Sharif Digital Repository

“How can we generate a random point with uniform distribution over a convex body ?” According to it’s applications, it’s important for a solution to this problem to be applicable in high dimensions. Here, we are interested in algorithms with polynomial order with respect to the dimension. All existing methods for dealing with this problem are based on the Markov chain Monte Carlo method, i.e. a random walk is constructed in such that its stationary distribution is the uniform distribution over. Then, after simulating “enough” steps of this random walk, the distribution of the resulting point is “approximately” uniform. The real problem in Monte Carlo method is analyzing its “mixing time”,... 

Random Polytopes, the first occurrence of which dates back to the famous Sylvester’s four points problem in the 1860s, is a branch of geometric probability, typically concerning the convex hull of some random points chosen from a convex subset of Rd. In this thesis we have studied some special kind of random polytopes; the one that is the convex hull of some independent random points chosen from a convex body (a convex, compact set with interior point) according to the uniform distribution. It was a new approach from A. Rényi and R. Sulanke in 1963 to consider this type when the number of random points tends to infinity.This thesis consists of three main parts: The first part is devoted to... 

The purpose of this thesis is to introduce and review a recent methods in simultaneous hypothesis testing. False discovery rates, Benjamini and Hochberg’s FDR Control Algorithm, is the great success story of the new methodology. Much of what follows is an attempt to explain that success in empirical Bayes terms.The later chapters are at pains to show the limitations of current largescale statistical practice: Which cases should be combined in a single analysis? How do we account for notions of relevance between cases? What is the correct null hypothesis? How do we handle correlations? Some helpful theory is provided in answer, but much of the argumentation is by example, with graphs and... 

With Cramer’s model we have a probability measure on the power set of N. This probability measure is concentrated on the set that its elements are that subsets of N which number of their elements up to a certain natural number is asymptotically equal with the number of primes up to the same number. Let Pc be a sample obtained from this probability measure and consider 8n 2 N, an counts the number of ways that ncan be represented as a multiplication of some elements of Pc, such that changing the arrangement of factors in a representation does not introduce a new one. In this thesis, we prove that limn!1 a1++an n almost surely exists and is positive  

Determinantal processes are a special family of stochastic processes that arise in physics (fermions), random matrices (eigenvalues), and in combinatorics (random spanning trees and non-intersecting paths). These processes have repelling property (points close to each other are chosen with low probability). Because of this repelling property, determinantal processes are approporiat for modeling some physical quantities (e.g. the position of electrons). Their probabilistic structure is described by operators on complex vector spaces and their eigenvalues. Determinantal processes have interesting properties, e.g. number of points in a region is a sum of independent Bernoulli random variables.... 

Travel time estimation is a central issue in the urban transportation industry and is the basis of many analyses and services in businesses related to this area. In the past few years, various statistical approaches have been devised to solve this problem. The purpose of this dissertation is to review existing methods by focusing on segment-based approaches for urban travel time estimation. A big challenge is the small amount of data in hand compared to the size of the urban network. Exploring historical data and extracting correlation between urban network segments leads to modeling the urban traffic condition and travel time estimation in one specific time interval of the day  

Birkhoff's ergodic theorem in dynamical systems and ergodic theory, and the strong law of large numbers in probability theory are among the fundamental theorems of the two fields, which are closely related. Thus Birkhoff's ergodic theorem directly yields the strong law of large numbers. Attempts were then made to express some limit theorems in probability theory in the form of dynamic systems, such as the central limit theorem, which was expressed in the form of dynamic systems, and even generalizations of It was also obtained. In this paper, we will investigate the above and similar connections between probability limit theorems and well-known theorems in ergodic theory  

This thesis is devoted to introduce coherent risk measures on general probability spaces. After studying their properties, we also will characterize them using functional analysis tools. First we describe some related economic concepts such as risk concept, risk management and risk measures. Then we will study Value at Risk (VaR) as an applicable risk measure and determine its advantages and disadvantages. The motivation for studying risk measures in an axiomatic point of view and also introducing coherent risk measures was that VaR doesn’t have the diversification property. In chapter 2 and 3, we introduced coherent risk measures comprehensively. We began the second chapter by the... 

Mankind according to his authority (or delusion of authority) always finds himself in a situation which need decision-¬making. Usually, he seeks to make the best possible decision. The basis for measuring the goodness of choices is different in different occasions. This measure could be level of enjoyment, economic profit, probability of reaching a goal, etc. These decisions have consequences such that the situations before and after the decisions are not the same. Most challenging decision¬-making situations are those which the decision¬maker has not the complete authority over the situation and the results of decisions are influenced by out of control factors. A significant part of... 

Reinforcement learning (RL) is a subfield of machine learning that expresses how to learn optimal actions in a wide range of unknown environments. Reinforcement learning problems are often phrased in terms of Markov decision processes (MDPs). However, being restricted to Markov environments to solve problems with limited state space is not an unreasonable assumption, but the main challenge is to consider these problems in as large a class of environments as possible, which includes any challenges that an agent may face in real world. Such agents are able to learn to play chess, wash dishes, invest in financial markets, and do many tasks that an intelligent human being can learn and do. In...