The largest eigenvalues of the sample covariance matrix. in the heavy-tail case
|
|
- Lorraine Gray
- 5 years ago
- Views:
Transcription
1 The largest eigenvalues of the sample covariance matrix 1 in the heavy-tail case Thomas Mikosch University of Copenhagen Joint work with Richard A. Davis (Columbia NY), Johannes Heiny (Aarhus University) 1 Suzhou, June
2 lower tail index Upper tail index Figure 1. Estimation of tail indices for S&P 500.
3 1. Heavy-tailed matrices: preliminaries 3 We consider p n matrices X = X n = ( X it )i,...,p;t=1,...,n, where all entries have the same distribution with heavy tails; X denotes a generic element with this distribution. We are interested in the asymptotic behavior of the largest eigenvalues and corresponding eigenvectors of the sample covariance matrices for p = p n : ( n XX = X it X jt, n = 1, 2,..., )i,j=1,...,p t=1
4 An excursion to regularly varying random variables. The random variable X and its distribution are regularly varying with index α > 0 if there exists a slowly varying function L and constants p ± such that p + + p = 1 and P(X > x) p + L(x) x α and P(X > x) p L(x) x α, x > 0. Then P(X 2 > x) L( x) x α/2. For iid copies (X i ) of X > 0, some slowly varying l. Embrechts and Goldie (1984): P(X 1 X 2 > x) l(x) x α.
5 5 If X, σ > 0 independent and E[σ α+δ ] <, then Breiman (1965) P(σ X > x) E[σ α ] P(X > x), x. A regularly varying random variable X > 0 and its distribution are subexponential: for some (any) n 2, P(X X n > x) n P(X > x), n. Other subexponential distributions: log-normal, Weibull F (x) = exp( x τ ), τ (0, 1). Assume X > 0. There exists a sequence x n such that P(X X n > x n ) n P(X > x n ), n, if and only if X is subexponential. Cline, Hsing (1998)
6 Heavy-tail large deviations. Define for iid (X i ) with a generic element X, S n = X X n, n 1. Assume E[X] = 0 if the first moment is finite. If x n, A is bounded away from zero, and x 1 n S n P 0, n, we call P(x 1 n S n A) a large deviation probability.
7 If (X i ) is iid regularly varying with index α > 0 then P(±S n > x) sup x γ n n P( X > x) p ± 0, n, 7 where γ n /a n for α (0, 2), n P( X > a n ) 1, γ n = c n log n for c > α 2, α > 2. A.V. Nagaev (1969), S.V. Nagaev (1979), C.C. Heyde (1969), Cline, Hsing (1998),... If α > 2, 0 < c < α 2, var(x) = 1, uniformly for x < c n log n, P(S n > x) Φ(x/ n), n.
8 8 Separating sequences for the normal and subexponential approximations to large deviation probabilities exist for general subexponential distributions.
9 2. Heavy-tailed matrices with iid entries: Finite p 9 We consider p n matrices X = X n = ( X it )i=1,...,p;t=1,...,n, where all entries have the same distribution with heavy tails L(x) L(x) P(X > x) p + and P(X < x) p x α x α for a slowly varying function L, some α > 0, p + + p = 1. Assume E[X] = 0 if expectation is finite. Therefore Gnedenko and Kolmogorov (1949), Feller (1971), Resnick (2007) ( n a 2 n X 2 it c ) d (α/2) n ξ i, α (0, 4), t=1 where P( X > a n ) n 1 and (ξ (γ) i ) are iid γ-stable, γ (0, 2],
10 10 and for i j, b 1 n n t=1 X itx jt d ξ (2 α) i,j, α (0, 4), where P( X 1 X 2 > b n ) n 1 for α (0, 2) and b n = n for α (2, 4). Since a 2 n n2/α and b n n 1/(2 α), ( a 2 n XX diag(xx ) ) 2 b2 n 2 a 4 n 1 i j p ( 1 b n n ) 2 P X it X jt 0. t=1 Therefore, by Hermann Weyl s inequality, a 2 n max λ(i) λ (i) (diag(xx )) i=1,...,p a 2 XX diag(xx P ) 0, 2 n where λ (1) (A) λ (p) (A) for any symmetric matrix A and λ (i) = λ (i) (XX ).
