Probability Theory - Part 2 Independent Random Variables
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Modes of Convergence in Probability Theory
Modes of Convergence in Probability Theory David Mandel November 5, 2015 Below, fix a probability space (Ω; F;P ) on which all random variables fXng and X are defined. All random variables are assumed to take values in R. Propositions marked with \F" denote results that rely on our finite measure space. That is, those marked results may not hold on a non-finite measure space. Since we already know uniform convergence =) pointwise convergence this proof is omitted, but we include a proof that shows pointwise convergence =) almost sure convergence, and hence uniform convergence =) almost sure convergence. The hierarchy we will show is diagrammed in Fig. 1, where some famous theorems that demonstrate the type of convergence are in parentheses: (SLLN) = strong long of large numbers, (WLLN) = weak law of large numbers, (CLT) ^ = central limit theorem. In parameter estimation, θn is said to be a consistent ^ estimator or θ if θn ! θ in probability. For example, by the SLLN, X¯n ! µ a.s., and hence X¯n ! µ in probability. Therefore the sample mean is a consistent estimator of the population mean. Figure 1: Hierarchy of modes of convergence in probability. 1 1 Definitions of Convergence 1.1 Modes from Calculus Definition Xn ! X pointwise if 8! 2 Ω, 8 > 0, 9N 2 N such that 8n ≥ N, jXn(!) − X(!)j < . Definition Xn ! X uniformly if 8 > 0, 9N 2 N such that 8! 2 Ω and 8n ≥ N, jXn(!) − X(!)j < . 1.2 Modes Unique to Measure Theory Definition Xn ! X in probability if 8 > 0, 8δ > 0, 9N 2 N such that 8n ≥ N, P (jXn − Xj ≥ ) < δ: Or, Xn ! X in probability if 8 > 0, lim P (jXn − Xj ≥ 0) = 0: n!1 The explicit epsilon-delta definition of convergence in probability is useful for proving a.s. -
Arxiv:2102.05840V2 [Math.PR]
SEQUENTIAL CONVERGENCE ON THE SPACE OF BOREL MEASURES LIANGANG MA Abstract We study equivalent descriptions of the vague, weak, setwise and total- variation (TV) convergence of sequences of Borel measures on metrizable and non-metrizable topological spaces in this work. On metrizable spaces, we give some equivalent conditions on the vague convergence of sequences of measures following Kallenberg, and some equivalent conditions on the TV convergence of sequences of measures following Feinberg-Kasyanov-Zgurovsky. There is usually some hierarchy structure on the equivalent descriptions of convergence in different modes, but not always. On non-metrizable spaces, we give examples to show that these conditions are seldom enough to guarantee any convergence of sequences of measures. There are some remarks on the attainability of the TV distance and more modes of sequential convergence at the end of the work. 1. Introduction Let X be a topological space with its Borel σ-algebra B. Consider the collection M˜ (X) of all the Borel measures on (X, B). When we consider the regularity of some mapping f : M˜ (X) → Y with Y being a topological space, some topology or even metric is necessary on the space M˜ (X) of Borel measures. Various notions of topology and metric grow out of arXiv:2102.05840v2 [math.PR] 28 Apr 2021 different situations on the space M˜ (X) in due course to deal with the corresponding concerns of regularity. In those topology and metric endowed on M˜ (X), it has been recognized that the vague, weak, setwise topology as well as the total-variation (TV) metric are highlighted notions on the topological and metric description of M˜ (X) in various circumstances, refer to [Kal, GR, Wul]. -
Large Deviations for Stochastic Navier-Stokes Equations With
Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2013 Large deviations for stochastic Navier-Stokes equations with nonlinear viscosities Ming Tao Louisiana State University and Agricultural and Mechanical College, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Applied Mathematics Commons Recommended Citation Tao, Ming, "Large deviations for stochastic Navier-Stokes equations with nonlinear viscosities" (2013). LSU Doctoral Dissertations. 1558. https://digitalcommons.lsu.edu/gradschool_dissertations/1558 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. LARGE DEVIATIONS FOR STOCHASTIC NAVIER-STOKES EQUATIONS WITH NONLINEAR VISCOSITIES A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by Ming Tao B.S. in Math. USTC, 2004 M.S. in Math. USTC, 2007 May 2013 Acknowledgements This dissertation would not be possible without several contributions. First of all, I am extremely grateful to my advisor, Professor Sundar, for his constant encouragement and guidance throughout this work. Meanwhile, I really wish to express my sincere thanks to all the committee members, Professors Kuo, Richardson, Sage, Stoltzfus and the Dean's representa- tive Professor Koppelman, for their help and suggestions on the corrections of this dissertaion. I also take this opportunity to thank the Mathematics department of Louisiana State University for providing me with a pleasant working environment and all the necessary facilities. -
