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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 −* Weak convergence n λn Lebegue measure on R ; the n is omitted if n = 1 µ^ Characteristic function of a measure µ −! Complete convergence @B The boundary of a set B B¯ The closer of a set B B◦ The interior of a set B Bc The complement of a set B B(Ω) σ-algebra generated by open sets of the implicit topology τ(Ω) n Bn σ-algebra generated by the usual topology on R Br(!) A ball of radius r about an element ! of an implicit metric space (Ω; d) C k(X; Y ) Space of continuous functions from X to Y with k continuous derivatives Cb(X; Y ) Space of bounded continuous functions from X to Y Cc(X; Y ) Space of continuous functions from X to Y with compact support Cp(X; Y ) Space of periodic continuous functions from X to Y C The complex numbers diag(α1; :::; αn) An n × n matrix with diagonal elements α1; :::; αn Fµ The distribution function of µ, that is, Fµ(x) = µ((−∞; x)) iff If and only if Lp(Ω; Ω0; µ) Space of functions f :Ω ! Ω0 such f p is µ-integrable P(F) Set of probability measures on F Pn Set of probability measures on Bn Q The rational numbers R(F) Set of Radon measures on the σ-algebra F Rn Set of Radon measures on Bn R The real numbers s:t: Such that U! 2 τ An open set in τ containing ! z Complex conjugate ii Ackowledgement I would like to thank Dr. Jason C. Murphy1 for motivating my studies in harmonic analysis which ultimately enabled me to write this work. Dr. Murphy also guided me through some proofs and replaced my doubts with clarity; his input was of great value. 1Jason Murphy: https://scholar.google.com/citations?user=32q4x cAAAAJ&hl=en iii Characteristic Functions Introduction Characteristic functions (CFs) are often used in problems involving convergence in distribution, independence of random variables, infinitely divisible distribu- tions, and stochastics [5]. The most famous use of characteristic functions is in the proof of the Central Limit Theorem, also known as the Fundamental Theo- rem of Statistics. Though less frequent, CFs have also been used in problems of nonparametric time series analysis [6] and in machine learning [7{9]. Moreover, CFs uniquely determine their distribution, much like the moment generating functions (MGFs), but the major difference is that CFs always exists, whereas MGFs can fail, e.g. the Cauchy distribution. This makes CFs more robust in general. In the following, I will present an introduction and basic properties of the Fourier-Stieltjes transform, it's inverse and relation to the Radon-Nikodym derivative, then go on to prove the L´evyContinuity Theorem, and finally a short presentation of measure convolutions. Much of the following presentation will be for probability measures and their distribution functions; however, some results can be generalized to (un)signed finite measures [1, 5, 10]. One can find an overview of background knowledge in the Appendix. 