11 11 Hence for α (0, 4), ξ (α/2) (1) ξ (α/2) (p) ( ) a 2 n λ(i) c n i=1,...,p d ( ξ (α/2) ) (i) i=1,...,p. In particular, n a 2 n (λ (1) n E[X2 ] 1 (2,4) (α) ) d max i=1,...,p ξ(α/2) i. The unit eigenvector V j associated with λ (j) = λ Lj can be localized: V j e Lj P 0.
12 12 We have for α (2, 4), with c n = n E[X 2 ] 1 ( p det(xx ) cn) p = a 2 n cp 1 n d = i=1 p i=1 p i=1 a 2 n (λ (i) c n ) ξ (α/2) (i) ξ (α/2) i d = p 2/α ξ (α/2) 1. i 1 j=1 λ (j) c n
13 3. Heavy-tailed matrices with iid entries: p increases with n 13 We consider p n matrices X = X n = ( X it )i,...,p;t=1,...,n, where the entries are iid and have a distribution with heavy tails P(X > x) p + x α and P(X < x) p x α for some α (0, 4), p + + p = 1, and p = p n = n β for some β > 0. Then for a np = n (1+β)/α, i.e., np P( X > a np ) 1, a 2 np XX diag(xx ) P 2 0, β (0, 1], X X diag(x X) P 2 0, β (1, ). a 2 np
14 14 One needs Nagaev-type large deviations: for i j and β 1, ( p 2 P a 2 np n ) X it X jt > ε p 2 n P( X 1 X 2 > εa 2 np ) P 0. t=1 Observe that XX and X X have the same positive eigenvalues. For β > 1, interchange the roles of n and p, and observe that n = p 1/β. For the moment, we consider the case β 1 only.
15 Write λ 1,..., λ p for the eigenvalues of XX and 15 λ (p) λ (1). We also write n D i = X 2 it = λ i(diag(xx )), i = 1,..., p, t=1 and for the ordered values By Weyl s inequality, 1 λ (i) D (i) 1 a 2 np max i=1,...,p D (p) D (1). a 2 np XX diag(xx ) 2 P 0, β (0, 1].
16 16 Limit theory for the order statistics of (D i ) can be handled by point process convergence towards a Poisson process: for example, p i=1 ε a 2 np (D i c n ) d i=1 ε Γ 2/α i = N Γ for Γ i = E E i, (E i ) iid standard exponential, if and only if Resnick (2007) p P(a 2 np (D i c n ) > x) x α/2, x > 0, p P(a 2 np (D i c n ) < x) 0, x > 0. This follows by Nagaev-type large deviations.
17 Centering with c n is only needed for α [2, 4) and if 17 (n p)/a 2 np 0. Centering is not needed if for example when min(β, β 1 ) ( (α/2 1) +, 1 ], (3.1) p/n γ (0, ). Under (3.1), this was proved by Soshnikov (2004,2006) for α (0, 2) and by Auffinger, Ben Arous, Péché (2009) for α (0, 4). They do not make explicit use of large deviation probabilities but prove their results by approximating the largest eigenvalue λ (i) by the corresponding ith order statistic X(i) 2 from (Xjt 2 ) k=1,...,p ;t=1,...,n.
18 18 Partial results (for particular choices of p) were proved in Davis, Pfaffel and Stelzer (2014), Davis, Mikosch, Pfaffel (2016) for (X it ) satisfying some linear dependence conditions. If α < 1, (p n ) could be chosen of almost exponential rate.
19 In the light-tailed case, limit results for eigenvalues of XX are very sensitive with respect to the growth rate of p. 19 For example, Johnstone (2001) showed for iid standard normal X it under (3.1) that n + p ( 1/ n + 1/ p ) 1/3 ( λ (1) ( n + p) 2 1 ) d Tracy-Widom distr. This result remains valid for iid entries X it whose first four moments match those of the standard normal, and all moments of X are finite Tao and Vu (2011).
20 20 In the iid regular variation case, using a.s. continuous mappings, folklore (Resnick (1987,2007)) applies to prove the joint convergence of the order statistics (possibly with centering c n for α [2, 4)) ( ) a 2 d ( 2/α np λ(1),..., λ (k) Γ 1,..., Γ 2/α ) k, in particular, P(Γ 2/α 1 x) = exp( x α/2 ), x > 0, is the Fréchet distribution Φ α/2. the joint convergence of the ratios of successive order statistics (λ (2) λ (1),..., λ (k 1) λ (k) ) d ((Γ 1 Γ 2 ) 2/α,..., (Γ k 1 Γ k ) 2/α ).
