DOCSLIB.ORG

Heavy Tailed Analysis Eurandom Summer 2005

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Heavy Tailed Analysis Eurandom Summer 2005 HEAVY TAILED ANALYSIS EURANDOM SUMMER 2005 SIDNEY RESNICK School of Operations Research and Industrial Engineering Cornell University Ithaca, NY 14853 USA [email protected] http://www.orie.cornell.edu/»sid & Eurandom http://www.eurandom.tue.nl/people/EURANDOM_chair/eurandom_chair.htm 1 80 SIDNEY RESNICK 11. The Laplace functional. The Laplace functional is a transform technique which is useful for manipulating distri- butions of random measures and point processes. When applied to Poisson processes and empirical measures, algebraic manipulations become familiar to ones used with either char- acteristic functions or Laplace transforms applied to sums of iid random variables. Continue to assume the state space of the random measures or point processes is the nice space E. For a non-negative, bounded measureable function f : E 7! R+ and for ¹ 2 M+(E), we continue to use the notation Z ¹(f) = f(x)d¹(x): x2E P For a point measure m = i ²xi , this is Z X m(f) = f(x)m(dx) = f(xi): x2E i If we think of f ranging over all the non-negative bounded functions, m(f) yields all the information contained in m; certainly we learn about the value of m on each set A 2 E since we can always set f = 1A. So integrals of measures with respect to arbitrary test functions contain as much information as evaluating the measures on arbitrary sets. De¯nition 3 (Laplace functional). Suppose B+ are the non-negative, bounded, measurable functions from E 7! R+ and let » : (­; A; P) 7! (M+(E); M+(E)) be a random measure. The Laplace functional of » is the non-negative function on B+ given by Z Z ¡1 ª»(f) = E expf¡»(f)g = expf¡N(!; f)dP(!) = expf¡m(f)P ± N (dm): ­ M+(E) Note if P is a probability measure on Mp(E), its Laplace functional is Z expf¡m(f)P (dm); f 2 B+: Mp(E) Proposition 10. The Laplace functional of » uniquely determines the distribution of », ¡1 namely P ± » on M+(E)). Proof. The topology (and hence the Borel σ-algebra on M+(E) is generated by basic open sets of the form f¹ 2 M+(E): ¹(fi) 2 Ii; i = 1; : : : ; kg; + where fi 2 CK (E) and Ii are bounded open intervals. (Review page 33. ) This class of sets is closed under ¯nite intersection and hence is a ¦-system generating the σ-algebra. So from (Resnick, 1998, page 38) is su±ces to show that P ± »¡1 is uniquely determined on this class by the Laplace functional. HEAVY TAIL ANALYSIS 81 Pk For f = i=1 ¸ifi, with ¸i > 0, we have Pk ¡»(f) ¡ ¸i»(fi) û(f) = Ee = Ee i=1 : So û determines the joint distribution of the random vector (»(f1);:::;»(fk)) and hence ¡1 P ± » f¹ : ¹(fi) 2 Ii; i = 1; : : : ; kg = P[(»(f1);:::;»(fk)) 2 ¢] is determined, as required. ¤ 11.1. The Laplace functional of the Poisson process. Recall De¯nition 2 of the Poisson process and the two parts 1 and 2 of the de¯nition. The Poisson process can be identi¯ed by the characteristic form of its Laplace functional as discussed next. Theorem 8 (Laplace functional of PRM). The distribution of PRM(¹) is uniquely deter- mined by 1 and 2 in De¯nition 2. Furthermore, the point process N is PRM(¹) i® its Laplace functional is of the form Z ¡f(x) (11.1) ªN (f) = expf¡ (1 ¡ e )¹(dx)g; f 2 B+: E So