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I have received a thorough training in probability, statistics as well as abstract mathematics during my studies at Charles University, Prague, Czech Republic where I have taken a number of courses ranging from real, complex and functional analysis, probability, stochastic processes, and statistics. I have further expanded my knowledge of probability and mathematical statistics during my doctoral studies at Michigan State University. I wrote my dissertation under the supervision of Prof. A.~V.~Skorokhod. The main topic was a study of properties of filtrations supporting only purely discontinuous martingales. Since becoming a faculty member I developed rather diverse research interests in both theoretical statistics and probability. Currently I pursue research projects in three main areas: Gaussian processes, application to engineering, and generalized fiducial statistics. The problems related to Gaussian processes are partially supported by an NSF grant DMS-0504737. One of the main questions of interest there are small deviations for Gaussian processes, i.e., understanding the behavior of the probability that a stochastic process X(t), 0<t<T stays in a small ball of radius e around the origin: P( | X| < e). Here the norm $|.|$ is applied to the sample paths of the stochastic process and $|X|$ is therefore a non-negative random variable. As $e$ tends to 0, this probability clearly tends to zero --- the question is at what rate? Usually even finding the logarithmic small ball rate is valuable. The logarithmic small ball rate is the rate at which-log P(|X|<e) tends to infinity as e tends 0. Questions of this type fall into the realm of what has come to be called "small ball" probabilities. In the past I have made contributions to the theory of small deviations under the L2 norm. In the future I am interested in looking at small deviations for processes with jumps. Another important area of my interest is application of stochastic processes to engineering. The first project is simultaneous tracking of multiple moving objects extracted from an image sequence. This is an important problem which finds numerous applications in science and engineering. My co-authors and I have established theoretical properties of a statistical model for tracking such moving objects, or targets, are investigated. This tracking model allows for birth, death, splitting and merging of targets, and uses a Markov model to decide the times at which such events occur. This model also assumes that the track traveled by each target behaves like a Gaussian process, and it estimates these tracks by maximum likelihood. One of the contributions of our contribution was in establishing the almost sure convergence of these maximum likelihood tracking estimates to the truth. In other words we proved that this model should work well under the conditions it was designed for provided we have enough data. A major technical challenge for proving this consistency result is to identify the correct track estimate amongst a group of similar (but incorrect) track estimates that are results of various combinations of target birth, death, splitting and/or merging. This consistency property of the tracking estimates was empirically verified by numerical experiments. To the best of our knowledge, this is the first time that a comprehensive study is performed for the large sample properties of a multiple target tracking method. The second project I am part of is concerned with the modeling and simulation of extremely large networks using time-dependent partial differential equations (PDEs). In many applications, numerical simulation is the tool of choice for the design and evaluation of large networks. However, the computational overhead associated with direct simulation severely limits the size and complexity of networks that can be studied in this fashion. Performing numerical simulations of large stochastic networks has been widely recognized as a major hurdle to future progress in understanding and evaluating large networks. Our modeling approach is based on asymptotic analysis of a stochastic system that provides a probabilistic description of the network dynamics. This approach appears particularly promising for networks like a wireless ad hoc network. In this kind of network, nodes send to and receive from other nodes that are within transmission range. Transmission success is affected by interference; e.g., nodes are often so simple that they can receive only one message at a time, and propagation losses are often modeled by a power law dependence on distance. In this situation, we believe that it is possible to formulate the flow of information through the network using hydrodynamic scaling limits for the behavior of the individual packets or particles. The technical details involve defining a probability structure that describes the likelihood of information from one node passing to nearby nodes and then passing from this local probability structure to a diffusion limit description of the motion. The team working on this project contains an electrical engineer, probabilist and pde specialist. A lot of my current effort is concerned with studying theoretical properties of generalized fiducial inference. R. A. Fisher's fiducial inference has been the subject of many discussions and controversies ever since he introduced the idea during the 1930's. The idea experienced a bumpy ride, to say the least, during its early years and one can safely say that it eventually fell into disfavor among mainstream statisticians. However, it appears to have made a resurgence recently under the label of generalized inference. In this new guise fiducial inference has proved to be a useful tool for deriving statistical procedures for problems where frequentist methods with good properties were previously unavailable. The aim of our work is to revisit the fiducial idea of Fisher from a fresh new angle. We do not attempt to derive a new ``paradox free theory of fiducial inference'' as we do not believe this is possible. Instead, with minimal assumptions we present a new simple fiducial recipe that can be applied to conduct statistical inference via the construction of generalized fiducial distributions. This recipe is designed to be fairly easily implementable in various practical applications, and can be applied regardless of the dimension of the parameter space (i.e., including nonparametric problems). We term the resulting inference generalized fiducial inference. From the very beginning our work has been motivated by important applications in areas such as pharmaceutical statistics and metrology. Several of my students are applying the generalized fiducial methodology to important applied problems with a great success. In addition to using generalized fiducial distribution in applications we have also analyzed its properties in some special cases. Generalized fiducial distribution often gives rise to statistical procedures that are asymptotically exact and more importantly possess very good approximate small sample properties. This is shown by a mounting evidence from several simulation studies many of which are cited below. We will investigate the theoretical underpinnings of this good behavior of generalized fiducial procedures using higher order asymptotic tools such as Edgeworth expansion. The flexibility of generalized fiducial inference also allows us to move beyond parametric problems. We plan to apply it to large sparse linear systems for which wavelet regression is a representative example. A distinctive characteristic for such a system is that, when comparing to the sample size, the number of parameters in the system is large and most of these parameters are statistically insignificant. Therefore the issue of model selection is inherently built-in for this class of problems. As a first step we investigate the use of generalized fiducial inference for constructing wavelet regression confidence intervals. A pilot study indeed shows some very promising preliminary results. To the best of our knowledge, this is the first time that fiducial ideas are being applied to a nonparametric or a model selection problem. In conclusion I plan to continue working on various theoretical problems arising on the boundary between probability and theoretical statistics. Most of my work is done in collaboration with a wide variety of people who bring in a diverse skill set. Additionally I love working with students and have supervised a number of MS project and Ph.D. dissertations on topics related to my research program. I plan to continue this mode of operation in the future.