21 the convergence of the ratio of largest eigenvalue to the 21 trace of XX for α (0, 2) 2/α λ (1) d Γ 1 λ λ p Γ 2/α 1 + Γ 2/α 2 + With these techniques one can handle results involving finitely many of the largest eigenvalues of XX. One cannot handle the smallest eigenvalues, determinants,.... The largest eigenvalues have very distinct sizes and the eigenvectors are localized: let V k be the unit eigenvector corresponding to λ (k) = λ Lk. Then V k e Lk P 0.
22 22 log(λ(i+1) λ(i)) i Figure 2. The logarithms of the ratios λ (i+1) /λ (i) for the S&P 500 series. We also show the 1, 50 and 99% quantiles (bottom, middle, top lines, respectively) of the variables log((γ i /Γ i+1 ) 2 ).
23 23 12 x 105 Eigenvalues and their ratios as in Lam & Yao 10 8 λ (i) i λ (i+1) /λ (i) i Figure 3. Eigenvalues and ratios λ (i+1) /λ (i) for S&P , p = 442
24 e 15 5e 16 0e+00 5e 16 Normal data Indices of components Size of components Pareto data Indices of components Size of components Figure 4. The components of the eigenvector V 1. Right: The case of iid Pareto(0.8) entries. Left: The case of iid standard normal entries. We choose p = 200 and n = 1, 000.
25 4. Stochastic volatility models 25 The main idea in the iid case is to exploit the heavier tails of X 2 in comparison with those of X 1 X 2. Similar behavior can be observed for regularly varying stochastic volatility models X it = σ it Z it, i, t Z, where (σ it ) is a strictly stationary random field 2 (with finite moments of order α + δ) independent of the iid random field (Z it ) assumed regularly varying with index α (0, 4). 2(log σ it ) is often assumed Gaussian.
26 26 By Breiman s lemma, (X it ) is (jointly) regularly varying and P(X 2 > x) E[σ α ] P(Z 2 > x) x α/2, P(X it X jt > x) E[(σ it σ jt ) α ] P(Z 1 Z 2 > x) x α. Nagaev-type large deviations for stochastic volatility models Mikosch and Wintenberger (2016) confirm that a 2 XX diag(xx ) P 2 0. and np p i=1 ε a 2 np (λ i c n ) d i=1 ε Γ 2/α i Then, essentially, the same results as for iid (X it ) hold..
27 Thinning of X 27 Similar techniques apply when σ it = σ (n) it assumes zero with high probability, e.g. (σ (n) it ) iid Bernoulli distributed for n 1 and q (n) 0 = P(σ (n) = 0) 1 and lim n n P(σ (n) = 1) = lim n n q (n) 1 > 0, and independent of (Z it ); see Auffinger, Tang (2016), q (n) 1 = n v, v (0, 1) New normalization (b n ) n p E[(σ (n) ) α ] P( Z > b n ) = n p q (n) 1 P( Z > b n) 1, n. or b n = a [ n p E[(σ (n) ) α ]]
28 28 Then, for β 1, using Nagaev-type large deviations if lim n n p (n) 1 = or regular variation if lim n n p (n) b 2 n sup i=1,...,p λ(i) λ (i) ( diag(xx ) ) b 2 XX diag(xx ) 2 0 n 1 is finite and p i=1 ε b 2 n (λ i c n ) d i=1 ε Γ 2/α i.
29 5. Another structure where the squares dominate: linear 29 processes. Davis, Pfaffel, Stelzer (2014), Davis, Mikosch, Pfaffel (2016) Special case: X it = θ 0 Z i,t + θ 1 Z i 1,t for iid (Z it ) with regularly varying Z with index α (0, 4), real coefficients θ i, E[Z] = 0 if finite. Choose (a n ) such that n P( Z > a n ) 1. Observe that n X 2 i,t = t=1 n t=1 ( θ 2 0 Z2 i,t + ) n θ2 1 Z2 i 1,t + 2θ0 θ 1 Z i,t Z i 1,t t=1 = θ 2 0 D i + θ 2 1 D i 1 + o P (a 2 n ).
