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Arxiv:1906.06684V1 [Math.NA]
Randomized Computation of Continuous Data: Is Brownian Motion Computable?∗ Willem L. Fouch´e1, Hyunwoo Lee2, Donghyun Lim2, Sewon Park2, Matthias Schr¨oder3, Martin Ziegler2 1 University of South Africa 2 KAIST 3 University of Birmingham Abstract. We consider randomized computation of continuous data in the sense of Computable Analysis. Our first contribution formally confirms that it is no loss of generality to take as sample space the Cantor space of infinite fair coin flips. This extends [Schr¨oder&Simpson’05] and [Hoyrup&Rojas’09] considering sequences of suitably and adaptively biased coins. Our second contribution is concerned with 1D Brownian Motion (aka Wiener Process), a prob- ability distribution on the space of continuous functions f : [0, 1] → R with f(0) = 0 whose computability has been conjectured [Davie&Fouch´e’2013; arXiv:1409.4667,§6]. We establish that this (higher-type) random variable is computable iff some/every computable family of moduli of continuity (as ordinary random variables) has a computable probability distribution with respect to the Wiener Measure. 1 Introduction Randomization is a powerful technique in classical (i.e. discrete) Computer Science: Many difficult problems have turned out to admit simple solutions by algorithms that ‘roll dice’ and are efficient/correct/optimal with high probability [DKM+94,BMadHS99,CS00,BV04]. Indeed, fair coin flips have been shown computationally universal [Wal77]. Over continuous data, well-known closely connected to topology [Grz57] [Wei00, 2.2+ 3], notions of proba- § § bilistic computation are more subtle [BGH15,Col15]. 1.1 Overview Section 2 resumes from [SS06] the question of how to represent Borel probability measures. -
A Guide on Probability Distributions
powered project A guide on probability distributions R-forge distributions Core Team University Year 2008-2009 LATEXpowered Mac OS' TeXShop edited Contents Introduction 4 I Discrete distributions 6 1 Classic discrete distribution 7 2 Not so-common discrete distribution 27 II Continuous distributions 34 3 Finite support distribution 35 4 The Gaussian family 47 5 Exponential distribution and its extensions 56 6 Chi-squared's ditribution and related extensions 75 7 Student and related distributions 84 8 Pareto family 88 9 Logistic ditribution and related extensions 108 10 Extrem Value Theory distributions 111 3 4 CONTENTS III Multivariate and generalized distributions 116 11 Generalization of common distributions 117 12 Multivariate distributions 132 13 Misc 134 Conclusion 135 Bibliography 135 A Mathematical tools 138 Introduction This guide is intended to provide a quite exhaustive (at least as I can) view on probability distri- butions. It is constructed in chapters of distribution family with a section for each distribution. Each section focuses on the tryptic: definition - estimation - application. Ultimate bibles for probability distributions are Wimmer & Altmann (1999) which lists 750 univariate discrete distributions and Johnson et al. (1994) which details continuous distributions. In the appendix, we recall the basics of probability distributions as well as \common" mathe- matical functions, cf. section A.2. And for all distribution, we use the following notations • X a random variable following a given distribution, • x a realization of this random variable, • f the density function (if it exists), • F the (cumulative) distribution function, • P (X = k) the mass probability function in k, • M the moment generating function (if it exists), • G the probability generating function (if it exists), • φ the characteristic function (if it exists), Finally all graphics are done the open source statistical software R and its numerous packages available on the Comprehensive R Archive Network (CRAN∗). -
Handbook on Probability Distributions
