# NSF/CBMS Conference Small Deviation Probabilities: Theory and Applications

## Wenbo V. Li, Lecturer

### June 4-8, 2012

University of Alabama in Huntsville

## Conference Information

All talks will be held in the Shelby Center for Science and Technology on the UAH campus.

- Overview of the Lectures
- Principal Lecturer
- Tentative Schedule
- Invited Talks
- Contributed Talks
- Participants
- Notes on Logistics
- Tentative Schedule and Abstracts (printer friendly)
- Conference Flyer
- 2012 CBMS Conferences

## Local Information

## Conference Organizers

- Dongsheng Wu: dongsheng.wu@uah.edu
- Kyle Siegrist: siegrist@math.uah.edu

Please contact us if you have questions or need additional information.

## Sponsors

We are very grateful to our sponsors for their support

## Description of Lectures

### Overview

Each day of the conference we will have two lectures presented by the principal lecturer. The purpose of these lectures is to present the state of the art of various powerful techniques on estimating small deviation probabilities, including traditional ones such as blocking, chaining, series expansions, Laplace/Taubirean theorems, classical Gaussian inequalities, Feynman-Kac formula, and newly developed ones such as metric entropy, weaker correlation and reverse Slepian type inequalities, determinantal approaches, etc. Major applications include strong limit theorems in probability and statistics, smoothness of density via Malliavin calculus, approximation quantities for stochastic processes, exit time and boundary crossing asymptotics, deviations for local times, lower tail behavior of Martingale limits for branching related processes, smallest singular values and Littlewood-Offer theory. The guiding philosophy of these lectures is that analysis of concrete processes is the most effective way to explain even the most general methods or abstract principles. The ten lectures will essentially cover the following:

- An overview of small deviation probabilities, basic estimates and techniques associated with independent random variables, together with various applications.
- Blocking techniques for the maximum partial sums and stable processes, following early work of Chung and recent refined work for weighted sup-norms.
- Gaussian processes and related Gaussian measures in Banach space setting. Two highlights are the precise connections between small ball provability and metric entropy estimates of the associated generating compact operators, and the estimates for dependent sums via the weaker correlation inequality. Their far-reaching implications are explored.
- More general techniques for both upper and lower bounds, including chaining, locally non-determinant method, sup via L2 method, Riesz representations for random fields, determinant method for smooth processes, Fourier analytic arguments.
- Techniques for the existence of precise constants, including orthogonal series expansions, the l2 comparison theorems, and scaling/subsdditive method.
- Small deviation probabilities of lower tail type and the one-sided exit asymptotics for both Markov and Gaussian processes.
- Lower tail behavior of Martingale limits for branching related processes.
- Smallest singular values and Littlewood-Offer theory.

One special feature is that, at the end of each lecture, several open and important problems related to techniques discussed will be offered. This will ensure that the participants, especially the new or recent entrants to the field, can start thinking and working on interesting problems and branch/connect into the area.

Besides the ten lectures presented by the principal lecturer, there will be several lectures presented by the additional invited speakers. They will be dealing mostly with the subjects where the use of small deviation probabilities is essential. There will also be afternoon sessions devotes to contributed talks.

### Lecture 1: Introduction, overview and applications

We first define the small deviation (value) probability in several settings, which basically study the asymptotic rate of approaching zero for rare events that {\it positive} random variables take smaller values. Many applications discussed in the scientific justification section are given. Benefits and differences of various formulations of small deviation probabilities are examined in details, together with connections to related fields.

### Lecture 2: Basic estimates and equivalent transformations

We first formulate several equivalent results for small deviation probability, including negative moments, exponential moments, Laplace transform and Taubirean theorems. The basic techniques involved are various useful inequalities, motivated from large deviation estimates. Some refinement of known results are given, including the classical Paley-Zegmund inequality. Applications to regularity and smoothness of probability laws via small deviation estimates of the determinant of Malliavin matrix are discussed in the setting of stochastic (partial) differential equations.

### Lecture 3: Techniques associated with independent variables

We start with probabilistic arguments for algebraic properties of small deviation probability, such as independent sums and products. These estimates are non-asymptotic and hence they can be applied in the setting of conditional probability. Separate treatments are analyzed for exponential and power decay rates. A newly discover symmetrization inequality is proved by Fourier analytic method. Littlewood and Offord type problems are discussed. We end with Komlos Conjecture on balancing vectors in discrepancy theory.

### Lecture 4: Blocking techniques for the sup-norm

We first present the vary useful blocking techniques for the maximum of the absolute value of partial sums in both upper and lower bound setting. The lower bound is more involved since the end position of each block has to be controlled also. The resulting estimates play a critical role in the Chung's type strong limit theorems for sample paths. Similar techniques are applied to weighted and/or controlled sup-norms for Brownian motion and stable processes. Applications to the two-sided exit time and Wichura type functional limit theorems are indicated.

### Lecture 5: Links between small ball probabilities and metric entropy

For a continuous centered Gaussian process, the generating linear operator is compact and so is the unit ball of the associated reproducing kernel Hilbert space. The fundamental links between small ball probability for Gaussian measure and the metric entropy are given and various far-reaching implications are explored. Several purely probabilistic results, obtained via the analytic connection without direct probabilistic proofs, are analyzed.