30 30 Here we used Nagaev-type large deviations and the fact that Z 2 has tail index α/2, while Z 1 Z 2 has tail index α. Similarly, n X i,t X i+1,t = θ 0 θ 1 t=1 n t=1 Z 2 i,t + o P (a 2 n ) = θ 0 θ 1 D i + o P (a 2 n ). This leads to the approximation ( n n t=1 X2 i,t t=1 X i,tx i+1,t ) n t=1 X i,tx i+1,t ( θ 2 0 θ 0 θ 1 θ 0 θ 1 θ 2 1 ) D i + n t=1 X2 i+1,t ( θ ) D i 1 + ( θ 2 0 ) D i+1.
31 The sample covariance matrix can be approximated by p XX D 2 i M i = o P (a 2 np ), where M 1 = i=1 θ 2 0 θ 0 θ θ 0 θ 1 θ , M 2 = Denote the order statistics of the D i D L i = D (i) θ 2 0 θ 0 θ θ 0 θ 1 θ ,... by D (1) D (p) and 31 Then a 2 np XX p D 2 P L i M Li 0. i=1
32 32 For k = k n slowly, a 2 np XX k D 2 P L i M Li 0. i=1 Since (D i ) is iid, (L 1,..., L p ) is a random permutation of (1,..., p), hence the event A k = { L i L j > 1, i j = 1,..., k} has probability close to one provided k 2 = o(p). On the set A k, the matrix k i=1 D L i M Li is block-diagonal with non-zero eigenvalues DL i (θ0 2 + θ2 1 ), i = 1,..., k. Here we used that M Li has rank 1 with non-zero eigenvalue equal to θ θ2 1.
33 By Weyl s inequality, λ (i) D L i (θ θ2 1 ) a 2 a 2 np max i=1,...,k np XX k D 2 P L i M Li 0. i=1 33 Extension to general linear structure: X it = k,l h kl Z k i,l t Use truncation of the coefficient matrix H = (h kl ) of the linear process. Then a 2 np max λ(i) δ (i) P 0, n, i=1,...,p where δ (1),..., δ (p) are the p ordered values (with respect to absolute value) of the set
34 34 {(D i E[D 1 ])v j, i = 1,..., k; j = 1, 2,...} for α (0, 2), (α (2, 4)) where (v j ) are the eigenvalues of H H = ( h il h jl )i,j=1,2,... l=0 The mapping theorem implies for suitable real or complex numbers (v j ) j=1 p i=1 ε a 2 np (D i ED 1 )v j d j=1 The limit is a Poisson cluster process. i=1 ε Γ 2/α i v j. An example: The separable case: We assume h kl = θ k c l. The matrix HH = ( ) l=0 c2 l θi θ j has rank r = 1 and i,j 0 v (1) = θ 2 k. l=0 c 2 l k=0
35 The limit point process is Poisson as in the iid case. For α < 2, ( ) d ( 2/α λ(1),..., λ (k) v(1) Γ 1,..., Γ 2/α ), a 2 np 2/α λ (1) d Γ 1 λ (1) + + λ (k) Γ 2/α k 2/α λ (1) d Γ 1 λ 1 + λ λ p Γ 2/α Γ 2/α 1 + Γ 2/α 2 +, k, 35
36 36 6. Final comments The asymptotic behavior of the largest eigenvalues of a heavy-tailed large sample covariance matrix XX are understood * if the diagonal diag(xx ) = ( n t=1 X 2 it ) i=1,...,p dominates the asymptotic behavior of the eigenvalues * for linear random fields X it = k,l h k,lz i k,t l with iid heavy-tailed (Z it ) Davis, Mikosch, Pfaffel (2016), Davis, Heiny, Mikosch, Xie (2017) * if the rows are iid. Davis, Pfaffel, Stelzer (2015) Nagaev-type large deviations are key techniques.
37 The eigenvalues of the sample covariance matrices of other 37 types of heavy-tailed multivariate time series are less understood.