PRM(¹) can be identi¯ed by the characteristic form of its Laplace functional. Proof. We ¯rst show (1) and (2) imply (11.1). If f = ¸1A where ¸ > 0 then because N(f) = ¸N(A) and N(A) is Poisson with parameter ¹(A) we get ³ ´ ¡¸N(A) ¡¸ ªN (f) =E e = expf(e ¡ 1)¹(A)g Z = expf¡ (1 ¡ e¡f(x))¹(dx)g E which is the correct form given in (11.1). Next suppose f has a somewhat more complex form Xk f = ¸i1Ai i=1 where ¸i ¸ 0;Ai 2 E; 1 · i · k and A1;:::;Ak are disjoint. Then à ( )! Xk ªN (f) =E exp ¡ ¸iN(Ai) i=1 Yk ³ ´ = E expf¡¸iN(Ai)g from independence i=1 Yk ½ Z ¾ ¡¸ 1 (x) = exp ¡ (1 ¡ e i Ai )¹(dx) from the previous step i=1 E 82 SIDNEY RESNICK ( ) Z Xk ¡¸ 1 (x) = exp (1 ¡ e i Ai )¹(dx) E ½Z i=1 ¾ P ¡ k ¸ 1 (x) = exp (1 ¡ e i=1 i Ai )¹(dx) ½ZE ¾ = exp (1 ¡ e¡f(x))¹(dx) E which again veri¯es (11.1). Now the last step is to take general f 2 B+ and verify (11.1) for such f. We may approximate f from below by simple fn of the form just considered. We may take for instance n Xn2 i ¡ 1 fn(x) = 1[ i¡1 ; i )(f(x)) + n1[n;1)(f(x)) 2n 2n 2n i=1 so that 0 · fn(x) " f(x): ¡f By monotone convergence N(fn) " N(f) and since e · 1 we get by dominated convergence that ªN (f) = lim ªN (fn): n!1 We have from the previous step that ½ Z ¾ ¡fn(x) ªN (fn) = exp ¡ (1 ¡ e )¹(dx) : E Since 1 ¡ e¡fn " 1 ¡ e¡f we conclude by monotone convergence that Z Z (1 ¡ e¡fn(x)¹(dx) " (1 ¡ e¡f(x)¹(dx) E E and thus we conclude (11.1) holds for any f 2 B+. Since the distribution of N is uniquely determined by ªN we have shown that 1 and 2 in De¯nition 2 determine the distribution of N. Conversely if the Laplace functional of N is given by (11.1) then N(A) must be Poisson distributed with parameter ¹(A) for any A 2 E which is readily checked by substituting f = ¸1A in (11.1) to get a Laplace transform of a Poisson distribution. Furthermore if Pk A1;:::;Ak are disjoint sets in E and f = i=1 ¸i1Ai then substituting in (11.1) gives ½ Z ¾ P P ¡ k ¸ N(A ) ¡ k ¸ 1 Ee i=1 i i = exp ¡ (1 ¡ e i=1 i Ai )d¹ ( E ) Z Xk ¡¸ 1 = exp ¡ (1 ¡ e i Ai )d¹ E i=1 HEAVY TAIL ANALYSIS 83 Yk ¡¸i = expf¡(1 ¡ e )¹(Ai)g i=1 Yk = Ee¡¸iN(Ai) i=1 and so the joint Laplace transform of (N(Ai); 1 · i · k) factors into a product of Laplace transforms and this shows independence. ¤ 11.1.1. Application: A general construction of the Poisson process. When E = [0; 1), we know the renewal theory construction yields a Poisson process with Lebesgue measure as the mean measure. Here is a general scheme for constructing a Poisson process with mean measure ¹. Start by supposing that ¹(E) < 1. De¯ne the probability measure F F (dx) = ¹(dx)=¹(E) on E. Let fXn; n ¸ 1g be iid random elements of E with common distribution F and let º be independent of fXng with a Poisson distribution with parameter ¹(E). De¯ne (P º ² ; if º ¸ 1 N = i=1 Xi 0; if º = 0: We claim N is PRM(¹). Verify this by computing the Laplace functional of N: Pº ¡ f(Xi) ªN (f) =Ee i=1 1 X P ¡ j f(X ) = Ee i=1 i P [º = j] j=0 1 X ¡ ¢j = Ee¡f(X1) P [º = j] j=0 © ª = exp ¹(E)(Ee¡f(X1) ¡ 1) ½ µ Z ¶¾ ¹(dx) = exp ¡¹(E) 1 ¡ e¡f(x) ¹(E) ½ Z E ¾ ¡ ¢ = exp ¡ 1 ¡ e¡f(x) ¹(dx) : E The Laplace functional has the correct form and hence N is indeed PRM(¹). Note that what the construction does is to toss points at random into E according to distribution ¹(dx)=¹(E); the number of points tossed