Noncommutative Ergodic Theorems for Connected Amenable Groups 3
NONCOMMUTATIVE ERGODIC THEOREMS FOR CONNECTED AMENABLE GROUPS MU SUN Abstract. This paper is devoted to the study of noncommutative ergodic theorems for con- nected amenable locally compact groups. For a dynamical system (M,τ,G,σ), where (M, τ) is a von Neumann algebra with a normal faithful finite trace and (G, σ) is a connected amenable locally compact group with a well defined representation on M, we try to find the largest non- commutative function spaces constructed from M on which the individual ergodic theorems hold. By using the Emerson-Greenleaf’s structure theorem, we transfer the key question to proving the ergodic theorems for Rd group actions. Splitting the Rd actions problem in two cases accord- ing to different multi-parameter convergence types—cube convergence and unrestricted conver- gence, we can give maximal ergodic inequalities on L1(M) and on noncommutative Orlicz space 2(d−1) L1 log L(M), each of which is deduced from the result already known in discrete case. Fi- 2(d−1) nally we give the individual ergodic theorems for G acting on L1(M) and on L1 log L(M), where the ergodic averages are taken along certain sequences of measurable subsets of G. 1. Introduction The study of ergodic theorems is an old branch of dynamical system theory which was started in 1931 by von Neumann and Birkhoff, having its origins in statistical mechanics. While new applications to mathematical physics continued to come in, the theory soon earned its own rights as an important chapter in functional analysis and probability. In the classical situation the sta- tionarity is described by a measure preserving transformation T , and one considers averages taken along a sequence f, f ◦ T, f ◦ T 2,.. -
Probability Theory Manjunath Krishnapur
Probability theory Manjunath Krishnapur DEPARTMENT OF MATHEMATICS,INDIAN INSTITUTE OF SCIENCE 2000 Mathematics Subject Classification. Primary ABSTRACT. These are lecture notes from the spring 2010 Probability theory class at IISc. There are so many books on this topic that it is pointless to add any more, so these are not really a substitute for a good (or even bad) book, but a record of the lectures for quick reference. I have freely borrowed a lot of material from various sources, like Durrett, Rogers and Williams, Kallenberg, etc. Thanks to all students who pointed out many mistakes in the notes/lectures. Contents Chapter 1. Measure theory 1 1.1. Probability space 1 1.2. The ‘standard trick of measure theory’! 4 1.3. Lebesgue measure 6 1.4. Non-measurable sets 8 1.5. Random variables 9 1.6. Borel Probability measures on Euclidean spaces 10 1.7. Examples of probability measures on the line 11 1.8. A metric on the space of probability measures on Rd 12 1.9. Compact subsets of P (Rd) 13 1.10. Absolute continuity and singularity 14 1.11. Expectation 16 1.12. Limit theorems for Expectation 17 1.13. Lebesgue integral versus Riemann integral 17 1.14. Lebesgue spaces: 18 1.15. Some inequalities for expectations 18 1.16. Change of variables 19 1.17. Distribution of the sum, product etc. 21 1.18. Mean, variance, moments 22 Chapter 2. Independent random variables 25 2.1. Product measures 25 2.2. Independence 26 2.3. Independent sequences of random variables 27 2.4. -
A Brief on Characteristic Functions
Missouri University of Science and Technology Scholars' Mine Graduate Student Research & Creative Works Student Research & Creative Works 01 Dec 2020 A Brief on Characteristic Functions Austin G. Vandegriffe Follow this and additional works at: https://scholarsmine.mst.edu/gradstudent_works Part of the Applied Mathematics Commons, and the Probability Commons Recommended Citation Vandegriffe, Austin G., "A Brief on Characteristic Functions" (2020). Graduate Student Research & Creative Works. 2. https://scholarsmine.mst.edu/gradstudent_works/2 This Presentation is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Graduate Student Research & Creative Works by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. A Brief on Characteristic Functions A Presentation for Harmonic Analysis Missouri S&T : Rolla, MO Presentation by Austin G. Vandegriffe 2020 Contents 1 Basic Properites of Characteristic Functions 1 2 Inversion Formula 5 3 Convergence & Continuity of Characteristic Functions 9 4 Convolution of Measures 13 Appendix A Topology 17 B Measure Theory 17 B.1 Basic Measure Theory . 17 B.2 Convergence in Measure & Its Consequences . 20 B.3 Derivatives of Measures . 22 C Analysis 25 i Notation 8 For all 8P For P-almost all, where P is a measure 9 There exists () If and only if U Disjoint union -