Assessing the dependence of high-dimensional time series via sample autocovariances and correlations
Assessing the dependence of high-dimensional time series via sample autocovariances and correlations Johannes Heiny University of Aarhus Joint work with Thomas Mikosch (Copenhagen), Richard Davis (Columbia),
More informationLimit theorems for dependent regularly varying functions of Markov chains
Limit theorems for functions of with extremal linear behavior Limit theorems for dependent regularly varying functions of In collaboration with T. Mikosch Olivier Wintenberger wintenberger@ceremade.dauphine.fr
More informationRandom matrices: Distribution of the least singular value (via Property Testing)
Random matrices: Distribution of the least singular value (via Property Testing) Van H. Vu Department of Mathematics Rutgers vanvu@math.rutgers.edu (joint work with T. Tao, UCLA) 1 Let ξ be a real or complex-valued
More informationNonlinear Time Series Modeling
Nonlinear Time Series Modeling Part II: Time Series Models in Finance Richard A. Davis Colorado State University (http://www.stat.colostate.edu/~rdavis/lectures) MaPhySto Workshop Copenhagen September
More informationStochastic volatility models with possible extremal clustering
Bernoulli 19(5A), 213, 1688 1713 DOI: 1.315/12-BEJ426 Stochastic volatility models with possible extremal clustering THOMAS MIKOSCH 1 and MOHSEN REZAPOUR 2 1 University of Copenhagen, Department of Mathematics,
More informationExtremogram and Ex-Periodogram for heavy-tailed time series
Extremogram and Ex-Periodogram for heavy-tailed time series 1 Thomas Mikosch University of Copenhagen Joint work with Richard A. Davis (Columbia) and Yuwei Zhao (Ulm) 1 Jussieu, April 9, 2014 1 2 Extremal
More informationExtremogram and ex-periodogram for heavy-tailed time series
Extremogram and ex-periodogram for heavy-tailed time series 1 Thomas Mikosch University of Copenhagen Joint work with Richard A. Davis (Columbia) and Yuwei Zhao (Ulm) 1 Zagreb, June 6, 2014 1 2 Extremal
More informationLARGE DEVIATION PROBABILITIES FOR SUMS OF HEAVY-TAILED DEPENDENT RANDOM VECTORS*
LARGE EVIATION PROBABILITIES FOR SUMS OF HEAVY-TAILE EPENENT RANOM VECTORS* Adam Jakubowski Alexander V. Nagaev Alexander Zaigraev Nicholas Copernicus University Faculty of Mathematics and Computer Science
More informationConcentration Inequalities for Random Matrices
Concentration Inequalities for Random Matrices M. Ledoux Institut de Mathématiques de Toulouse, France exponential tail inequalities classical theme in probability and statistics quantify the asymptotic
More informationRegular Variation and Extreme Events for Stochastic Processes
1 Regular Variation and Extreme Events for Stochastic Processes FILIP LINDSKOG Royal Institute of Technology, Stockholm 2005 based on joint work with Henrik Hult www.math.kth.se/ lindskog 2 Extremes for
More informationExponential tail inequalities for eigenvalues of random matrices
Exponential tail inequalities for eigenvalues of random matrices M. Ledoux Institut de Mathématiques de Toulouse, France exponential tail inequalities classical theme in probability and statistics quantify
More informationHeavy Tailed Time Series with Extremal Independence
Heavy Tailed Time Series with Extremal Independence Rafa l Kulik and Philippe Soulier Conference in honour of Prof. Herold Dehling Bochum January 16, 2015 Rafa l Kulik and Philippe Soulier Regular variation
More informationStochastic volatility models: tails and memory
: tails and memory Rafa l Kulik and Philippe Soulier Conference in honour of Prof. Murad Taqqu 19 April 2012 Rafa l Kulik and Philippe Soulier Plan Model assumptions; Limit theorems for partial sums and
More informationPractical conditions on Markov chains for weak convergence of tail empirical processes
Practical conditions on Markov chains for weak convergence of tail empirical processes Olivier Wintenberger University of Copenhagen and Paris VI Joint work with Rafa l Kulik and Philippe Soulier Toronto,
More informationOn the estimation of the heavy tail exponent in time series using the max spectrum. Stilian A. Stoev
On the estimation of the heavy tail exponent in time series using the max spectrum Stilian A. Stoev (sstoev@umich.edu) University of Michigan, Ann Arbor, U.S.A. JSM, Salt Lake City, 007 joint work with:
More informationLarge deviations for random walks under subexponentiality: the big-jump domain
Large deviations under subexponentiality p. Large deviations for random walks under subexponentiality: the big-jump domain Ton Dieker, IBM Watson Research Center joint work with D. Denisov (Heriot-Watt,
More informationTail process and its role in limit theorems Bojan Basrak, University of Zagreb