is Poisson with parameter ¹(E). When the condition ¹(E) < 1 fails, we proceed as follows to make a minor modi¯cation in the foregoing construction: Decompose E into disjoint bits E1; E2;::: so that E = [iEi and ¹(Ei) < 1 for each i. Let ¹i(dx) = ¹(dx)1Ei (x) and let Ni be PRM(¹i) on E (do the 84 SIDNEY RESNICK constructionP just outlined) and arrange things so the collection fNig is independent. De¯ne N := i Ni and N is PRM(¹) since Y ªN (f) = ªNi (f) i Y ½ Z ¾ ¡ ¡f(x)¢ = exp ¡ 1 ¡ e ¹i(dx) Ei i ( ) X Z ¡ ¡f(x)¢ = exp ¡ 1 ¡ e ¹i(dx) E ( i ) Z X ¡ ¡f(x)¢ = exp ¡ 1 ¡ e ¹i(dx) E ½ Z i ¾ ¡ ¢ = exp ¡ 1 ¡ e¡f(x) ¹(dx) E P since i ¹i = ¹: This completes the construction. 11.2. Weak convergence of point processes and random measures. We now discuss weak convergence in M+(E) and specialize to give criterion for weak convergence to a Poisson process. 11.2.1. Basic criterion via Laplace functionals. Since M+(E) can be metrized as a complete separable metric space, the theory of weak convergence applies. A useful criterion for our purposes is in terms of Laplace functionals. Theorem 9 (Convergence criterion). Let f»n; n ¸ 0g be random elements of M+(E). Then »n ) »0 in Mp(E); i® ¡»n(f) ¡»0(f) + (11.2) ûn (f) = Ee ! Ee = û0 (f); 8f 2 CK (E): + So weak convergence is characterized by convergence of Laplace functionals on CK (E). Proof. Here is the proof of the easy half of the equivalence. Suppose »n ) »0 in M+(E). The map M+(E) 7! [0; 1) de¯ned by ¹ 7! ¹(f) is continuous, so the continuous mapping theorem gives »n(f) ) »0(f) in R.
Load more

Recommended publications
  • The Concentration of Measure Phenomenon
    http://dx.doi.org/10.1090/surv/089 Selected Titles in This Series 89 Michel Ledoux, The concentration of measure phenomenon, 2001 88 Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, 2001 87 Bruno Poizat, Stable groups, 2001 86 Stanley N. Burris, Number theoretic density and logical limit laws, 2001 85 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, 2001 84 Laszlo Fuchs and Luigi Salce, Modules over non-Noetherian domains, 2001 83 Sigurdur Helgason, Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions, 2000 82 Goro Shimura, Arithmeticity in the theory of automorphic forms, 2000 81 Michael E. Taylor, Tools for PDE: Pseudodifferential operators, paradifferential operators, and layer potentials, 2000 80 Lindsay N. Childs, Taming wild extensions: Hopf algebras and local Galois module theory, 2000 79 Joseph A. Cima and William T. Ross, The backward shift on the Hardy space, 2000 78 Boris A. Kupershmidt, KP or mKP: Noncommutative mathematics of Lagrangian, Hamiltonian, and integrable systems, 2000 77 Fumio Hiai and Denes Petz, The semicircle law, free random variables and entropy, 2000 76 Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmuller theory, 2000 75 Greg Hjorth, Classification and orbit equivalence relations, 2000 74 Daniel W. Stroock, An introduction to the analysis of paths on a Riemannian manifold, 2000 73 John Locker, Spectral theory of non-self-adjoint two-point differential operators, 2000 72 Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, 1999 71 Lajos Pukanszky, Characters of connected Lie groups, 1999 70 Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems and differential equations, 1999 69 C.
