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#### On Mixing Time for Some Markov Chain Monte Carlo

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

Markov chains are memoryless stochastic processes that undergoes transitions from one state to another state on a state space having the property that, given the present,the future is conditionally independent of the past. Under general conditions, the markov chain has a stationary distribution and the probability distribution of the markov chain, independent of the staring state, converges to it’s stationary distribution.

We use this fact to construct markov chain monte carlo, which are a class of algorithms for sampling from probability distributions based on constructing a markov chain that has the desired distribution as its stationary distribution. The state of a chain after a large...

We use this fact to construct markov chain monte carlo, which are a class of algorithms for sampling from probability distributions based on constructing a markov chain that has the desired distribution as its stationary distribution. The state of a chain after a large...

#### Cramér’s Model for Random Primes

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

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

#### Philosophical Theories of Probability

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

We can note to at least four very different interpretations of probability that have become popular in the twentieth century. These interpretations are as follows:1. Logical theory: According to this theory, the probability of each event will be equal to the degree of rational belief in that event. In addition, these rational beliefs are assumed to be equal for each individual in the same situation. This is the difference between logical theory and subjective theory.2. Subjective theory: as mentioned, in Subjective theory like Logical theory, probability of an event equals to degree of rational belief in that event, except that Subjective theory allows the differences between the beliefs...

#### Random Polytopes

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

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...

#### Generating Random Points in a Convex Body in High Dimensions

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

“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”,...

#### Irregularities of Some Random Point Processes

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

The spherical ensemble is a well-studied determinantal process with a fixed number of points on the sphere. The points of this process correspond to the generalized eigenvalues of two appropriately chosen random matrices, mapped to the surface of the sphere by stereographic projection. This model can be considered as a spherical analogue for other random matrix models on the unit circle and complex plane such as the circular unitary ensemble or the Ginibre ensemble, and is one of the most natural constructions of a (statistically) rotation invariant point process with repelling property on the sphere. In this dissertation we study the spherical ensemble and its local repelling property by...

#### Determinantal Processes

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

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....

#### Coherent Risk Measures on General Probability Spaces

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

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...

#### Simultaneous Hypothesis Testing and False Discovery Rate

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

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...

#### False Discovery Rate for Large Scale Hypothesis Testing

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

The chapter 1 begins the discussion of a theory of large-scale simultaneous hypothesis testing now under development in the statistics literature. Furthermore,this chapter introduces the False Discovery Rate (FDR) and Empirical Bayes approach. In chapter 2, the frequentist viewpoints to the simultaneous hypothesis testing is mentioned. apter 3 describes the break through paper of the Benjamini and Hochberg published in 1995. Chapter 4 provides new criteria for error and represents an outstanding method of controlling FDR by J.D. Storey. The first part of chapter 5 discusses a paper related to control of FDR for variable selection in linear model setting by E.Candes and R. Barber. In the rest...

#### Statistical Methodes for Urban Travel Time Estimation

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

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

#### Markov Decision Process with Timeconsuming Transition

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

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...

#### Generative Models and their Role in Development of Generality in AI

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

In this thesis Generative Models in Deep Learning are discussed, especially Generative Models which are based on latent variables. Deep Generative Models have key role in developing Artificial Intelligence, particularly in developments of general cognition and perception in AI. In this thesis, this role for Generative Models and their applications in cognition development, and also the mathematical foundation of generative models are discussed

#### Phase Transition in Convex Optimization Problems with Random Data

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

In the behavior of many convex optimization problems with random constraints in high dimensions, sudden changes or phase transitions have been observed in terms of the number of constraints. A well-known example of this is the problem of reconstructing a thin vector or a low-order matrix based on a number of random linear observations. In both cases, methods based on convex optimization have been developed, observed, and proved that when the number of observations from a certain threshold becomes more (less), the answer to the problem with a probability of close to one (zero) is correct and the original matrix is reconstructed. Recently, results have been obtained that explain why this...

#### Regularization Methods for Improving Data Efficiency in Reinforcement Learning

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

Reinforcement learning is a successful model of learning that has received a lot of attention in recent years and has had significant achievements. However, methods based on reinforcement require a lot of data. Therefore, it is important to find ideas to keep learning at a high level despite the lack of data. Many of these ideas are known as statistical regularity. In this thesis, we study methods to enhance the learning rate, including methods for sharing neural network weights between value function and policy networks. In this thesis we will try to gain a more general understanding of the regularization in reinforcement learning and increase the learning rate by implementing these methods...

#### General Reinforcement Learning

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

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...

#### Investigating the Relationship between Limit Theorems in Probability Theory and Ergodic Theory

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor)
Abstract

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

#### On The Existence of Arithmetic Progressions In Subsets of Integers

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor) ; Hatami, Omid (Supervisor)
Abstract

Suppose that A is a large subset of N. It is interesting to think about the arithmetic progressions in A.In 1936, Erdos and Turan conjectured that for > 0 and k 2 N, there exists N = N(k; ) that for all subsets A {1; 2; : : : ;N}, if lAl N, A has a nontrivial arithmetic progression of length k. Roth proved the conjecture for k = 3 in 1953. In 1969, Szemeredi proved the case k = 4 and in 1975, he gave a combinatorial proof for the general case. In 1977, using ergodic theory, Furstenberg gave a different proof for the Erdos-Turan conjecture (or Szemeredi Theorem!) and finally Gowers found another proof for the Szemeredi theorem, which was an elegant generalization of the Roth’s proof for k =...

#### Continuum Scaling Limit of Critical Percolation

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor) ; Rouhani, Shahin (Supervisor)
Abstract

Percolation is a simple probabilistic model which exhibits a phase transition. Here, we study this critical model from properties of random curves which in the scaling limit, appear as features seen on the macroscopic scale, in situations where the microscopic scale is taken to zero. Among the principal questions are the construction of the scaling limit, and the discription of some of the emergent properties, in particular the behavior under conformal maps Over the past few years, SLE has been developed as a valuable new tool to study the random paths of the scaling limit of two-dimensional critical models, and it is believed that SLE is the conformally invariant scaling limit of these...

#### Discrete Time vs Continuous Time Stock-price Dynamics and Implications for Option Pricing

, M.Sc. Thesis Sharif University of Technology ; Alishahi, Kasra (Supervisor) ; Zamani, Shiva (Supervisor)
Abstract

In the present paper we construct stock price processes with the same marginal log- normal law as that of a geometric Brownian motion and also with the same transition density (and returns’ distributions) between any two instants in a given discrete-time grid. We then illustrate how option prices based on such processes differ from Black and Scholes’, in that option prices can be either arbitrarily close to the option intrinsic value or arbitrarily close to the underlying stock price. We also explain that this is due to the particular way one models the stock-price process in between the grid time instants which are relevant for trading