### Lecture 6: Small deviation (ball) estimates for sums of correlated Gaussian elements

We treat the sum of two {\it not} necessarily independent Gaussian random vectors in a separable Banach space. The main ingredients are Anderson's inequality and the weaker correlation inequality developed by the lecturer. Various applicants are provided to show the power of the method. As a direct consequence, under the sup-norm or Lp-norm, Brownian motion and Brownian bridge have exact the same small ball behavior at the log level, and so do Brownian sheets and various tied down Brownian sheets including Kiefer process.

### Lecture 7: More lower bound techniques

We first establish a commonly used general lower bound estimate for the supremum of non-differentiable Gaussian process via the chaining argument, as well as improvements for smooth Gaussian processes. Then we present a connection between small ball probabilities, discovered recently, that can be used to estimate small ball probabilities under any norm via a relatively easier L2-norm estimate.

### Lecture 8: More upper bound techniques

We present three techniques: locally non-determinant method, determinant method for smooth processes and Riesz representations for Gaussian random fields. The key ideas are illustrated by several important processes: fractional Brownian motion, L-process (infinitely differentiable), and Brownian and/or Slepian sheet.

### Lecture 9: Evaluation and existence of precise constants

Most of techniques discussed so far are for the asymptotic decay rate (up to a constant factor). Here we present a few known methods in which the exact constants can be obtained or shown to exist. In the Hilbert space l2, the full asymptotic formula is developed. And with the help of a comparison result, most small deviation probabilities under the L2-norm can thus be treated, and in particular when the Karhunen-Loeve expansion for a given Gaussian process can be found in some reasonable form. This is the case for Brownian motion, fractional Brownian motion and Brownian sheets, etc. A scaling argument, similar to the well known subadditive method, is established for the sup-norm of the fractional Brownian motion

### Lecture 10: Lower tail probabilities and one-sided exit asymptotics

There are only a handful of known examples (specific Gaussian processes) for the one-sided exit asymptotics and it is intellectual challenging to work out more examples in order to find a theory. We focus on extending classical results from Brownian motion to the fractional Brownian motions. The main motivations are not only the importance of these processes, but also the need to find proofs that rely upon general principles at a more fundamental level by moving away from crucial properties (such as the Markov property) of Brownian motion. Fractional Brownian motion might not be an object of central mathematical importance but abstract principles are.

