Probability I Fall 2011 Contents
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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. -
Weak Topologies
Weak topologies David Lecomte May 23, 2006 1 Preliminaries from general topology In this section, we are given a set X, a collection of topological spaces (Yi)i∈I and a collection of maps (fi)i∈I such that each fi maps X into Yi. We wish to define a topology on X that makes all the fi’s continuous. And we want to do this in the cheapest way, that is: there should be no more open sets in X than required for this purpose. −1 Obviously, all the fi (Oi), where Oi is an open set in Yi should be open in X. Then finite intersections of those should also be open. And then any union of finite intersections should be open. By this process, we have created as few open sets as required. Yet it is not clear that the collection obtained is closed under finite intersections. It actually is, as a consequence of the following lemma: Lemma 1 Let X be a set and let O ⊂ P(X) be a collection of subsets of X, such that • ∅ and X are in O; • O is closed under finite intersections. Then T = { O | O ⊂ O} is a topology on X. OS∈O Proof: By definition, T contains X and ∅ since those were already in O. Furthermore, T is closed under unions, again by definition. So all that’s left is to check that T is closed under finite intersections. Let A1 and A2 be two elements of T . Then there exist O1 and O2, subsets of O, such that A = O and A = O 1 [ 2 [ O∈O1 O∈O2 1 It is then easy to check by double inclusion that A ∩ A = O ∩ O 1 2 [ 1 2 O1∈O1 O2∈O2 Letting O denote the collection {O1 ∩ O2 | O1 ∈ O1 O2 ∈ O2}, which is a subset of O since the latter is closed under finite intersections, we get A ∩ A = O 1 2 [ O∈O This set belongs to T . -
Distinguished Property in Tensor Products and Weak* Dual Spaces
axioms Article Distinguished Property in Tensor Products and Weak* Dual Spaces Salvador López-Alfonso 1 , Manuel López-Pellicer 2,* and Santiago Moll-López 3 1 Department of Architectural Constructions, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] 2 Emeritus and IUMPA, Universitat Politècnica de València, 46022 Valencia, Spain 3 Department of Applied Mathematics, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] * Correspondence: [email protected] 0 Abstract: A local convex space E is said to be distinguished if its strong dual Eb has the topology 0 0 0 0 b(E , (Eb) ), i.e., if Eb is barrelled. The distinguished property of the local convex space Cp(X) of real- valued functions on a Tychonoff space X, equipped with the pointwise topology on X, has recently aroused great interest among analysts and Cp-theorists, obtaining very interesting properties and nice characterizations. For instance, it has recently been obtained that a space Cp(X) is distinguished if and only if any function f 2 RX belongs to the pointwise closure of a pointwise bounded set in C(X). The extensively studied distinguished properties in the injective tensor products Cp(X) ⊗# E and in Cp(X, E) contrasts with the few distinguished properties of injective tensor products related to the dual space Lp(X) of Cp(X) endowed with the weak* topology, as well as to the weak* dual of Cp(X, E). To partially fill this gap, some distinguished properties in the injective tensor product space Lp(X) ⊗# E are presented and a characterization of the distinguished property of the weak* dual of Cp(X, E) for wide classes of spaces X and E is provided. -
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]. -
Let H Be a Hilbert Space. on B(H), There Is a Whole Zoo of Topologies
Let H be a Hilbert space. On B(H), there is a whole zoo of topologies weaker than the norm topology – and all of them are considered when it comes to von Neumann algebras. It is, however, a good idea to concentrate on one of them right from the definition. My choice – and Murphy’s [Mur90, Chapter 4] – is the strong (or strong operator=STOP) topology: Definition. A von Neumann algebra is a ∗–subalgebra A ⊂ B(H) of operators acting nonde- generately(!) on a Hilbert space H that is strongly closed in B(H). (Every norm convergent sequence converges strongly, so A is a C∗–algebra.) This does not mean that one has not to know the other topologies; on the contrary, one has to know them very well, too. But it does mean that proof techniques are focused on the strong topology; if we use a different topology to prove something, then we do this only if there is a specific reason for doing so. One reason why it is not sufficient to worry only about the strong topology, is that the strong topology (unlike the norm topology of a C∗–algebra) is not determined by the algebraic structure alone: There are “good” algebraic isomorphisms between von Neumann algebras that do not respect their strong topologies. A striking feature of the strong topology on B(H) is that B(H) is order complete: Theorem (Vigier). If aλ λ2Λ is an increasing self-adjoint net in B(H) and bounded above (9c 2 B(H): aλ ≤ c8λ), then aλ converges strongly in B(H), obviously to its least upper bound in B(H). -
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,.. -
Chapter 14. Duality for Normed Linear Spaces