R powered R-forge project Handbook on probability distributions R-forge distributions Core Team University Year 2009-2010 LATEXpowered Mac OS' TeXShop edited Contents Introduction 4 I Discrete distributions 6 1 Classic discrete distribution 7 2 Not so-common discrete distribution 27 II Continuous distributions 34 3 Finite support distribution 35 4 The Gaussian family 47 5 Exponential distribution and its extensions 56 6 Chi-squared's ditribution and related extensions 75 7 Student and related distributions 84 8 Pareto family 88 9 Logistic distribution and related extensions 108 10 Extrem Value Theory distributions 111 3 4 CONTENTS III Multivariate and generalized distributions 116 11 Generalization of common distributions 117 12 Multivariate distributions 133 13 Misc 135 Conclusion 137 Bibliography 137 A Mathematical tools 141 Introduction This guide is intended to provide a quite exhaustive (at least as I can) view on probability distri- butions. It is constructed in chapters of distribution family with a section for each distribution. Each section focuses on the tryptic: definition - estimation - application. Ultimate bibles for probability distributions are Wimmer & Altmann (1999) which lists 750 univariate discrete distributions and Johnson et al. (1994) which details continuous distributions. In the appendix, we recall the basics of probability distributions as well as \common" mathe- matical functions, cf. section A.2. And for all distribution, we use the following notations • X a random variable following a given distribution, • x a realization of this random variable, • f the density function (if it exists), • F the (cumulative) distribution function, • P (X = k) the mass probability function in k, • M the moment generating function (if it exists), • G the probability generating function (if it exists), • φ the characteristic function (if it exists), Finally all graphics are done the open source statistical software R and its numerous packages available on the Comprehensive R Archive Network (CRAN∗). -
A Complete Bibliography of Publications in Statistics and Probability Letters: 2010–2019
A Complete Bibliography of Publications in Statistics and Probability Letters: 2010{2019 Nelson H. F. Beebe University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA Tel: +1 801 581 5254 FAX: +1 801 581 4148 E-mail: [email protected], [email protected], [email protected] (Internet) WWW URL: http://www.math.utah.edu/~beebe/ 24 April 2020 Version 1.24 Title word cross-reference (1; 2) [Yan19a]. (α, β)[Kay15].(1; 1) [BO19]. (k1;k2) [UK18]. (n − k +1) [MF15, ZY10a]. (q; d) [BG16]. (X + Y ) [AV13b]. 1 [Hel19, Red17, SC15, Sze10]. 1=2[Efr10].f2g [Sma15, Jac13, JKD15a, JKD15b, Kha14, LM17, MK10, Ose14a, WL11]. 2k [Lu16a, Lu16b]. 2n−p [Yan13]. 2 × 2 [BB12, MBTC13]. 2 × c [SB16]. 3 [CGLN17, HWB10]. 3x + 1 [DP16]. > [HB18]. > 1=2 [DO11a]. [0; 1] [LP19a]. L1 q (` ) [Ose15a]. k [BML14]. r [VCM14]. A [CDS17, MPA12, PPM18, KDW15]. A1 [Ose13a]. α [AH12b, hCyP15, GT14, LT19b, MY14, MU13, Mic11, MM14b, PP14b, Sre12, Yan17b, ZZT17, ZYS19, ZZ13b, ZLC18, ZZ19a]. AR(1) [CL10, Mar16, PPN10]. AR(p) [WN14b]. ARH(1) [RMAL19].´ β [EM10, JW16, Kre18, Poi19, RBY10, Su10, XXY17, YZ16]. β !1[JW16]. 1 C[0; 1] [Mor18]. C[0; 1] [KM19]. C [LL17c]. D [CDS17, DS15, Jac11, KHN12, Sin19, Sma14, ZP12, Dav12b, OA15,¨ Rez15]. 1 2 Ds [KHN12]. D \ L [BS12a]. δ [Ery12, PB15a]. d ≥ 3 [DH19]. DS [WZ17]. d × R [BS19b]. E [DS15, HR19a, JKD14, KDW15]. E(fNOD)[CKF17].`1 N [AH11]. `1 [Ose19]. `p [Zen14]. [BCD19]. exp(x)[BM11].F [MMM13]. -
List of Glossary Terms
List of Glossary Terms A Ageofaclusterorfuzzyrule 1054 A correlated equilibrium 3064 Agent 58, 76, 105, 1767, 2578, 3004 Amechanism 1837 Agent architecture 105 Anearestneighbor 790 Agent based models 2940 A social choice function 1837 Agent (or software agent) 2999 A stochastic game 3064 Agent-based computational models 2898 Astrategy 3064 Agent-based model 58 Abelian group 2780 Agent-based modeling 1767 Absolute temperature 940 Agent-based modeling (ABM) 39 Absorbing state 1080 Agent-based simulation 18, 88 Abstract game 675 Aggregation 862 Abstract game of network formation with respect to Aggregation operators 122 irreflexive dominance 2044 Aging 2564, 2611 Abstract game of network formation with respect to path Algebra 2925 dominance 2044 Algebraic models 2898 Accuracy 161, 827 Algorithm 2496 Accuracy (rate) 862 Algorithmic complexity of object x 132 Achievable mate 