### References

- Alon, N. and Spencer, J. (1992). The Probablistic Method, Wiley-Interscience Series.
- Aurzada, F., Gao, F., Kuhn, T., Li, W.V. and Shao, Q. (2011). Small ball probability for a family of smooth Gaussian processes, to appear.
- Berlinet, A. and Thomas-Agnan, C. (2004).
*Reproducing Kernel Hilbert Spaces in Probability and Statistics. Kluwer Acadmic.* - Bronski, J. C.(2003). Asymptotics of Karhunen-Loeve eigenvalues and tight constants for probability distributions of passive scalar transport.
*Comm. Math. Phys.*238, 563--582. - Bronski, J. C.(2003). Small ball constants and tight eigenvalue asymptotics for fractional Brownian motions.
*J. Theoret. Probab.*16, 87--100. - Chazelle, B. (2000).
*The Discrepancy MEthod: Randomness and Complexity,*Cambridge University Press. - Chen, X. and Li, W.V. (2003). Quadratic functionals and small ball probabilities for the m-fold integrated Brownian motion,
*Annals of Probability,***31**, 1052-1077. - Chen, X. and Li, W.V. (2011). Small value probablity and partition functions for intersection local times, preprint.
- Chen, X., Li, W.V., Rosinski, J. and Shao, Q.M. (2011). Large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes, to appear in
*Annals of Probability.* - Chu, W., Li, W.V. and Ren, Y. (2011). Small value probabilities for Galton-Watson processes with immigration and emigration, preprint.
- Creutzig, J. (2002). Relations between classical, average, and probabilistic Kolmogorov widths.
*J. Complexity*18, 287--303. - Dereich, S.; Fehringer, F.; Matoussi, A.; Scheutzow, M.; (2003). On the link between small ball probabilities and the quantization problem for Gaussian measures on Banach spaces.
*J. Theoret. Probab.*16, 249--265. - Gao, F. and Li, W.V. (2007), Small deviation probabilities for Slepian Gaussian random fields,
*Transactions of AMS,***359**, 1339-1350. - Khoshnevisan, D. and Shi, Z. (1998). Gaussian measure of a small ball and capacity in Wiener space, Asymptotic methods in Probability and statistics. 453-465.
- Klartag, B.; Vershynin, R. (2007). Small ball probability and Dvoretzky's theorem. Isreal J. Math. 157, 193--207.
- Kuelbs, J. and Li, W.V. (1993). Metric entropy and the small ball problem for Gaussian measures.
*J. Funct. Anal.***116**, 133-157. - Latala, R.; Oleszkiewicz, K. (2005). Small ball probability estimates in terms of widths. Studia Math. 169, 305--314.
- Lawler, G. and Limic, V. (2010).
*Random Walk: A Modern Introduction,*Cambridge Studies in Advanced Mathematics (No. 123). - Ledoux, M. (1996).
*Isoperimetry and Gaussian Analysis,*Lectures on Probability Theory and Statistics, Lecture Notes in Math.**1648**165--294, Springer-Verlag. - Ledoux, M. and Talagrand, M. (1991). Probability on Banach Spaces, Springer, Berlin.
- Li, W.V. (1999), A Gaussian correlation inequality and its applications to small ball probabilities.
*Electronic Commun. in Probab.***4**, 111-118. - Li, W.V. (2003), The first exit time of Brownian motion from unbounded domain, Ann. Prob.
**31**, 1078-1096. - Li, W.V. (2010+). Small Value Probabilities: Techniques and Applications, in preparation.
- Li, W.V. (2011). Lectures on small value probabilities: techniques and applications, handouts for topic courses given in US and China.
- Li, W.V. and Linde, W. (1998). Existence of small ball constants for fractional Brownian motions. C.R. Acad. Sci. Paris,
**326**, 1329-1334. - Li, W.V. and Shao, Q.M. (2001). Gaussian processes: inequalities, small ball probabilities and applications. Handbook of Statistics, Vol.
**19**,*Stochastic processes: Theory and methods,*Edited by C.R. Rao and D. Shanbhag, 533-598. - Li, W.V. and Shao, Q.M. (2002), A normal comparison inequality and its applications, Prob. Th. Rel. Fields.
**122**, 494-508. - Li, W.V. and Shao, Q.M. (2004), Lower tail probabilities for Gaussian processes, Ann. Prob.
**32**, 216-242. - Li. W.V. and Zhang, C-H. (2010+), Lower tail estimates for the maximum of partial sums via the Wiener-Hopf equation, in preparation.
- Lifshits, M. (1999). Asymptotic behavior of small ball probabilities. In: Probability theory and Mathematical Statistics. p.453-468.
- Lifshits, M. and Shi, Z. (2002). The first time of Brownian motion from a parabolic domain, Bernoulli, 8, 745-765.
- Milman, V. D.; Szarek, S. J.; A geometric lemma and duality of entropy numbers. Geometric aspects of functional analysis. 191--222,
*Lecture Notes in Math.*1745, Springer, Berlin, 2000. - Mas, A. (2011). Representation of small ball probabilities in Hilbert space and lower bound in regression for functional data, preprint. arXiv:0901.0264v2.
- Molchan, G.M. (1999). Maximum of a fractional Brownian motion: probabilities of small values.
*Comm. Math. Phys.,***205**, 97--111. - Nualart, D. (2010). Malliavin Calculus and its Applications, AMS, CBMS Conference Series.
- Rudelson,M and Vershynin, R. (2011), Non-asymptotic theory of random matrices: extreme singular values, Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010.
- Sinai, Ya. G. (1997). Distribution of the maximum of a fractional Brownian motion.
*Russian Math. Surveys,***52**, 359-378. - Talagrand, M. (1993). New Gaussian estimates for enlarged balls. Geometric and Funct. Anal.,
**3**, 502-526. - Talagrand, M. (1994). The small ball problem for the Brownian sheet. Ann. Probab.
**22**, 1331-1354. - Vanden-Eijnden, E. (2001). Non-Gaussian invariant measures for the Majda model of decaying turbulent transport. Comm. Pure Appl. Math.
**54**, 1146--1167. - van der Vaart, A.W. and van Zanten, J.H. (2007). Bayesian inference with rescaled Gaussian process priors. Electronic J. Statist.
**1**, 433-448. - van der Vaart, A.W. and van Zanten, J.H. (2008). Rates of contraction of posterior distributions based on Gaussian process priors.
*Ann. Statist.***36**, 1435-1463.

## Principle Lecturer

Professor Wenbo V. Li started his professional career at the University of Delaware in 1992 immediately after his Ph.D. and moved through the ranks to Full Professor in 2002. In 2006 he was elected a Fellow of the Institute of Mathematical Statistics (IMS) with citation for distinguished research in the theory of Gaussian processes and in using this theory to solve many important problems in diverse areas of probability.

Professor Li has contributed extensively to probabilistic aspects of geometric functional analysis, high dimensional probability, Gaussian processes/fields, stochastic inequalities, and small deviation/value probabilities and their applications. His current research interests involve probability estimates of rare events, and also related stochastic modeling and probabilistic analysis.

Professor Li is also actively engaged in other national and international activities. He has served as Associate Editors for four international journals. He served as Chair (09-10) and member (07-09) of the IMS Committee on Travel Awards, and also as a co-organizer or co-editor of meetings and conferences over 15 times.

Last but not least, Professor Li is an outstanding speaker. In the past five years alone, he has delivered over 100 talks at major international conferences, schools, and workshops to promote the use of probabilistic methods and analysis. He is widely known as a lecturer who conveys an unbounded and infectious enthusiasm for his subject.