Sequences and Series of Functions, Convergence, Power Series
6: SEQUENCES AND SERIES OF FUNCTIONS, CONVERGENCE STEVEN HEILMAN Contents 1. Review 1 2. Sequences of Functions 2 3. Uniform Convergence and Continuity 3 4. Series of Functions and the Weierstrass M-test 5 5. Uniform Convergence and Integration 6 6. Uniform Convergence and Differentiation 7 7. Uniform Approximation by Polynomials 9 8. Power Series 10 9. The Exponential and Logarithm 15 10. Trigonometric Functions 17 11. Appendix: Notation 20 1. Review Remark 1.1. From now on, unless otherwise specified, Rn refers to Euclidean space Rn n with n ≥ 1 a positive integer, and where we use the metric d`2 on R . In particular, R refers to the metric space R equipped with the metric d(x; y) = jx − yj. (j) 1 Proposition 1.2. Let (X; d) be a metric space. Let (x )j=k be a sequence of elements of X. 0 (j) 1 Let x; x be elements of X. Assume that the sequence (x )j=k converges to x with respect to (j) 1 0 0 d. Assume also that the sequence (x )j=k converges to x with respect to d. Then x = x . Proposition 1.3. Let a < b be real numbers, and let f :[a; b] ! R be a function which is both continuous and strictly monotone increasing. Then f is a bijection from [a; b] to [f(a); f(b)], and the inverse function f −1 :[f(a); f(b)] ! [a; b] is also continuous and strictly monotone increasing. Theorem 1.4 (Inverse Function Theorem). Let X; Y be subsets of R. -
Basic Functional Analysis Master 1 UPMC MM005
Basic Functional Analysis Master 1 UPMC MM005 Jean-Fran¸coisBabadjian, Didier Smets and Franck Sueur October 18, 2011 2 Contents 1 Topology 5 1.1 Basic definitions . 5 1.1.1 General topology . 5 1.1.2 Metric spaces . 6 1.2 Completeness . 7 1.2.1 Definition . 7 1.2.2 Banach fixed point theorem for contraction mapping . 7 1.2.3 Baire's theorem . 7 1.2.4 Extension of uniformly continuous functions . 8 1.2.5 Banach spaces and algebra . 8 1.3 Compactness . 11 1.4 Separability . 12 2 Spaces of continuous functions 13 2.1 Basic definitions . 13 2.2 Completeness . 13 2.3 Compactness . 14 2.4 Separability . 15 3 Measure theory and Lebesgue integration 19 3.1 Measurable spaces and measurable functions . 19 3.2 Positive measures . 20 3.3 Definition and properties of the Lebesgue integral . 21 3.3.1 Lebesgue integral of non negative measurable functions . 21 3.3.2 Lebesgue integral of real valued measurable functions . 23 3.4 Modes of convergence . 25 3.4.1 Definitions and relationships . 25 3.4.2 Equi-integrability . 27 3.5 Positive Radon measures . 29 3.6 Construction of the Lebesgue measure . 34 4 Lebesgue spaces 39 4.1 First definitions and properties . 39 4.2 Completeness . 41 4.3 Density and separability . 42 4.4 Convolution . 42 4.4.1 Definition and Young's inequality . 43 4.4.2 Mollifier . 44 4.5 A compactness result . 45 5 Continuous linear maps 47 5.1 Space of continuous linear maps . 47 5.2 Uniform boundedness principle{Banach-Steinhaus theorem . -
The Gap Between Gromov-Vague and Gromov-Hausdorff-Vague Topology
THE GAP BETWEEN GROMOV-VAGUE AND GROMOV-HAUSDORFF-VAGUE TOPOLOGY SIVA ATHREYA, WOLFGANG LOHR,¨ AND ANITA WINTER Abstract. In [ALW15] an invariance principle is stated for a class of strong Markov processes on tree-like metric measure spaces. It is shown that if the underlying spaces converge Gromov vaguely, then the processes converge in the sense of finite dimensional distributions. Further, if the underly- ing spaces converge Gromov-Hausdorff vaguely, then the processes converge weakly in path space. In this paper we systematically introduce and study the Gromov-vague and the Gromov-Hausdorff-vague topology on the space of equivalence classes of metric boundedly finite measure spaces. The latter topology is closely related to the Gromov-Hausdorff-Prohorov metric which is defined on different equivalence classes of metric measure spaces. We explain the necessity of these two topologies via several examples, and close the gap between them. That is, we show that convergence in Gromov- vague topology implies convergence in Gromov-Hausdorff-vague topology if and only if the so-called lower mass-bound property is satisfied. Further- more, we prove and disprove Polishness of several spaces of metric measure spaces in the topologies mentioned above (summarized in Figure 1). As an application, we consider the Galton-Watson tree with critical off- spring distribution of finite variance conditioned to not get extinct, and construct the so-called Kallenberg-Kesten tree as the weak limit in Gromov- Hausdorff-vague topology when the edge length are scaled down to go to zero. Contents 1. Introduction 1 2. The Gromov-vague topology 6 3. -
Review of Local and Global Existence Results for Stochastic Pdes with Lévy Noise