    [Show full text]
  • Electron. Commun. Probab. 0 (2016), No. 0, DOI: 10.1214/ECP
    Electron. Commun. Probab. 0 (2016), no. 0, 1–16. ELECTRONIC DOI: 10.1214/ECP.10.1214/YY-TN COMMUNICATIONS ISSN: 1083-589X in PROBABILITY Point processes in a metric space* Yuwei Zhao† Abstract As a useful and elegant tool of extreme value theory, the study of point processes on a metric space is important and necessary for the analyses of heavy-tailed functional data. This paper focuses on the definition and properties of such point processes. A complete convergence result for a regularly varying iid sequence in a metric space is proved as an example of the application in extreme value theory. Keywords: point process; functional data; M0-convergence; regular variation. AMS MSC 2010: 60G55; 60G70;60G57. Submitted to ECP on NA, final version accepted on NA. 1 Introduction Point process theory is considered as a useful tool in the analyses of extremal events of stochastic processes due to its close connection to the concept of regular variation, which plays an important role in modelling extremal events. For example, for an iid copies (Xi)i>1 of a random element X, it is well known that regular variation of X is equivalent to the convergence of point processes n X Nn = δ(i=n;Xi=an) i=1 towards a suitable Poisson point process. This result is referred to as the complete convergence result; see for example [17]. A large number of literatures focus on the extension of this result and its applications; see [9, 10, 1, 3], to name a few. There are also a large amount of literatures on the applications of regular variation in extreme value theory, such as [12, 18, 2, 11], to name a few.
    [Show full text]
  • Stochastic Ordering of Interferences in Large-Scale Wireless Networks
    1 Stochastic Ordering of Interference in Large-scale Wireless Networks Junghoon Lee and Cihan Tepedelenlioglu,˘ Member, IEEE Abstract Stochastic orders are binary relations defined on probability distributions which capture intuitive notions like being larger or being more variable. This paper introduces stochastic ordering of interference distributions in large-scale networks modeled as point processes. Interference is a major performance- limiting factor in most wireless networks, thus it is important to characterize its statistics. Since closed- form results for the distribution of interference for such networks are only available in limited cases, it is of interest to compare network interference using stochastic orders, for two different point processes with different fading or path-loss scenarios between the interferers and the receiver. In this paper, conditions on the fading distribution and path-loss model are given to establish stochastic ordering between interferences. Moreover, Laplace functional ordering is defined between point processes and applied for comparing interference. Monte-Carlo simulations are used to supplement our analytical results. arXiv:1204.6341v2 [cs.IT] 24 Oct 2012 Index Terms Interference, point process, stochastic order. The authors are with the School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA. (Email:{junghoon.lee,cihan}@asu.edu). April 19, 2021 DRAFT 2 I. INTRODUCTION Since interference is the main performance-limiting factor in most wireless networks, it is crucial to characterize its statistics. The interference mainly depends on the fading channel (interfering power distribution), the path-loss model (signal attenuation with distance), and network geometry (spatial distribution of concurrently transmitting nodes). The spatial location of the interferers can be modeled either deterministically or stochastically.