1 Basic Properites of Characteristic Functions Definition 1.1 (Characteristic Function / Fourier{Stieltjes Transform). Let n X : (Ω; F; P) ! (R ; Bn; h•; •i) and µ = X#P, the pushforward measure / distribution of X. The characteristic function or Fourier{Stieltjes transform of µ is Z Z iht;X(!)i iht;xi µ^(t) = e dP(!) = e dµ(x) n Ω R where equality comes from Theorem B.7. In terms of the Fourier transform, if µ admits a Radon{Nikodym derivative f with respect to the n-Lebesgue measure λn (Theorem B.26), then Z Z iht;xi iht;xi ^ µ^(t) = e dµ(x) = e f(x) dλn(x) = f(t) n n R R That is, the characteristic function is nothing but the Fourier transform of the density of µ if one exists. Theorem 1.2 (Properties of the Characteristic Function). Let µ be as above, then i) µ^(0) = 1 ii) jµ^(t)j 5 1 1 iii) µ^(−t) = µ^(t) iv) µ^ is uniformly continuous on Rn v) (µ\1 + µ2) =µ ^1 +µ ^2 Proof. i) Z Z ih0;xi n µ^(0) = e dµ(x) = dµ(x) = µ(R ) = 1 n n R R ii) Z ih0;xi jµ^(t)j = e dµ(x) n R Z ih0;xi 5 je j dµ(x) n R Z 5 1 dµ(x) n R n = µ(R ) = 1 iii) Z µ^(−t) = eih-t;xi dµ(x) n R Z = e-iht;xi dµ(x) n R Z = eiht;xi dµ(x) nn since eix = e−ix n R Z = eiht;xi dµ(x) n R = µ^(t) iv) Let δ > 0 Z iht+δ;xi iht;xi jµ^(t + δ) − µ^(t)j = e − e dµ(x) n R 2 Z iht;xi+ihδ;xi iht;xi = e − e dµ(x) n R Z iht;xi ihδ;xi iht;xi = e · e − e dµ(x) n R Z iht;xi ihδ;xi iht;xi 5 e · e − e dµ(x) n R Z iht;xi ihδ;xi = e e − 1 dµ(x) n R Z ihδ;xi = e − 1 dµ(x) n R ihδ;xi δ#0 5 sup e − 1 −−! 0 n x2R v) Z iht;xi (µ\1 + µ2) = e d[µ1 + µ2](x) n R Z Z iht;xi iht;xi = e dµ1(x) + e dµ2(x) n n R R =µ ^1 +µ ^2 Theorem 1.3. Let (Rn; B(Rn); λ) be as usual, µ 2 P(B(Rn)), and f 2 L1(Rn; C; λ), then Z Z ^ f dµ = fµ^ dλn n n R R Proof. This follows from Fubini's theorem 0 1 Z Z Z ^ iπht;xi f dµ = @ e f(x) dλn(x)A dµ(t) n n n R R R 0 1 Z Z iπht;xi = @ e f(x) dµ(t)A dλn(x) n n R R 0 1 Z Z iπht;xi = f(x) @ e dµ(t)A dλn(x) n n R R 3 Z = [f · µ^](x) dλn(x) n R Theorem 1.4 (Uniqueness). If µ1; µ2 2 Pn, then µ1 = µ2 () µ^1 =µ ^2 : Note: the equality of the characteristic functions is over the real numbers. Proof. \ =) ": This is obvious Z Z iht;xi iht;xi µ^1(t) = e dµ1(x) = e dµ2(x) =µ ^2(t) n n R R \ (= ": Supposeµ ^1(t) =µ ^2(t) for all t, then Z Z iht;xi iht;xi n e d[µ1 − µ2](x) = e dµ¯(x) = 0 8t 2 R n n R R N P iht(1;:::;n);•i then for all trigonometric polynomials γN (•) = c(t1;:::;tn)e t1;:::;tn=−N we have Z γN (x)dµ¯(x) = 0 n R and so for the uniform limit γ(•) in N, which are the continuous periodic func- tions (Theorem C.6), we have Z γ(x)dµ¯(x) = 0 n R Let c(•) be a continuous function which vanishes outside a fixed, bounded ⊗n support S, choose m such that S ⊆ (−m; m] , and choose γm(•) to be a continuous periodic function with period 2m such that γm(x) = c(x) 8x 2 (−m; m]⊗n. Sinceµ ¯ is the difference of of two monotone increasing functions (Fµ = Fµ1 − Fµ2 ), it is a function of bounded variation; hence 8 > 0 9 m s.t. µ¯(x) < for jxj = m1 and we have Z m"1 Z Z 0 = γm dµ¯ −−−! c dµ¯ = c dµ¯ = 0 Rn Rn S Since c was arbitary, we have, by Theorem B.20 and taking c to be indicator functions on sets in Bn, that µ¯ = µ1 − µ2 = 0 and so µ1 = µ2. 4 2 Inversion Formula n n Theorem 2.1 (Inversion). Let (R ; Bn; λ) be as usual; µ 2 Pn; a; b 2 R (a < b) and let (a; b) = fx 2 Rn : a < x < b)g, then 2 n 3 Z -itj aj -itj bj -n -n Y e − e µ((a; b)) + 2 µ(fa; bg) = (2π) lim 4 5 µ^(t) dλn(t) k"1 it j=1 j (-k;k) T n there k = [k1; :::; kn] and (-k; k) = fx 2 R : −k 5 x 5 kg.
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