## Tentative Schedule

Sunday, June 3 |
||

Time |
Event |
Location |

6:30-8:00 PM | Informal Reception (Cash Bar) | Bevill Center Hotel |

Monday, June 4 (Chair: Kyle Siegrist) |
||

Time |
Event |
Location |

8:00-8:45 | On-site Registration and Coffee | Shelby Center Lobby |

8:45-9:00 | Opening Remarks, Dr. Jack Fix (Dean, UAH College of Science) | Shelby Center 109 |

9:00-10:00 | Lecture 1, Wenbo V. Li | Shelby Center 109 |

10:00-10:15 | Coffee Break | Shelby Center 301 |

10:15-11:15 | Lecture 2, Wenbo V. Li | Shelby Center 109 |

11:15-11:30 | Coffee Break | Shelby Center 301 |

11:30-12:30 | Laplace asymptotics and Brownian functionals, Xia Chen (University of Tennessee Knoxville) | Shelby Center 109 |

12:30-2:00 | Lunch | Shelby Center 301 |

2:00-3:00 | Lecture 3, Wenbo V. Li | Shelby Center 109 |

3:00-3:15 | Coffee Break | Shelby Center 301 |

3:15-3:35 | Symmetry Breaking in Quasi 1-D Coloumb Systems, Paul Jung (University of Alabama Birmingham) | Shelby Center 109 |

3:40-4:00 | Almost sure asymptotics for Ornstein-Uhlenbeck processes of Poisson potential, Fei Xing (University of Tennessee Knoxville) | Shelby Center 109 |

4:05-4:25 | Semiparametric bounds on completely monotone functions, Guoqing Liu (Harbin Institute of Technology) | Shelby Center 109 |

Tuesday, June 5 (Chair: Zhijian Wu) |
||

Time |
Event | Location |

9:00-9:50 | Lecture 4, Wenbo V. Li | Shelby Center 109 |

9:55-10:15 | A quick primer on entropic limit theorems, Mokshay Madimam (Yale University) | Shelby Center 109 |

10:15-10:45 | Coffee Break | Shelby Center 301 |

10:45-11:35 | Lecture 5, Wenbo V. Li | Shelby Center 109 |

11:40-12:30 | A CLT for empirical processes and empirical quantile processes involving time dependent data, James Kuelbs (University of Wisconson, Madison) | Shelby Center 109 |

12:30-2:00 | Lunch | Garden View Cafe |

2:00-2:50 | Second-order chaos and processes on Heisenberg-like groups, Tai Melcher (University of Virginia) | Shelby Center 109 |

2:50-3:20 | Coffee Break | Shelby Center 301 |

3:20-3:40 | Small value probabilities for conitnuous state branching processes with immigration, Weijuan Chu (Nanjing University) | Shelby Center 109 |

3:45-4:05 | Posterior consistency of the Bayesian approach to linear ill-posed inverse problems, Sergios Agapiou (University of Warwick) | Shelby Center 109 |

4:10-4:30 | Strong analytic solutions of fractional Cauchy problems, Jebessa B Mijena (Auburn University) | Shelby Center 109 |

Wednesday, June 6 (Chair: Shannon Starr) |
||

Time |
Event |
Location |

9:00-9:50 | Lecture 6, Wenbo V. Li | Shelby Center 109 |

9:55-10:15 | On the Gaussian Correlation Conjecture, Joel Zinn (Texas A&M University) | Shelby Center 109 |

10:15-10:45 | Coffee Break | Shelby Center 301 |

10:45-11:35 | Lecture 7, Wenbo V. Li | Shelby Center 109 |

11:40-12:30 | Metric entropy in learning theory and small deviations, Thomas Kühn (Universität Leipzig) | Shelby Center 109 |

12:30-2:00 | Lunch | Shelby Center 301 |

2:00-2:50 | Interplay between convex geometry, bracketing entropy and small ball probability, Frank Gao (University of Idaho) | Shelby Center 109 |

2:50-3:20 | Coffee Break | Shelby Center 301 |

3:20-3:40 | Most likely paths of short falls in certain hedging problems, Zhijian Wu (University of Alabama) | Shelby Center 109 |

3:45-4:05 | The first exit time of a Brownian motion from the minimum and maximum parabolic domains, Dawei Lu (Dalian University) | Shelby Center 109 |

4:10-4:30 | First passage times of Levy processes over a one-sided moving boundary, Tanja Kramm (Teschnishe Universität, Berlin) | Shelby Center 109 |

4:35-4:55 | Small-ball probabilities for the volume of random convex sets, Peter Pivovarov (Texas A&M University) | Shelby Center 109 |

Thursday, June 7 (Chair: Paul Jung) |
||

Time |
Event |
Location |

9:00-9:50 | Lecture 8, Wenbo V. Li | Shelby Center 109 |

9:55-10:15 | Thinning Invariant Sequences, Shannon Starr (University of Rochester) | Shelby Center 109 |

10:15-10:45 | Coffee Break | Shelby Center 301 |

10:45-11:35 | Lecture 9, Wenbo V. Li | Shelby Center 109 |

11:40-12:30 | Malliavin calculus and convergence in density, Yaozhong Hu (University of Kansas) | Shelby Center 109 |

12:30-2:00 | Lunch | Garden View Cafe |

2:00-2:50 | Littlewood-Offord estimates and applications to random matrix theory, Hoi Nguyen (University of Pennsylvania) | Shelby Center 109 |

2:50-3:20 | Coffee Break | Shelby Center 301 |

3:20-3:40 | Survival Probabilities of weighted randon walks, Christoph Baumgarten (Technische Universität, Berlin) | Shelby Center 109 |

3:45-4:05 | Central limit theorem for an additive functional of the fractional Brownian motion, Jangjun Xu (University of Kansas) | Shelby Center 109 |

4:10-4:30 | Exact small deviation asymptotics for some Brownian functionals, Ruslan Pusev (Saint Petersburg State University) | Shelby Center 109 |

