*
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• December 4*
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• Two-type Stochastic model for concentration in yeast cell*

We study the model proposed by Altschuler, Angenent and Wu for the dynamics of particles in a yeast cell. In this model there are two types of particles, A and B, which interact with each other. We are specifically interested in the clustering of particles on the membrane. Each cluster is called a clan. For any finite population size _N_, clan sizes follow a Markov Chain over the space of measures. We show that under suitable scaling these clan sizes follow a measure-valued diffusion process in the infinite population limit. Moreover the ratio of A and B type particles converges to a constant for every clan. We also obtain the stationary distribution for clan sizes.

• December 11*
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I*nvariant measures for independent particles in a dynamical random environment*

It is known that the invariant measures for a collection of independent random walks on Z^d  are i.i.d. Poissons. In this talk, we will consider a system of independent particles in a space-time random environment. We will show that the spatially ergodic invariant distributions for the particle process are mixtures of inhomogeneous Poisson product measures that depend on the past of the environment.

This is joint work with Prof. Timo Seppalainen.

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