    [Show full text]
  • Lecture-27: Properties of Poisson Point Processes
    Lecture-27: Properties of Poisson point processes 1 Laplace functional For a realization of a simple point process S, we observe that dN(x) = 0 for all x 2/ S and dN(x) = d dx1fx2Sg. Hence, for any function f : R ! R, we have Z f (x)dN(x) = f (S )1 = f (S ). ∑ i fSi2Ag ∑ i x2A i2N Si2A Definition 1.1. The Laplace functional L of a point process S and associated counting process N is defined for all non-negative function f : Rd ! R as Z LS( f ) , Eexp − f (x)dN(x) . Rd Remark . f (x) = k t 1 For simple function ∑i=1 i1fx2Aig, we can write the Laplace functional k Z ! k ! LS( f ) = Eexp − ∑ ti dN(x) = Eexp − ∑ ti N(Ai) , i=1 Ai i=1 as a function of the vector (t1,t2,...,tk), a joint Laplace transform of the random vector (N(A1),..., N(Ak)). This way, one can compute all finite dimensional distribution of the counting process N. Proposition 1.2. The Laplace functional of the Poisson process with intensity measure L is Z − f (x) LS( f ) = exp − (1 − e )dL(x) . Rd d Proof. For a bounded Borel measurable set A 2 B(R ), consider g(x) = f (x)1fx2Ag. Then, Z Z LS(g) = Eexp(− g(x)dN(x)) = Eexp(− f (x)dN(x)). Rd A ( ) = L ( ) = E − ( ) Clearly dN x dx1fx2Sg and hence we can write S g exp ∑Si2S\A f Si . We know that the probability of N(A) = jS(A)j = n points in set A is given by L(A)n PfN(A) = ng = e−L(A) .
    [Show full text]
  • Arxiv:0803.1266V3 [Math-Ph] 10 Nov 2009 N N Mrvmn Eursabte Nesadn Fted the of Understanding Randomness
    DIFFRACTION OF STOCHASTIC POINT SETS: EXPLICITLY COMPUTABLE EXAMPLES MICHAEL BAAKE, MATTHIAS BIRKNER, AND ROBERT V. MOODY Abstract. Stochastic point processes relevant to the theory of long-range aperiodic order are considered that display diffraction spectra of mixed type, with special emphasis on explic- itly computable cases together with a unified approach of reasonable generality. The latter is based on the classical theory of point processes and the Palm distribution. Several pairs of autocorrelation and diffraction measures are discussed which show a duality structure analogous to that of the Poisson summation formula for lattice Dirac combs. 1. Introduction The discoveries of quasicrystals [53], aperiodic tilings [47, 39], and complex metallic alloys [60] have greatly increased our awareness that there is a substantial difference between the notions of periodicity and long-range order. Although pinning an exact definition to the con- cept of long-range order is not yet possible (nor perhaps desirable at this intermediate stage, compare the discussion in [56]), there is still some general agreement that the appearance of a substantial point-like component in the diffraction of a structure is a strong, though not a necessary, indicator of the phenomenon. Mathematically, the diffraction – say of a point set Λ in R3 – is the measure γ on R3 which is the Fourier transform of the volume averaged autocorrelation γ of Λ (or, more precisely, of its Dirac comb δΛ = x∈Λ δx). Over the past 20 years or so, considerableb effort has been put into understanding the mathematics of diffraction, especially conditions P under which Λ is pure point diffractive, in the sense that γ is a pure point measure, compare [32, 16, 55, 2, 8, 29, 42, 43].