4:35-4:55 | Some results on the excursion probabilities of Gaussian randon fields, Dan Cheng (Michigan State University) | Shelby Center 109 |

5:30-7:00 | Conference Reception (Cash Bar) | Bevill Center |

Friday, June 8 (Chair: Dongsheng Wu) |
||

Time |
Event |
Location |

9:00-9:50 | Lecture 10, Wenbo V. Li | Shelby Center 109 |

9:50-10:20 | Coffee Break | Shelby Center 301 |

10:20-11:10 | Small ball properties and fractal properties of Gaussian random fields, Yimin Xiao (Michigan State University) | Shelby Center 109 |

11:15-12:05 | Discrepancy, small balls, and harmonic analysis, Michael Lacey (Georgia Institute of Technology) | Shelby Center 109 |

12:05-12:30 | Group Picture | TBA |

12:30-2:00 | Lunch | Shelby Center 301 |

2:00-3:30 | Problems and Discussion | Shelby Center 109 |

## Invited Talks

*Laplace asymptotics and Brownian functionals*

Xia Chen, University of Tennessee

#### Abstract

The method of Laplace transformation is also known as Tauberian theorem or time-exponentiation, and has been applied to a variety of the problems associated to Brownian motions (and other models) such as small ball probabilities, large deviations, moment computation of local and intersection local times, and ray-knight theorem. In this talk, examples will be given to demonstrate how this powerful tool is used in different settings and related questions will be asked.

*Interplay between convex geometry, bracketing entropy and small ball probability*

Frank Gao, University of Idaho

#### Abstract

In this talk, I will present a number of examples/problems in convex geometry and in bracketing entropy where small ball probability is a powerful tool, and examples where tools from these areas are useful to estimate small ball probability.

*Malliavin calculus and convergence in density*

Yaozhong Hu, University of Kansas

#### Abstract

The classical central limit theorem in probability theory states that if X1,⋯,Xn are iid with mean μ, then (under some mild conditions)

Fn=n−√(X1+⋯+Xnn−μ)

converges in distribution to a normal distribution. This is true for many other random variables Fn such as multiple Wiener-Ito integrals. In this talk we shall discuss under what condition, Fn have densities and the densities of Fn converge to the normal density. More precisely, we will consider the problem of finding conditions such that there are integrable positive functions fn(x) such that

P(a≤Fn≤b)=∫bafn(x)dx

and

limn→∞∫∞−∞|fn(x)−ϕ(x)|pdx=0

for all p≥1, where ϕ(x)=12π√e−x22 is the density of standard normal. The tool that we use is the Malliavin calculus. This is an ongoing joint work with Fei Lu and David Nualart.

*Small and large value probabilities and related limit theorems*

James Kuelbs, University of Wisconsin Madison

#### Abstract

The interplay of large and small value probabilities and functional limit theorems of Strassen and Chung-Wichura type, respectively, is examined. Applications to fractional Brownian motions, symmetric stable processes with stationary independent increments, Lévy area processes, and also multiple generations of Galton-Watson branching processes are included.

*Metric entropy in learning theory and small deviations*

Thomas Kühn, Universität Leipzig

#### Abstract

In the first part of the talk I will give a short introduction into learning theory, in order to show the importance of metric entropy in this very active field. A particular problem - which is related, e.g., to support vector machines - consists in finding good upper bounds for entropy numbers (or covering numbers) in reproducing kernel Hilbert spaces. In the second part I will determine the exact asymptotic behaviour of covering numbers in Gaussian RKHSs. On one hand, this improves earlier results by Ding-Xuan Zhou, and on the other hand it has an interpretation in terms of small deviations of certain smooth Gaussian processes. This part is based on my paper Covering numbers in Gaussian reproducing kernel Hilbert spaces J. Complexity 27 (2011), 489--499.

*Discrepancy, Small Balls, and Harmonic Analysis*

Michael Lacey, Georgia Institute of Technology

#### Abstract

We will survey the remarkably close connection with small ball problems in probability theory, and the classical bounds associated with the Discrepancy function. Tools to analysize these questions arise from Harmonic Analysis, but don't seem strong enough to complete the proofs of outstanding conjectures in the subject.

*Second-order chaos and processes on Heisenberg-like groups*

Tai Melcher, University of Virginia

#### Abstract

Smoothness properties of measures in infinite-dimensional spaces, in particular the laws of Brownian motions in these spaces, have been the subject of much research in various settings, including certain curved examples. We will consider the setting of infinite-dimensional Heisenberg-like groups. The Brownian motions in this case may be realized as a flat infinite-dimensional Brownian motion along with its second-order chaos. We will discuss recent smoothness results for the law of these processes; reverse log Sobolev inequalities play a critical role in the proof of these results. I will provide all basic definitions as well as some background to put these results in context. This is joint work with F. Baudoin and M. Gordina.

*Littlewood-Offord estimates and applications to random matrix theory*

Hoi Nguyen, University of Pennsylvania

#### Abstract

In the first half of the talk I will introduce several versions of the Erdos and Littlewood-Offord inequality. In the second half I will then present an application to bound the least singular value and to establish the circular law in random matrix theory.