Tail process and its role in limit theorems Bojan Basrak, University of Zagreb The Fields Institute Toronto, May 2016 based on the joint work (in progress) with Philippe Soulier, Azra Tafro, Hrvoje Planinić
More informationFluctuations from the Semicircle Law Lecture 4
Fluctuations from the Semicircle Law Lecture 4 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 23, 2014 Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, 2014
More informationMFM Practitioner Module: Quantitiative Risk Management. John Dodson. October 14, 2015
MFM Practitioner Module: Quantitiative Risk Management October 14, 2015 The n-block maxima 1 is a random variable defined as M n max (X 1,..., X n ) for i.i.d. random variables X i with distribution function
More informationExtreme Value Analysis and Spatial Extremes
Extreme Value Analysis and Department of Statistics Purdue University 11/07/2013 Outline Motivation 1 Motivation 2 Extreme Value Theorem and 3 Bayesian Hierarchical Models Copula Models Max-stable Models
More informationThe sample autocorrelations of financial time series models
The sample autocorrelations of financial time series models Richard A. Davis 1 Colorado State University Thomas Mikosch University of Groningen and EURANDOM Abstract In this chapter we review some of the
More informationPainlevé Representations for Distribution Functions for Next-Largest, Next-Next-Largest, etc., Eigenvalues of GOE, GUE and GSE
Painlevé Representations for Distribution Functions for Next-Largest, Next-Next-Largest, etc., Eigenvalues of GOE, GUE and GSE Craig A. Tracy UC Davis RHPIA 2005 SISSA, Trieste 1 Figure 1: Paul Painlevé,
More informationComparison Method in Random Matrix Theory
Comparison Method in Random Matrix Theory Jun Yin UW-Madison Valparaíso, Chile, July - 2015 Joint work with A. Knowles. 1 Some random matrices Wigner Matrix: H is N N square matrix, H : H ij = H ji, EH
More informationRandom regular digraphs: singularity and spectrum
Random regular digraphs: singularity and spectrum Nick Cook, UCLA Probability Seminar, Stanford University November 2, 2015 Universality Circular law Singularity probability Talk outline 1 Universality
More informationMax stable Processes & Random Fields: Representations, Models, and Prediction
Max stable Processes & Random Fields: Representations, Models, and Prediction Stilian Stoev University of Michigan, Ann Arbor March 2, 2011 Based on joint works with Yizao Wang and Murad S. Taqqu. 1 Preliminaries
More informationParameter estimation in linear Gaussian covariance models
Parameter estimation in linear Gaussian covariance models Caroline Uhler (IST Austria) Joint work with Piotr Zwiernik (UC Berkeley) and Donald Richards (Penn State University) Big Data Reunion Workshop
More informationPart IA Probability. Definitions. Based on lectures by R. Weber Notes taken by Dexter Chua. Lent 2015
Part IA Probability Definitions Based on lectures by R. Weber Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.
More information2. Matrix Algebra and Random Vectors
2. Matrix Algebra and Random Vectors 2.1 Introduction Multivariate data can be conveniently display as array of numbers. In general, a rectangular array of numbers with, for instance, n rows and p columns
More information1 Tridiagonal matrices
Lecture Notes: β-ensembles Bálint Virág Notes with Diane Holcomb 1 Tridiagonal matrices Definition 1. Suppose you have a symmetric matrix A, we can define its spectral measure (at the first coordinate
More informationUpper Bound for Intermediate Singular Values of Random Sub-Gaussian Matrices 1
Upper Bound for Intermediate Singular Values of Random Sub-Gaussian Matrices 1 Feng Wei 2 University of Michigan July 29, 2016 1 This presentation is based a project under the supervision of M. Rudelson.
More informationMarkov operators, classical orthogonal polynomial ensembles, and random matrices
Markov operators, classical orthogonal polynomial ensembles, and random matrices M. Ledoux, Institut de Mathématiques de Toulouse, France 5ecm Amsterdam, July 2008 recent study of random matrix and random
More informationFinite-time Ruin Probability of Renewal Model with Risky Investment and Subexponential Claims
Proceedings of the World Congress on Engineering 29 Vol II WCE 29, July 1-3, 29, London, U.K. Finite-time Ruin Probability of Renewal Model with Risky Investment and Subexponential Claims Tao Jiang Abstract
More informationEstimation of the functional Weibull-tail coefficient
1/ 29 Estimation of the functional Weibull-tail coefficient Stéphane Girard Inria Grenoble Rhône-Alpes & LJK, France http://mistis.inrialpes.fr/people/girard/ June 2016 joint work with Laurent Gardes,
More informationNon white sample covariance matrices.