    [Show full text]
  • Concentration of Measure for the Analysis of Randomised Algorithms
    Concentration of Measure for the Analysis of Randomised Algorithms Devdatt P. Dubhashi Alessandro Panconesi October 21, 2005 DRAFT 2 DRAFT Contents 1 Chernoff–Hoeffding Bounds 17 1.1 Whatis“ConcentrationofMeasure”?. 17 1.2 TheBinomialDistribution . 19 1.3 TheChernoffBound ......................... 19 1.4 HeterogeneousVariables . 21 1.5 TheHoeffdingExtension . 22 1.6 UsefulFormsoftheBound. 22 1.7 AVarianceBound .......................... 24 1.8 BibliographicNotes. .. .. 26 1.9 Problems................................ 27 2 Interlude: Probabilistic Recurrences 33 3 Applying the CH-bounds 39 3.1 Probabilistic Amplification . 39 3.2 LoadBalancing ............................ 40 3.3 DataStructures:SkipLists . 41 3.3.1 SkipLists: TheDataStructure . 41 3.3.2 Skip Lists: Randomization makes it easy . 42 3 DRAFT 4 CONTENTS 3.3.3 Quicksort ........................... 45 3.4 PacketRouting ............................ 47 3.5 RandomizedRounding . 49 3.6 BibliographicNotes. .. .. 49 3.7 Problems................................ 50 4 Chernoff-Hoeffding Bounds in Dependent Settings 55 4.1 NegativeDependence. 55 4.2 LocalDependence........................... 59 4.3 Janson’sInequality . .. .. 61 4.4 LimitedIndependence . 64 4.5 MarkovDependence ......................... 67 4.5.1 Definitions........................... 67 4.5.2 StatementoftheBound . 68 4.5.3 Application: Probability Amplification . 68 4.6 BibliographicNotes. .. .. 70 4.7 Problems................................ 71 5 Martingales and Azuma’s Inequality 73 5.1 Review of Conditional Probabilities and Expectations . ...... 74 5.2 Martingales and Azuma’s Inequality. 76 5.3 Generalizing Martingales and Azuma’s Inequality. ..... 81 5.4 The Method of Bounded Differences . 83 5.5 BibliographicNotes. .. .. 87 5.6 Problems................................ 88 DRAFT CONTENTS 5 6 The Method of Bounded Differences 91 6.1 Chernoff–HoeffdingRevisited . 91 6.2 Stochastic Optimization: Bin Packing . 92 6.3 Game Theory and Blackwell’s Approachability Theorem .
    [Show full text]
  • Lectures on the Poisson Process
    Lectures on the Poisson Process Gunter¨ Last and Mathew Penrose, Version from 21 August 2017, Commercial reproduction prohibited, to be published as IMS Textbook by Cambridge University Press, c Gunter¨ Last and Mathew Penrose Gunter¨ Last Institut f¨urStochastik, Karlsruhe Institute of Technology, Englerstraße 2, D-76128 Karlsruhe, Germany, [email protected] Mathew Penrose Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom, [email protected] Contents Preface page x List of Symbols xiii 1 Poisson and Other Discrete Distributions 1 1.1 The Poisson Distribution1 1.2 Relationships Between Poisson and Binomial Distributions3 1.3 The Poisson Limit Theorem4 1.4 The Negative Binomial Distribution5 1.5 Exercises7 2 Point Processes 9 2.1 Fundamentals9 2.2 Campbell’s Formula 12 2.3 Distribution of a Point Process 14 2.4 Point Processes on Metric Spaces 16 2.5 Exercises 18 3 Poisson Processes 19 3.1 Definition of the Poisson Process 19 3.2 Existence of Poisson Processes 20 3.3 Laplace Functional of the Poisson Process 23 3.4 Exercises 24 4 The Mecke Equation and Factorial Measures 26 4.1 The Mecke Equation 26 4.2 Factorial Measures and the Multivariate Mecke Equation 28 4.3 Janossy Measures 32 4.4 Factorial Moment Measures 34 4.5 Exercises 36 v vi Contents 5 Mappings, Markings and Thinnings 38 5.1 Mappings and Restrictions 38 5.2 The Marking Theorem 39 5.3 Thinnings 42 5.4 Exercises 44 6 Characterisations of the Poisson Process 46 6.1 Borel Spaces 46 6.2 Simple Point Processes 49 6.3 Renyi’s´ Theorem 50
    [Show full text]
  • Stochastic Geometry and Wireless Networks: Volume I Theory Contents
    Foundations and TrendsR in Networking Vol. 3, Nos. 3-4 (2009) 249–449 c 2009 F. Baccelli and B. Blaszczyszyn DOI: 10.1561/1300000006 Stochastic Geometry and Wireless Networks: Volume I Theory By Fran¸cois Baccelli and Bartlomiej Blaszczyszyn Contents I Classical Stochastic Geometry 259 1 Poisson Point Process 261 1.1 Definition and Characterizations 262 1.2 Laplace Functional 266 1.3 Operations Preserving the Poisson Law 269 1.4 Palm Theory 276 1.5 Strong Markov Property 282 1.6 Stationarity and Ergodicity 284 2 Marked Point Processes and Shot-Noise Fields 291 2.1 Marked Point Processes 291 2.2 Shot-Noise 300 2.3 Interference Field as Shot-Noise 305 2.4 Extremal Shot-Noise 316 3 Boolean Model 318 3.1 