Probability Commons^™

Scaling Limits Of Critical Observables Through The High Dimensional Incipient Infinite Cluster, Pranav Chinmay

Dissertations, Theses, and Capstone Projects

We give a general construction of the incipient infinite cluster in high dimensional percolation, and use it as a decoupling tool to rigorize geometric heuristics for analyzing the asymptotics of observables at criticality. Examples include demonstrating the limiting distribution of the chemical distance and full-strength asymptotics for k-point functions, which constitute foundational inputs for scaling limit results associated to critical clusters.

All Games Have Equilibria, M. Ali Khan, Arthur Paul Pedersen, Maxwell B. Stinchcombe

Publications and Research

Research on Nash equilibrium existence for infinite games has grown into a patchwork of technical preconditions and counterexamples. This paper presents a unified program in equilibrium theory by revising the predominant model of mixed strategies based on countable additivity. A game is specified by a nonempty set of players and, for each player, a nonempty action set and a bounded von Neumann-Morgenstern utility function. Every such game is shown to admit a Nash equilibrium in finitely additive mixed strategies. In addition, the equilibrium correspondence for any such game is shown to be nonempty, compact-valued, and upper hemicontinuous, and the same …

All Games Have Equilibria, Arthur Paul Pedersen, M. Ali Khan, Maxwell B. Stinchcombe

Publications and Research

Research on Nash equilibrium existence for infinite games has grown into a patchwork of technical preconditions and counterexamples. This paper presents a unified program in equilibrium theory by revising the predominant model of mixed strategies based on countable additivity. A game is specified by a nonempty set of players and, for each player, a nonempty action set and a bounded von Neumann-Morgenstern utility function. Every such game is shown to admit a Nash equilibrium in finitely additive mixed strategies. In addition, the equilibrium correspondence for any such game is shown to be nonempty, compact-valued, and upper hemicontinuous, and the same …

Limit Theorems For L-Functions In Analytic Number Theory, Asher Roberts

Dissertations, Theses, and Capstone Projects

We use the method of Radziwill and Soundararajan to prove Selberg’s central limit theorem for the real part of the logarithm of the Riemann zeta function on the critical line in the multivariate case. This gives an alternate proof of a result of Bourgade. An upshot of the method is to determine a rate of convergence in the sense of the Dudley distance. This is the same rate Selberg claims using the Kolmogorov distance. We also achieve the same rate of convergence in the case of Dirichlet L-functions. Assuming the Riemann hypothesis, we improve the rate of convergence by using …

Application Of Randomness In Finance, Jose Sanchez, Daanial Ahmad, Satyanand Singh

Publications and Research

Brownian Motion which is also considered to be a Wiener process and can be thought of as a random walk. In our project we had briefly discussed the fluctuations of financial indices and related it to Brownian Motion and the modeling of Stock prices.

V-Slam And Sensor Fusion For Ground Robots, Ejup Hoxha

Dissertations and Theses

In underground, underwater and indoor environments, a robot has to rely solely on its on-board sensors to sense and understand its surroundings. This is the main reason why SLAM gained the popularity it has today. In recent years, we have seen excellent improvement on accuracy of localization using cameras and combinations of different sensors, especially camera-IMU (VIO) fusion. Incorporating more sensors leads to improvement of accuracy,but also robustness of SLAM. However, while testing SLAM in our ground robots, we have seen a decrease in performance quality when using the same algorithms on flying vehicles.We have an additional sensor for ground …

One-Dimensional Excited Random Walk With Unboundedly Many Excitations Per Site, Omar Chakhtoun

Dissertations, Theses, and Capstone Projects

We study a discrete time excited random walk on the integers lattice requiring a tail decay estimate on the number of excitations per site and extend the existing framework, methods, and results to a wider class of excited random walks.

We give criteria for recurrence versus transience, ballisticity versus zero linear speed, completely classify limit laws in the transient regime, and establish a functional limit laws in the recurrence regime.

An Overview And Evaluation Of Synthetc: A Statistical Model For Extra-Tropical Cyclones, Rafael Uryayev

Dissertations and Theses

Extratropical cyclones (ETCs) are the most common weather phenomena affecting the United States, Canada, and Europe. They can pose serious hazards over large swaths of area. In this thesis, a statistical model of ETCs, called SynthETC, is discussed. The model accounts for the for genesis, track path, termination, and intensity of statistically generated ETCs. Genesis is modeled as a Poisson process, whose mean is determined by climate and historical information. Tracks are modeled as a regression-mean determined by climate and historical information plus a stochastic component. Lysis is modeled using logistic regression, with climate states as covariates. Intensity is modeled …

Season-Ahead Forecasting Of Water Storage And Irrigation Requirements – An Application To The Southwest Monsoon In India, Arun Ravindranath, Naresh Devineni, Upmanu Lall, Paulina Concha Larrauri

Publications and Research

Water risk management is a ubiquitous challenge faced by stakeholders in the water or agricultural sector. We present a methodological framework for forecasting water storage requirements and present an application of this methodology to risk assessment in India. The application focused on forecasting crop water stress for potatoes grown during the monsoon season in the Satara district of Maharashtra. Pre-season large-scale climate predictors used to forecast water stress were selected based on an exhaustive search method that evaluates for highest ranked probability skill score and lowest root-mean-squared error in a leave-one-out cross-validation mode. Adaptive forecasts were made in the years …

Burden Of Atopic Dermatitis In The United States: Analysis Of Healthcare Claims Data In The Commercial, Medicare, And Medi-Cal Databases, Sulena Shrestha, Raymond Miao, Li Wang, Jingdong Chao, Huseyin Yuce, Wenhui Wei

Publications and Research

Comparative data on the burden of atopic dermatitis (AD) in adults relative to the general population are limited. We performed a large-scale evaluation of the burden of disease among US adults with AD relative to matched non-AD controls, encompassing comorbidities, healthcare resource utilization (HCRU), and costs, using healthcare claims data. The impact of AD disease severity on these outcomes was also evaluated.

Stochastic Processes And Their Applications To Change Point Detection Problems, Heng Yang

Dissertations, Theses, and Capstone Projects

This dissertation addresses the change point detection problem when either the post-change distribution has uncertainty or the post-change distribution is time inhomogeneous. In the case of post-change distribution uncertainty, attention is drawn to the construction of a family of composite stopping times. It is shown that the proposed composite stopping time has third order optimality in the detection problem with Wiener observations and also provides information to distinguish the different values of post-change drift. In the case of post-change distribution uncertainty, a computationally efficient decision rule with low-complexity based on Cumulative Sum (CUSUM) algorithm is also introduced. In the time …

Simulation Insights Using R, Boyan Kostadinov

Publications and Research

This article attempts to introduce the reader to computational thinking and solving problems involving randomness. The main technique being employed is the Monte Carlo method, using the freely available software R for Statistical Computing. The author illustrates the computer simulation approach by focusing on several problems of increasing difficulty. The simulation techniques and the specific problems discussed in this article would be of interest to STEM students and instructors, teaching courses in Monte Carlo simulations, stochastic modeling, probability and statistics. The R code for all problems is discussed in full detail so that the reader can get a taste of …

Computational Insight With Monte Carlo Simulations, Boyan Kostadinov

Publications and Research

We introduce Monte Carlo simulations for estimating areas by playing a game of "darts". We also introduce simulations of random walks. We use compact, vectorized programming, based on the R language, for all computer simulations and visualizations, aimed at high school students. This presentation is based on the Invited, prime time lecture given at the summer camp for gifted high school students at City College of New York, July 13, 2011.