3235 Algorithmic self-assembly of DNA tiles 1894 Action profile 3023 Allowable set of partners 3234 Action set 3023 Almost equicontinuous CA 914, 3212 Action type 2656 Alphabet of a cellular automaton 1 Activation function 813 ˛-Level set and support 1240 Activator 622 Alternating independent-2-paths 2953 Active membranes 1851 Alternating k-stars 2953 Actors 2029 Alternating k-triangles 2953 Adaptation 39 Amorphous computer 147 Adaptive system 1619 Analog 3187 Additive cellular automata 1 Analog circuit 3260 Additively separable preferences 3235 Ancilla qubits 2478 Adiabatic switching 1998 Anisotropic elements 754 Adjacency matrix 1746, 3114 Annealed law 2564 Adjacent 2864, -
Probability Theory Oral Exam Study Notes
Probability Theory Oral Exam study notes Notes transcribed by Mihai Nica Abstract. These are some study notes that I made while studying for my oral exams on the topic of Probability Theory. I took these notes from a few dierent sources, and they are not in any particular order. They tend to move around a lot. They also skip some of the basics of measure theory which are covered in the real analysis notes. Please be extremely caution with these notes: they are rough notes and were originally only for me to help me study. They are not complete and likely have errors. I have made them available to help other students on their oral exams (Note: I specialized in probability theory, so these go a bit further into a few topics that most people would do for their oral exam). See also the sections on Conditional Expectation and the Law of the Iterated Logarithm from my Limit Theorem II notes. Contents Independence and Weak Law of Large Numbers 5 1.1. Independence 5 1.2. Weak Law of Large Numbers 9 Borel Cantelli Lemmas 12 2.3. Borel Cantelli Lemmas 12 2.4. Bounded Convergence Theorem 15 Central Limit Theorems 17 3.5. The De Moivre-Laplace Theorem 17 3.6. Weak Convergence 17 3.7. Characteristic Functions 24 3.8. The moment problem 31 3.9. The Central Limit Theorem 33 3.10. Other Facts about CLT results 35 3.11. Law of the Iterated Log 36 Moment Methods 39 4.12. Basics of the Moment Method 39 4.13. -
Distribution Theory & Its Summability Perspective
1 2 Summer Workshop on Distribution Theory & its Summability Perspective 50 45 40 35 30 25 Frequencies 20 15 10 5 0 | 16 17 18 18.9 20 21 22 Center of mass Amount of Drink Mix (in ounces) M. Kazım Khan Kent State University (USA) Place: Ankara University, Department of Mathematics Dates: 16 May - 27 May 2011 Supported by: The Scientific and Technical Research Council of Turkey (TUBITAK)¨ 4 Preface This is a collection of lecture notes I gave at Ankara University, department of math- ematics, during the last two weeks of May, 2011. I am greatful to Professor Cihan Orhan and the Scientific and Technical Research Council of Turkey (TUBITAK)¨ for the invitation and support. The primary focus of the lectures was to introduce the basic components of distribution theory and bring out how summability theory plays out its role in it. I did not assume any prior knowledge of probability theory on the part of the participants. Therefore, the first few lectures were completely devoted to building the language of probability and distribution theory. These are then used freely in the rest of the lectures. To save some time, I did not prove most of these results. Then a few lectures deal with Fourier inversion theory specifically from the summability perspective. The next batch consists of convergence concepts, where I introduce the weak and the strong laws of large numbers. Again long proofs were omitted. A noteable exception deals with the results that involve the uniformly in- tegrable sequence spaces. Since this is a new concept from summability perspective, I have tried to sketch some of the proofs. -
2017 Timo Koski Department of Math