Boolean Model as a Coverage Process 318 3.2 Boolean Model as a Connectivity Model 329 4 Voronoi Tessellation 338 4.1 Introduction 338 4.2 The Inverse Formula of Palm Calculus 340 4.3 The Neveu Exchange Formula 343 4.4 Neighbors in the Voronoi Tessellation, Delaunay Triangulation 345 4.5 The Voronoi Tessellation Model for Cellular Access Networks 347 Bibliographical Notes on Part I 350 II Signal-to-Interference Ratio Stochastic Geometry 351 5 Signal-to-Interference Ratio Cells 353 5.1 Introduction 353 5.2 The Signal-to-Interference Ratio Cell is Well-Defined 355 5.3 Standard Stochastic Scenario and First Order Cell Characteristics 355 5.4 Fading in Signal-to-Interference Ratio Cell and Higher Order Characteristics 360 5.5 Noise or Interference Limited Cell: Towards a Boolean or Voronoi Shape 362 6 Interacting Signal-to-Interference Ratio Cells
    [Show full text]
  • Instructional Conference on Combinatorial Aspects of Mathematical Analysis Edinburgh, 25 March-5 April 2002
    Instructional Conference on Combinatorial Aspects of Mathematical Analysis Edinburgh, 25 March-5 April 2002 and Summer School on Singular Phenomena and Scaling in Mathematical Models Bonn, 10-13 June, 2003 MEASURE CONCENTRATION, TRANSPORTATION COST, AND FUNCTIONAL INEQUALITIES M. Ledoux University of Toulouse, France Abstract. — In these lectures, we present a triple description of the concentration of measure phenomenon, geometric (through Brunn- Minkoswki inequalities), measure-theoretic (through transportation cost inequalities) and functional (through logarithmic Sobolev inequalities), and investigate the relationships between these various viewpoints. Spe- cial emphasis is put on optimal mass transportation and the dual hy- percontractive bounds on solutions of Hamilton-Jacobi equations that offer a unified treatment of these various aspects. These notes survey recent developments around the concentration of measure phenomenon through various descriptions, geometric, measure theoretic and functional. These descriptions aim to analyze measure concentration for both product and (strictly) log-concave measures, with a special emphasis on dimension free bounds. Inequalities independent of the number of variables are indeed a key information in the study of a number of models in probability theory and statistical mechanics, with a view towards infinite dimensional analysis. To this task, we review the geometric tool of Brunn- Minkowski inequalities, transportation cost inequalities, and functional logarithmic Sobolev inequalities and semigroup methods. Connections are developed on the basis of optimal mass transportation and dual hypercontractive bounds on solutions of Hamilton-Jacobi equations, providing a synthetic view of these recent developments. Results and methods are only outlined in the simplest and basic setting. References to recent PDE extensions are briefly discussed in the last part of the notes.
    [Show full text]
  • Bayesian Nonparametrics Spring 2014
    Stat 362: Bayesian Nonparametrics Spring 2014 Bayesian nonparametrics 5 Feature models 5.1 Motivation A feature allocation table is a binary matrix which characterizes a set of objects (rows), with different features (columns). For example, consider Table 1. Female protagonist Action Comedy Biopic Gattaca 0 0 0 0 Side effects 1 1 0 0 The Iron Lady 1 0 0 1 Skyfall 0 1 0 0 Zero Dark Thirty 1 1 0 0 Table 1: Example of a feature allocation table for movies Given a feature allocation table (Zi`; 1 ≤ i ≤ n; 1 ≤ ` ≤ m), we can model a variety of data associated to the objects. The simplest example would be a linear regression model with normal random effects for each feature. Example (Gaussian linear model). Suppose there is a data point Xi for each object i. We assume m ! iid X Xi j Zi; (µ`) ∼ N Zi`µ`; 1 i = 1; : : : ; n `=1 iid µ` ∼ N (0; 1) ` = 1; : : : ; m: For example, Xi could be the amount of money grossed by movie i, and µ` an additive effect for each feature. We will be concerned with the situation where the features are not known and neither is the feature allocation table. We will assign a prior to (Zi`) and infer the latent features that best describe the data. Thus, our task has more in common with Principal Components Analysis than with linear regression, even though the resulting models can be useful for prediction. 1 Stat 362: Bayesian Nonparametrics Spring 2014 Clustering vs. feature allocation. We have discussed the Dirichlet Pro- cess as a tool for clustering data.