*Small Ball Properties and Fractal Propereties of Gaussian Random Fields*

Yimin Xiao, Michigan State University

#### Abstract

Small ball probabilities are very important for investigating fine structires of the sample functions of Gaussian random fields. In this talk we present applications of small ball probabilities in establishing results on exact Hausdorff measure functions, exact packing measure functions and multifractal analysis for Gaussian random fields.

## Contributed Talks

*Posterior consistency of the Bayesian approach to linear ill-posed inverse problems*

Sergios Agapiou, University of Warwick

#### Abstract

We consider the Bayesian approach to a family of linear inverse problems in a separable Hilbert space setting, with Gaussian prior and noise distribution. An alternative method of identifying the posterior distribution using its precision operator is presented. Working with the unbounded precision operator, enables us to use partial differential equations (PDE) methodology to study posterior consistency in a frequentist sense and in particular to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution in the small noise limit. Our methods assume a relatively weak relation between the prior covariance operator, the forward operator and the noise covariance operator, more precisely, we assume that appropriate powers of these operators induce equivalent norms. We compare our results to known sharp rates of convergence in the case where the forward operator and the prior and noise covariances are all simultaneously diagonalizable, and confirm that the PDE method provides the same rates in many situations.

*Survival probabilities of weighted random walks*

Christoph Baumgarten, Technische Universität Berlin

#### Abstract

I will present some results on the asymptotic behaviour of the probability that a weighted sum of centered i.i.d. random variables Xk does not exceed a constant barrier, i.e.

P(supn=1,...,N∑k=1nσ(k)Xk≤x),x≥0,

where σ is a positive function with σ(x)→∞ as x→∞. For regular random walks it is well-known that this survival probability decays like N−1/2 as N→∞ if X1 is centered and has finite variance.

First I discuss the case of a polynomial weight function and determine the rate of decay of the above probability for Gaussian Xk. This rate is shown to be universal over a larger class of distributions that obey suitable moment conditions.

Finally we discuss the case of an exponential weight function. The mentioned universality does not hold in this setup anymore so that the rate of decay has to be determined separately for different distributions of the Xk. I present some results in the Gaussian framework where the survival probability corresponds to that of a discrete Ornstein-Uhlenbeck process.

The talk is based on a joint work with Frank Aurzada (TU Berlin).

*Some results on the excursion probabilities of Gaussian random fields*

Dan Cheng, Michigan State University

#### Abstract

In this talk, we consider the excursion probabilities of two types of Gaussian random fields:those with stationary increments and smooth sample functions, and those with anisotropic and non-smooth (or fractal) sample functions. For the first type Gaussian random fields, it is shown that the ''Expected Euler Characteristic Heuristic'' still holds; and for the second type of Gaussian random fields, we prove an asymptotic result which extends those of Pickands (1969), Chan and Lai (2006).

This talk is based on joint work with Yimin Xiao.

*Small value probabilities for continuous state branching processes with immigration*

Weijuan Chu, Nanjing University

#### Abstract

In this paper, we consider the small value probability of supercritical continuous state branching processes with immigration, Zt,t≥0. It is well known that under some condition on the branching mechanism and immigration mechanism, e−mtZt converges to a non-degenerate finite and positive limit W as t tends to infinity, with proper positive constant m. Our goal is to estimate the asymptotic behavior of P(W≤ε) as ε→0+ by studying the Laplace transform of W. We also reprove the small value probability of W via the prolific backbone decomposition for continuous state branching processes in the non-subordinator case.

*Symmetry breaking in Quasi-1D Coulomb systems}\*

Paul Jung, University of Alabama at Birmingham

#### Abstract

Quasi one-dimensional systems are systems of particles in domains which are of infinite extent in one direction and of uniformly bounded size in all other directions, e.g. on a cylinder of infinite length. The main result proven here is that for such particle systems with Coulomb interactions and neutralizing background, the so-called jellium, at any temperature and at any finite-strip width there is translation symmetry breaking. This extends the previous result on Laughlin states in thin, two-dimensional strips by Jansen, Lieb and Seiler (2009). The structural argument which is used here bypasses the question of whether the translation symmetry breaking is manifest already at the level of the one particle density function. It is akin to that employed by Aizenman and Martin (1980) for a similar statement concerning symmetry breaking at all temperatures in strictly one-dimensional Coulomb systems. The extension is enabled through bounds which establish tightness of finite-volume charge fluctuations.

*First passage times of Lévy processes over a one-sided moving boundary*

Tanja Kramm, Technische Universität, Berlin)

#### Abstract

In this talk we discuss the one-sided exit problem with a moving boundary for Lévy processes. The main focus of this talk concerns the question: For which functions f does the one-sided exit problem with a constant boundary, i.e.

P(X(t)≤1, t≤T)=T−δ+o(1),as T→∞

with some δ≥0 imply

P(X(t)≤1±f(t), t≤T)=T−δ+o(1),as T→∞ ?

Our main result states that if the boundary f behaves asymptotically as tγ for some γ<1/2 then the probability that the process stays below the boundary behaves as in the case with a constant boundary. Both positive and negative boundaries are considered.

After presenting our main results we compare it to previously known ones and briefly sketch the main idea of the proof.

The talk is based on joint work with Frank Aurzada (Berlin) and Mladen Savov (Oxford).