Lecture Notes: Probability and Random Processes at KTH for sf2940 Probability Theory Edition: 2017 Timo Koski Department of Mathematics KTH Royal Institute of Technology Stockholm, Sweden 2 Contents Foreword 9 1 Probability Spaces and Random Variables 11 1.1 Introduction...................................... .......... 11 1.2 Terminology and Notations in Elementary Set Theory . .............. 11 1.3 AlgebrasofSets.................................... .......... 14 1.4 ProbabilitySpace.................................... ......... 19 1.4.1 Probability Measures . ...... 19 1.4.2 Continuity from below and Continuity from above . .......... 21 1.4.3 Why Do We Need Sigma-Fields? . .... 23 1.4.4 P - Negligible Events and P -AlmostSureProperties . 25 1.5 Random Variables and Distribution Functions . ............ 25 1.5.1 Randomness?..................................... ...... 25 1.5.2 Random Variables and Sigma Fields Generated by Random Variables ........... 26 1.5.3 Distribution Functions . ...... 28 1.6 Independence of Random Variables and Sigma Fields, I.I.D. r.v.’s . ............... 29 1.7 TheBorel-CantelliLemmas ............................. .......... 30 1.8 ExpectedValueofaRandomVariable . ........... 32 1.8.1 A First Definition and Some Developments . ........ 32 1.8.2 TheGeneralDefinition............................... ....... 34 1.8.3 The Law of the Unconscious Statistician . ......... 34 1.8.4 Three Inequalities for Expectations . ........... 35 1.8.5 Limits and Integrals . ...... 37 1.9 Appendix: lim sup xn and lim inf xn ................................. -
Quantization for Infinite Affine Transformations
Quantization for infinite affine transformations Do˘ganC¸¨omez NDSU Department of Mathematics, Analysis Seminar January 26, 2021 Outline 1. Main problems of quantization theory 2. Probability measures and distributions 3. Probabilities associated to affine maps 4. Some relevant results 5. Infinite affine systems Outline: 1. Main Problems of quantization theory 2. Probability measures and distributions 3. Probabilities associated to affine maps 4. Some relevant results 5. Infinite affine systems 6. Main results (i): optimal sets and associated errors 7. Main results (ii): quantization dimension and coefficients Do˘ganC¸¨omez Quantization for infinite affine transformations Quantization: The process of approximating a given probability measure µ with a discrete probability measure of finite support. It concerns determining an appropriate partitioning of the underlying space for the discrete measure, and error analysis. Hence, main goals of the theory are: 1. find the optimal discrete measure that yields \good" approximation of the given probability to within an allowable margin of error, and 2. estimate the rate at which some specified measure of error goes to 0 as n ! 1: d (Throughout, all the measures considered will be on R with Euclidean norm k k:) Outline 1. Main problems of quantization theory 2. Probability measures and distributions 3. Probabilities associated to affine maps 4. Some relevant results 5. Infinite affine systems Introduction Do˘ganC¸¨omez Quantization for infinite affine transformations It concerns determining an appropriate partitioning of the underlying space for the discrete measure, and error analysis. Hence, main goals of the theory are: 1. find the optimal discrete measure that yields \good" approximation of the given probability to within an allowable margin of error, and 2. -
Arxiv:1808.01079V4 [Math.DS] 30 Jan 2019 Dimensional Persistent Homology for Xn
A FRACTAL DIMENSION FOR MEASURES VIA PERSISTENT HOMOLOGY HENRY ADAMS, MANUCHEHR AMINIAN, ELIN FARNELL, MICHAEL KIRBY, JOSHUA MIRTH, RACHEL NEVILLE, CHRIS PETERSON, AND CLAYTON SHONKWILER i Abstract. We use persistent homology in order to define a family of fractal dimensions, denoted dimPH(µ) for each homological dimension i ≥ 0, assigned to a probability measure µ on a metric space. The case of 0-dimensional homology (i = 0) relates to work by Michael J Steele (1988) studying the total length of a minimal spanning tree on a random sampling of points. Indeed, if µ is supported on a compact subset of m 0 Euclidean space R for m ≥ 2, then Steele's work implies that dimPH(µ) = m if the absolutely continuous 0 part of µ has positive mass, and otherwise dimPH(µ) < m. Experiments suggest that similar results may be true for higher-dimensional homology 0 < i < m, though this is an open question. Our fractal dimension is defined by considering a limit, as the number of points n goes to infinity, of the total sum of the i- dimensional persistent homology interval lengths for n random points selected from µ in an i.i.d. fashion. To some measures µ, we are able to assign a finer invariant, a curve measuring the limiting distribution of persistent homology interval lengths as the number of points goes to infinity. We prove this limiting curve exists in the case of 0-dimensional homology when µ is the uniform distribution over the unit interval, and conjecture that it exists when µ is the rescaled probability measure for a compact set in Euclidean space with positive Lebesgue measure. -