    [Show full text]
  • Course Notes for EE 87021 Advanced Topics in Random Wireless Networks
    Course Notes for EE 87021 Advanced Topics in Random Wireless Networks Martin Haenggi November 1, 2010 c Martin Haenggi, 2010 ii Contents I Point Process Theory 1 1 Introduction 3 1.1 Motivation . 3 1.2 Asymptotic Notation . 3 2 Description of Point Processes 5 2.1 One-dimensional Point Processes . 5 2.2 General Point Processes . 5 2.3 Basic Point Processes . 6 2.3.1 One-dimensional Poisson processes . 6 2.3.2 Spatial Poisson processes . 7 2.3.3 General Poisson point processes . 7 2.4 Distributional Characterization . 7 2.4.1 The distribution of a point process . 8 2.4.2 Comparison with numerical random variables . 8 2.4.3 Distribution of a point process viewed as a random set . 9 2.4.4 Finite-dimensional distributions and capacity functional . 9 2.4.5 Measurable decomposition . 10 2.4.6 Intensity measure . 11 2.5 Properties of Point Processes . 11 2.6 Point Process Transformations . 12 2.6.1 Shaping an inhomogeneous Poisson point process . 12 2.7 Distances . 14 2.8 Marked point processes . 14 3 Sums over Point Processes 17 3.1 Notation . 17 3.2 Campbell’s Theorem for the Mean . 18 3.3 The Probability Generating Functional . 19 3.3.1 The moment-generating function of the sum . 19 3.3.2 The characteristic or Laplace functional for the Poisson point process . 21 3.3.3 The probability generating functional for the Poisson point process . 21 3.3.4 Relationship between moment-generating function and the functionals . 22 3.4 Applications . 23 3.4.1 Mean interference in stationary point process .
    [Show full text]
  • Notes on the Poisson Point Process
    Notes on the Poisson point process Paul Keeler March 20, 2018 This work is licensed under a “CC BY-SA 3.0” license. Abstract The Poisson point process is a type of random object in mathematics known as a point process. For over a century this point process has been the focus of much study and application. This survey aims to give an accessible but detailed account of the Poisson point process by covering its history, mathematical defini- tions in a number of settings, and key properties as well detailing various termi- nology and applications of the point process, with recommendations for further reading. 1 Introduction In probability, statistics and related fields, a Poisson point process or a Pois- son process (also called a Poisson random measure or a Poisson point field) is a type of random object known as a point process that consists of randomly po- sitioned points located on some underlying mathematical space [51, 54]. The Poisson point process has interesting mathematical properties [51][54, pages xv- xvi], which has led to it being frequently defined in Euclidean space and used as a mathematical model for seemingly random objects in numerous disciplines such as astronomy [4], biology [62], ecology [84], geology [17], physics [74], im- age processing [10], and telecommunications [33]. For example, defined on the real line, it can be interpreted as a stochastic process 1 called the Poisson pro- cess [83, pages 7–8][2][70, pages 94–95], forming an example of a Levy´ process and a Markov process [73, page ix] [2]. In this setting, the Poisson process plays 1Point processes and stochastic processes are usually considered as different objects, with point processes being associated or related to certain stochastic processes [20, page 194][18, page 3][83, pages 7–8][25, page 580].
    [Show full text]