*Semiparametric bounds on completely monotone functions*

Guoqing Liu, Harbin Institute of Technology, China

#### Abstract

Given any random variable S≥0 and a completely monotone function f(s), various bounds are derived on the mean and variance of f(S). The techniques are based on domination by exponential functions, Cauchy-Schwarz inequality and symmetrization method for variance.

*The first exit time of a Brownian motion from the minimum and maximum parabolic domains*

Dawei Lu, Dalian University of Technology

#### Abstract

Consider a Brownian motion starting at an interior point of the minimum or maximum parabolic domains, namely, Dmin={(x,y1,y2):∥x∥1. Let τ(Dmin) and τ(Dmax) denote the first times the Brownian motion exits from Dmin and Dmax. Estimates with exact constants for the asymptotics of logP(τ(Dmin)>t) and logP(τ(Dmax)>t) are given as t→∞, depending on the relationship between p1 and p2, respectively. The proof methods are based on Gordon's inequality and early works of Li, Lifshits, and Shi in the single general parabolic domain case.

*A Quick Primer on Entropic Limit Theorems*

Mokshay Madiman, Yale University

#### Abstract

We attempt to provide a basic but quick, and hopefully enlightening, introduction to the information-theoretic understanding of the central limit theorem (and of other limit theorems).

*Strong analytic solutions of fractional Cauchy problems*

Jebessa B. Mijena, Auburn University

#### Abstract

Fractional derivatives can be used to model time delays in a diffusion process. When the order of the fractional derivative is distributed over the unit interval, it is useful for modeling a mixture of delay sources. In some special cases distributed order derivative can be used to model ultra-slow diffusion. We extend the results of Baeumer and Meerschaert in the single order fractional derivative case to distributed order fractional derivative case. In particular, we develop the strong analytic solutions of distributed order fractional Cauchy problems.

*Small-ball probabilities for the volume of random convex sets*

Peter Pivovarov, Texas A&M University

#### Abstract

The focus of the talk will be distributional inequalities for the volume of random convex sets. Typical models include convex hulls and Minkowski sums of line segments generated by independent random points. I will outline an approach to small-deviation estimates that makes use of rearrangement inequalities and tools from classical convexity such as intrinsic volumes and natural generalizations. This is joint work with Grigoris Paouris.

*Exact small deviation asymptotics for some Brownian functionals*

Ruslan Pusev, Saint Petersburg State University

#### Abstract

We find exact small deviation asymptotics with respect to weighted L2-norm for a rather wide class of Gaussian processes. Our approach does not require the knowledge of eigenfunctions of the covariance operator of a weighted process. Such a peculiarity of the method makes it possible to generalize many previous results in this area. We also obtain new relations connected to exact small deviation asymptotics for a Brownian excursion, a Brownian meander, and Bessel processes and bridges.

*Thinning Invariant Sequences*

Shannon Starr, University of Rochester

#### Abstract

I will discuss joint work with Ang Wei from University of Rochester. For a sequence (X1,X2,...) we consider Bernoulli-p thinning to be the new random sequence we get where we keep each point with probability p, independent of all others, and then left-justify the points that we keep to get (Y1,Y2,...). We characterize stationarity for this process.

*Most likely paths of shortfalls in certain hedging problems*

Zhijian Wu, University of Alabama Tuscaloosa

#### Abstract

With or without the constraint of the terminal risk, an optimal strategy to minimize the running risk in hedging a long-term commitment with short-term futures can be solved explicitly if the underline stock follows the simple stochastic differential equation

dSt=μdt+σdBt

where Bt is the standard Brownian motion. In this talk, the most likely paths of shortfalls associated with the hedging are discussed. We typically focus on the shortfalls corresponding to the optimal strategies established to minimize the running risk with or without the terminal constraint. These paths give information about how risky events occur instead of just their probability of occurrence.

*Almost sure asymptotics for Ornstein-Uhlenbeck processes of Poisson potenial*

Fei Xing, University of Tennessee Knoxville

#### Abstract

The objective of this paper is to study the long time behavior of the following exponential moment:

Exexp{±∫t0V(X(s))ds}

where {X(s)} is a d-dimensional Ornstein-Uhlenbeck process and {V(x)}x∈Rd is a homogeneous ergodic random field which will be defined in the paper. It turns out that the positive/negative exponential moment has ect growth/decay rate, which is different from the Brownian motion model studied by Carmona and Molchanov (1995) for positive exponential moment and Sznitman (1993) for negative exponential moment.

*Central limit theorem for an additive functional of the fractional Brownian motion*

Fangjun Xu, University of Kansas

#### Abstract

We prove a central limit theorem for an additive functional of the d-dimensional fractional Brownian motion with Hurst index H∈(11+d,1d), using the method of moments, extending the result by Papanicolaou, Stroock and Varadhan in the case of the standard Brownian motion.