Non white sample covariance matrices. S. Péché, Université Grenoble 1, joint work with O. Ledoit, Uni. Zurich 17-21/05/2010, Université Marne la Vallée Workshop Probability and Geometry in High Dimensions
More informationPrecise Large Deviations for Sums of Negatively Dependent Random Variables with Common Long-Tailed Distributions
This article was downloaded by: [University of Aegean] On: 19 May 2013, At: 11:54 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer
More informationLarge sample distribution for fully functional periodicity tests
Large sample distribution for fully functional periodicity tests Siegfried Hörmann Institute for Statistics Graz University of Technology Based on joint work with Piotr Kokoszka (Colorado State) and Gilles
More informationAsymptotic Statistics-III. Changliang Zou
Asymptotic Statistics-III Changliang Zou The multivariate central limit theorem Theorem (Multivariate CLT for iid case) Let X i be iid random p-vectors with mean µ and and covariance matrix Σ. Then n (
More informationSTAT 200C: High-dimensional Statistics
STAT 200C: High-dimensional Statistics Arash A. Amini May 30, 2018 1 / 59 Classical case: n d. Asymptotic assumption: d is fixed and n. Basic tools: LLN and CLT. High-dimensional setting: n d, e.g. n/d
More informationStein s Method and Characteristic Functions
Stein s Method and Characteristic Functions Alexander Tikhomirov Komi Science Center of Ural Division of RAS, Syktyvkar, Russia; Singapore, NUS, 18-29 May 2015 Workshop New Directions in Stein s method
More informationPart IA Probability. Theorems. Based on lectures by R. Weber Notes taken by Dexter Chua. Lent 2015
Part IA Probability Theorems Based on lectures by R. Weber Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.
More informationAsymptotics and Simulation of Heavy-Tailed Processes
Asymptotics and Simulation of Heavy-Tailed Processes Department of Mathematics Stockholm, Sweden Workshop on Heavy-tailed Distributions and Extreme Value Theory ISI Kolkata January 14-17, 2013 Outline
More informationSTA205 Probability: Week 8 R. Wolpert
INFINITE COIN-TOSS AND THE LAWS OF LARGE NUMBERS The traditional interpretation of the probability of an event E is its asymptotic frequency: the limit as n of the fraction of n repeated, similar, and
More informationOn the Estimation and Application of Max-Stable Processes
On the Estimation and Application of Max-Stable Processes Zhengjun Zhang Department of Statistics University of Wisconsin Madison, WI 53706, USA Co-author: Richard Smith EVA 2009, Fort Collins, CO Z. Zhang
More informationThe circular law. Lewis Memorial Lecture / DIMACS minicourse March 19, Terence Tao (UCLA)
The circular law Lewis Memorial Lecture / DIMACS minicourse March 19, 2008 Terence Tao (UCLA) 1 Eigenvalue distributions Let M = (a ij ) 1 i n;1 j n be a square matrix. Then one has n (generalised) eigenvalues
More informationGARCH processes probabilistic properties (Part 1)
GARCH processes probabilistic properties (Part 1) Alexander Lindner Centre of Mathematical Sciences Technical University of Munich D 85747 Garching Germany lindner@ma.tum.de http://www-m1.ma.tum.de/m4/pers/lindner/
More informationInternational Conference on Applied Probability and Computational Methods in Applied Sciences
International Conference on Applied Probability and Computational Methods in Applied Sciences Organizing Committee Zhiliang Ying, Columbia University and Shanghai Center for Mathematical Sciences Jingchen
More informationInvertibility of random matrices
University of Michigan February 2011, Princeton University Origins of Random Matrix Theory Statistics (Wishart matrices) PCA of a multivariate Gaussian distribution. [Gaël Varoquaux s blog gael-varoquaux.info]
More informationMath 576: Quantitative Risk Management
Math 576: Quantitative Risk Management Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 11 Haijun Li Math 576: Quantitative Risk Management Week 11 1 / 21 Outline 1
More informationThe Subexponential Product Convolution of Two Weibull-type Distributions
The Subexponential Product Convolution of Two Weibull-type Distributions Yan Liu School of Mathematics and Statistics Wuhan University Wuhan, Hubei 4372, P.R. China E-mail: yanliu@whu.edu.cn Qihe Tang
More informationPoisson Cluster process as a model for teletraffic arrivals and its extremes
Poisson Cluster process as a model for teletraffic arrivals and its extremes Barbara González-Arévalo, University of Louisiana Thomas Mikosch, University of Copenhagen Gennady Samorodnitsky, Cornell University
More informationEstimation of large dimensional sparse covariance matrices
Estimation of large dimensional sparse covariance matrices Department of Statistics UC, Berkeley May 5, 2009 Sample covariance matrix and its eigenvalues Data: n p matrix X n (independent identically distributed)
More informationarxiv: v3 [math-ph] 21 Jun 2012
LOCAL MARCHKO-PASTUR LAW AT TH HARD DG OF SAMPL COVARIAC MATRICS CLAUDIO CACCIAPUOTI, AA MALTSV, AD BJAMI SCHLI arxiv:206.730v3 [math-ph] 2 Jun 202 Abstract. Let X be a matrix whose entries are i.i.d.