Al-Kashi Background Statistics
Al-Kashi Background Statistics http://www.ar-php.org/stats/al-kashi/ PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Sat, 01 Feb 2014 07:11:01 UTC Contents Articles Summary Statistics 1 Mean 1 Median 6 Mode (statistics) 15 Variance 20 Standard deviation 32 Coefficient of variation 46 Skewness 49 Kurtosis 55 Ranking 61 Graphics 65 Box plot 65 Histogram 68 Q–Q plot 74 Ternary plot 79 Distributions 84 Normal distribution 84 Student's t-distribution 113 F-distribution 128 Feature scaling 131 Correlation and Regression 134 Covariance 134 Correlation and dependence 138 Regression analysis 144 Path analysis (statistics) 154 Analysis 156 Moving average 156 Student's t-test 162 Contingency table 171 Analysis of variance 174 Principal component analysis 189 Diversity index 205 Diversity index 205 Clustering 210 Hierarchical clustering 210 K-means clustering 215 Matrix 225 Matrix (mathematics) 225 Matrix addition 247 Matrix multiplication 249 Transpose 263 Determinant 266 Minor (linear algebra) 284 Adjugate matrix 287 Invertible matrix 291 Eigenvalues and eigenvectors 298 System of linear equations 316 References Article Sources and Contributors 326 Image Sources, Licenses and Contributors 332 Article Licenses License 335 1 Summary Statistics Mean In mathematics, mean has several different definitions depending on the context. In probability and statistics, mean and expected value are used synonymously to refer to one measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution. In the case of a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability P(x), and then adding all these products together, giving .[1] An analogous formula applies to the case of a continuous probability distribution. -
Lecture 11: Random Variables: Types and CDF Lecturer: Dr
EE5110: Probability Foundations for Electrical Engineers July-November 2015 Lecture 11: Random Variables: Types and CDF Lecturer: Dr. Krishna Jagannathan Scribe: Sudharsan, Gopal, Arjun B, Debayani In this lecture, we will focus on the types of random variables. Random variables are categorized into various types, depending on the nature of the measure PX induced on the real line (or to be more precise, on the Borel σ-algebra). Indeed, there are three fundamentally different types of measures possible on the real line. According to an important theorem in measure theory, called the Lebesgue decomposition theorem (see Theorem 12.1.1 of [2]), any probability measure on R can be uniquely decomposed into a sum of these three types of measures. The three fundamental types of measure are • Discrete, • Continuous, and • Singular. In other words, there are three ‘pure type’ random variables, namely discrete random variables, continuous random variables, and singular random variables. It is also possible to ‘mix and match’ these three types to get four kinds of mixed random variables, altogether resulting in seven types of random variables. Of the three fundamental types of random variables, only the discrete and continuous random variables are important for practical applications in the field of engineering and statistics. Singular random variables are largely of academic interest. Therefore, we will spend most of our effort in studying discrete and continuous random variables, although we will define and give an example of a singular random variable. 11.1 Discrete Random Variables Definition 11.1 Discrete Random Variable: A random variable X is said to be discrete if it takes values in a countable subset of R with probability 1.