14.1. Linear Functionals, Bounded Linear Functionals, and Weak Topologies 1 Chapter 14. Duality for Normed Linear Spaces Note. In Section 8.1, we defined a linear functional on a normed linear space, a bounded linear functional, and the functional norm. In Proposition 8.1 (the proof is Exercise 8.2) it is shown that the collection of bounded linear functionals themselves form a normed linear space called the dual space of X, denoted X∗. In Chapters 14 and 15 we consider the mapping from X × X∗ → R defined by (x, ψ) 7→ ψ(x) to “uncover the analytic, geometric, and topological properties of Banach spaces.” The “departure point for this exploration” is the Hahn-Banach Theorem which is started and proved in Section 14.2 (Royden and Fitzpatrick, page 271). Section 14.1. Linear Functionals, Bounded Linear Functionals, and Weak Topologies Note. In this section we consider the linear space of all real valued linear function- als on linear space X (without requiring X to be named or the functionals to be bounded), denoted X]. We also consider a new topology on a normed linear space called the weak topology (the old topology which was induced by the norm we now may call the strong topology). For the deal X∗ of normed linear space X, the weak topology is called the weak-∗ topology. 14.1. Linear Functionals, Bounded Linear Functionals, and Weak Topologies 2 Note. Recall that if Y and Z are subspaces of a linear space then Y + Z is also a subspace of X (by Exercise 13.2) and that if Y ∩ Z = {0} then Y + Z is denoted T ⊕ Z and is called the direct sum of Y and Z. -
The Banach-Alaoglu Theorem for Topological Vector Spaces
The Banach-Alaoglu theorem for topological vector spaces Christiaan van den Brink a thesis submitted to the Department of Mathematics at Utrecht University in partial fulfillment of the requirements for the degree of Bachelor in Mathematics Supervisor: Fabian Ziltener date of submission 06-06-2019 Abstract In this thesis we generalize the Banach-Alaoglu theorem to topological vector spaces. the theorem then states that the polar, which lies in the dual space, of a neighbourhood around zero is weak* compact. We give motivation for the non-triviality of this theorem in this more general case. Later on, we show that the polar is sequentially compact if the space is separable. If our space is normed, then we show that the polar of the unit ball is the closed unit ball in the dual space. Finally, we introduce the notion of nets and we use these to prove the main theorem. i ii Acknowledgments A huge thanks goes out to my supervisor Fabian Ziltener for guiding me through the process of writing a bachelor thesis. I would also like to thank my girlfriend, family and my pet who have supported me all the way. iii iv Contents 1 Introduction 1 1.1 Motivation and main result . .1 1.2 Remarks and related works . .2 1.3 Organization of this thesis . .2 2 Introduction to Topological vector spaces 4 2.1 Topological vector spaces . .4 2.1.1 Definition of topological vector space . .4 2.1.2 The topology of a TVS . .6 2.2 Dual spaces . .9 2.2.1 Continuous functionals . -
Weak Topologies Weak-Type Topologies on Vector Spaces. Let X
Weak topologies Weak-type topologies on vector spaces. Let X be a vector space with the algebraic dual X]. Let Y ½ X] be a subspace. We want to de¯ne a topology σ on X in order to make continuous all elements of Y . Fix x0 2 X. If σ is such a topology, then the sets of the form x0 V";g := fx 2 X : jg(x) ¡ g(x0)j < "g = fx 2 X : jg(x ¡ x0)j < "g (" > 0; g 2 Y ) are open neighborhoods of x0. But this family is not a basis of σ-neighborhoods of x0, since the intersection of two of its members does not necessarily contain another member of the family. This is the reason why we instead consider the sets of the form x0 (1) V";g1;:::;gn := fx 2 X : jgi(x ¡ x0)j < "; i = 1; : : : ; ng (" > 0; n 2 N; gi 2 Y ) : x0 x0 It is easy to see that the intersection V \ V 0 of two of such sets ";g1;:::;gn " ;h1;:::;hm x0 00 0 contains V 00 where " = minf"; " g. " ;g1;:::;gn;h1;:::;hm Theorem 0.1. Let X be a vector space, and Y ½ X] a subspace which separates the points of X (that is, a so-called total subspace). 1. There exists a (unique) topology on X such that, for each x0 2 X, the sets (1) form a basis of neighborhoods of x0. This topology, denoted by σ(X; Y ), is called the weak topology determined by Y . 2. σ(X; Y ) is the weakest topology on X that makes continuous all elements of Y . -
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. -
Arxiv:Math/0405137V1 [Math.RT] 7 May 2004 Oino Disbecniuu Ersnain Fsc Group)
Draft: May 4, 2004 LOCALLY ANALYTIC VECTORS IN REPRESENTATIONS OF LOCALLY p-ADIC ANALYTIC GROUPS Matthew Emerton Northwestern University Contents 0. Terminology, notation and conventions 7 1. Non-archimedean functional analysis 10 2. Non-archimedean function theory 27 3. Continuous, analytic, and locally analytic vectors 43 4. Smooth, locally finite, and locally algebraic vectors 71 5. Rings of distributions 81 6. Admissible locally analytic representations 100 7. Representationsofcertainproductgroups 124 References 135 Recent years have seen the emergence of a new branch of representation theory: the theory of representations of locally p-adic analytic groups on locally convex p-adic topological vector spaces (or “locally analytic representation theory”, for short). Examples of such representations are provided by finite dimensional alge- braic representations of p-adic reductive groups, and also by smooth representations of such groups (on p-adic vector spaces). One might call these the “classical” ex- amples of such representations. One of the main interests of the theory (from the point of view of number theory) is that it provides a setting in which one can study p-adic completions of the classical representations [6], or construct “p-adic interpolations” of them (for example, by defining locally analytic analogues of the principal series, as in [20], or by constructing representations via the cohomology of arithmetic quotients of symmetric spaces, as in [9]). Locally analytic representation theory also plays an important role in the analysis of p-adic symmetric spaces; indeed, this analysis provided the original motivation for its development. The first “non-classical” examples in the theory were found by Morita, in his analysis of the p-adic upper half-plane (the p-adic symmetric arXiv:math/0405137v1 [math.RT] 7 May 2004 space attached to GL2(Qp)) [15], and further examples were found by Schneider and Teitelbaum in their analytic investigations of the p-adic symmetric spaces of GLn(Qp) (for arbitrary n) [19]. -
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 .