*On the Gaussian Correlation Conjecture*

Joel Zinn, Texas A&M University

#### Abstract

## Participants

- Sergios Agapiou (University of Warwick)
- Shangbing Ai (University of Alabama in Huntsville)
- Gagik Amirkhanyan (Georgia Institute of Technology)
- Fawwaz Batayneh (Louisiana State University)
- Christoph Baumgarten (Technische Universität Berlin)
- Xia Chen (University of Tennessee Knoxville)
- Dan Cheng (Michigan State University)
- Stephane Chretien (Universite de Franche Comte)
- Weijuan Chu (Nanjing University)
- Frank Gao (University of Idaho)
- Xiaoqin Guo (University of Minnesota)
- Yunzhu He (University of Alabama in Huntsville)
- Yaozhong Hu (University of Kansas)
- Paul Jung (University of Alabama at Birmingham)
- Tanja Kramm (Technische Universität Berlin)
- Thomas Kühn (Universität Leipzig)
- James Kuelbs (University of Wisconsin Madison)
- Michael Lacey (Georgia Institute of Technology)
- Jiange Li (University of Delaware)
- Wenbo V. Li (University of Delaware)
- Guoqing Liu (Harbin Institute of Technology)
- Jin Liu (University of Delaware)
- Yanghui Liu (University of Kansas)
- Dawei Lu (Dalian University of Technology)
- Fei Lu (University of Kansas)
- Mokshay Madiman (Yale University)
- Tai Melcher (University of Virginia)
- Jebessa B. Mijena (Auburn University)
- Erkan Nane (Auburn University
- Hoi H. Nguyen (University of Philadelphia)
- Michael Palmer (University of Alabama in Huntsville)
- David Pan (University of Alabama in Huntsville)
- Moongyu Park (University of Alabama in Huntsville)
- Peter Pivovarov (Texas A&M University)
- Ruslan Pusev (Saint-Petersburg State University)
- Timothy Shellington (University of Alabama in Huntsville)
- Kyle Siegrist (University of Alabama in Huntsville)
- Shannon Starr (University of Rochester)
- Meg Walters (University of Rochester)
- Dongsheng Wu (University of Alabama in Huntsville)
- Zhijian Wu (University of Alabama Tuscaloosa)
- Yimin Xiao (Michigan State University)
- Fei Xing (University of Tennessee Knoxville)
- Fangjun Xu (University of Kansas)
- Peng Xu (University of Delaware)
- Joel Zinn (Texas A&M University)

## Notes on Logistics

- If you haven't emailed a signed copy of your Letter of Invitation to Ms. Tami Lang, please do so ASAP. The letters were sent on May 14. If you decide to not attend the conference, please also let us know. Please keep all your original receipts for reimbursement; we cannot reimburse you without them.
- We will schedule airport pick up on June 3. Based on the responses we received, we tentatively plan to meet you at the luggage claim area of the airport at 12:00 PM, 2:00 PM, 4:00 PM, 6:00 PM, and 9:30 PM.
- For the participants who will stay in the Charger Village Residence Hall, you may check-in at the front desk of the dorm. The desk will be open from 11:00 AM to 11:00 PM on June 3. Please mention that you are with the CBMS conference. As stated earlier, you will stay in a single bedroom in a suite of 4 bedrooms. A standard linen package (one fitted and one flat sheet, blanket, pillow and pillow case, one bath towel, one hand towel and washcloth) will be provided. Please do not lose your dorm key; there is a charge of $100.00 for lost keys. The address for Charger Village is 601 John Wright Drive, Huntsville, AL 35805 (campus map). The phone number is 256.824.3200.
- For the participants who will stay at the Huntsville Marriott, Wenbo has rented a seven passage van to commute from/to the conference. Please meet at the lobby area of the hotel at 8:00 AM on June 4 for travel to the conference. The address for the Huntsville Mariott is 5 Tranquility Base, Huntsville, Alabama 35805. The phone number is 800.228.9290. The driving time from the Marriott to the Shelby Center is about 5 minutes.
- If you have a car, you may park in the two story parking garage (at John Wright Drive), which is located next to the Charger Village, opposite to the University Fitness Center, and about two minutes away (on foot) from the Shelby Center (campus map). Campus security has agreed to not issue tickets in the parking garage during your stay. If you get a ticket there by chance, please bring it to us for cancelling.
- We will have an informal reception (with a cash bar) on Sunday evening (June 3) from 6:30 PM to 8:00 PM at the Bevill Center Hotel, which is on campus, about 5-7 minutes walking distance away from the Shelby Center/Charger Village (campus map).
- You will receive a wireless internet connection access code (for both the Shelby Center and Charger Village), a name tag, and conference materials during the onsite registration between 8:00 AM and 8:45 AM on Monday, June 4 in front of Room 109, Shelby Center. Openning remarks will begin at 8:45 AM, followed by the first lecture at 9:00 AM.
- You are welcome to visit the University Library but you will not be able to check out materials. You may use the University Fitness Center by purchasing a Day Pass ($10/day), with a picture ID (campus map). A grocery store, Big Brothers, is located 2 blocks from Charger Village on Holmes Avenue.
- Please contact the conference organizers Dongsheng Wu (256.497.9202) or Kyle Siegrist (256.652.5903) if you have questions or need assistance.

## Accomodations

**Charger Village Residence Hall**

John Wright Drive

Located on the UAH campus, adjacent to the Shelby Center

256.824.3200

550 Sparkman Drive, Huntsville, AL 35899

Located on the UAH Campus

256.721.9428

5 Tranquility Base, Huntsville, Alabama 35805

Located 1.3 miles from UAH campus

800.228.9290

4801 Governors House Drive, Huntsville, AL 35805

Located 1.4 miles from UAH campus

256.430.1778

- Details
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