DISCRETE AND CONTINUOUS doi:10.3934/dcds.2020241 DYNAMICAL SYSTEMS Volume 40, Number 10, October 2020 pp. 5639–5710 REVIEW OF LOCAL AND GLOBAL EXISTENCE RESULTS FOR STOCHASTIC PDES WITH LÉVY NOISE Justin Cyr1, Phuong Nguyen1;2, Sisi Tang1 and Roger Temam1;∗ 1 Department of Mathematics Indiana University Swain East Bloomington, IN 47405, USA 2 Department of Mathematics and Statistics Texas Tech University Lubbock, TX 79409, USA (Communicated by Nikolay Tzvetkov) Abstract. This article is a review of Lévy processes, stochastic integration and existence results for stochastic differential equations and stochastic partial differential equations driven by Lévy noise. An abstract PDE of the typical type encountered in fluid mechanics is considered in a stochastic setting driven by a general Lévy noise. Existence and uniqueness of a local pathwise solution is established as a demonstration of general techniques in the area. 1. Introduction. In this article we review results of existence and uniqueness of local pathwise solutions to stochastic partial differential equations (SPDEs) driven by a Lévy noise. Lévy processes are canonical generalizations of Wiener processes that possess jump discontinuities. Lévy noise is fitting in stochastic models of fluid dynamics as a way to represent interactions that occur as abrupt jolts at random times, such as bursts of wind. Models based on SPDEs with Lévy noise have been used to describe observational records in paleoclimatology in [14], with the jump events of the Lévy noise proposed as representing abrupt triggers for shifts between glacial and warm climate states. The articles [40] and [24] use processes with jumps to model phase transitions in bursts of wind that contribute to the dynamics of El Niño. -
Almost Uniform Convergence Versus Pointwise Convergence
ALMOST UNIFORM CONVERGENCE VERSUS POINTWISE CONVERGENCE JOHN W. BRACE1 In many an example of a function space whose topology is the topology of almost uniform convergence it is observed that the same topology is obtained in a natural way by considering pointwise con- vergence of extensions of the functions on a larger domain [l; 2]. This paper displays necessary conditions and sufficient conditions for the above situation to occur. Consider a linear space G(5, F) of functions with domain 5 and range in a real or complex locally convex linear topological space £. Assume that there are sufficient functions in G(5, £) to distinguish between points of 5. Let Sß denote the closure of the image of 5 in the cartesian product space X{g(5): g£G(5, £)}. Theorems 4.1 and 4.2 of reference [2] give the following theorem. Theorem. If g(5) is relatively compact for every g in G(5, £), then pointwise convergence of the extended functions on Sß is equivalent to almost uniform converqence on S. When almost uniform convergence is known to be equivalent to pointwise convergence on a larger domain the situation can usually be converted to one of equivalence of the two modes of convergence on the same domain by means of Theorem 4.1 of [2]. In the new formulation the following theorem is applicable. In preparation for the theorem, let £(5, £) denote all bounded real valued functions on S which are uniformly continuous for the uniformity which G(5, F) generates on 5. G(5, £) will be called a full linear space if for every/ in £(5, £) and every g in GiS, F) the function fg obtained from their pointwise product is a member of G (5, £). -
ACM 217 Handout: Completeness of Lp the Following
ACM 217 Handout: Completeness of Lp The following statements should have been part of section 1.5 in the lecture notes; unfortunately, I forgot to include them. p Definition 1. A sequence {Xn} of random variables in L , p ≥ 1 is called a Cauchy p sequence (in L ) if supm,n≥N kXm − Xnkp → 0 as N → ∞. p p Proposition 2 (Completeness of L ). Let Xn be a Cauchy sequence in L . Then there p p exists a random variable X∞ ∈ L such that Xn → X∞ in L . When is such a result useful? All our previous convergence theorems, such as the dominated convergence theorem etc., assume that we already know that our sequence converges to a particular random variable Xn → X in some sense; they tell us how to convert between the various modes of convergence. However, we are often just given a sequence Xn, and we still need to establish that Xn converges to something. Proving that Xn is a Cauchy sequence is one way to show that the sequence converges, without knowing in advance what it converges to. We will encounter another way to show that a sequence converges in the next chapter (the martingale convergence theorem). Remark 3. As you know from your calculus course, Rn also has the property that any Cauchy sequence converges: if supm,n≥N |xm − xn| → 0 as N → ∞ for some n n sequence {xn} ⊂ R , then there is some x∞ ∈ R such that xn → x∞. In fact, many (but not all) metric spaces have this property, so it is not shocking that it is true also for Lp.