More informationRegularly varying functions
July 6, 26 PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome 79(93) (26), od do Regularly varying functions Anders Hedegaard Jessen and Thomas Mikosch In memoriam Tatjana Ostrogorski. Abstract.
More informationAsymptotics of random sums of heavy-tailed negatively dependent random variables with applications
Asymptotics of random sums of heavy-tailed negatively dependent random variables with applications Remigijus Leipus (with Yang Yang, Yuebao Wang, Jonas Šiaulys) CIRM, Luminy, April 26-30, 2010 1. Preliminaries
More informationIII. Quantum ergodicity on graphs, perspectives
III. Quantum ergodicity on graphs, perspectives Nalini Anantharaman Université de Strasbourg 24 août 2016 Yesterday we focussed on the case of large regular (discrete) graphs. Let G = (V, E) be a (q +
More informationBootstrapping high dimensional vector: interplay between dependence and dimensionality
Bootstrapping high dimensional vector: interplay between dependence and dimensionality Xianyang Zhang Joint work with Guang Cheng University of Missouri-Columbia LDHD: Transition Workshop, 2014 Xianyang
More informationLeast singular value of random matrices. Lewis Memorial Lecture / DIMACS minicourse March 18, Terence Tao (UCLA)
Least singular value of random matrices Lewis Memorial Lecture / DIMACS minicourse March 18, 2008 Terence Tao (UCLA) 1 Extreme singular values Let M = (a ij ) 1 i n;1 j m be a square or rectangular matrix
More informationBasic concepts of probability theory
Basic concepts of probability theory Random variable discrete/continuous random variable Transform Z transform, Laplace transform Distribution Geometric, mixed-geometric, Binomial, Poisson, exponential,
More informationPhenomena in high dimensions in geometric analysis, random matrices, and computational geometry Roscoff, France, June 25-29, 2012
Phenomena in high dimensions in geometric analysis, random matrices, and computational geometry Roscoff, France, June 25-29, 202 BOUNDS AND ASYMPTOTICS FOR FISHER INFORMATION IN THE CENTRAL LIMIT THEOREM
More informationRuin Probabilities of a Discrete-time Multi-risk Model
Ruin Probabilities of a Discrete-time Multi-risk Model Andrius Grigutis, Agneška Korvel, Jonas Šiaulys Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 4, Vilnius LT-035, Lithuania
More informationThe purpose of this section is to derive the asymptotic distribution of the Pearson chi-square statistic. k (n j np j ) 2. np j.
Chapter 9 Pearson s chi-square test 9. Null hypothesis asymptotics Let X, X 2, be independent from a multinomial(, p) distribution, where p is a k-vector with nonnegative entries that sum to one. That
More informationBasic concepts of probability theory
Basic concepts of probability theory Random variable discrete/continuous random variable Transform Z transform, Laplace transform Distribution Geometric, mixed-geometric, Binomial, Poisson, exponential,
More informationLarge sample covariance matrices and the T 2 statistic
Large sample covariance matrices and the T 2 statistic EURANDOM, the Netherlands Joint work with W. Zhou Outline 1 2 Basic setting Let {X ij }, i, j =, be i.i.d. r.v. Write n s j = (X 1j,, X pj ) T and
More information1. Let A be a 2 2 nonzero real matrix. Which of the following is true?
1. Let A be a 2 2 nonzero real matrix. Which of the following is true? (A) A has a nonzero eigenvalue. (B) A 2 has at least one positive entry. (C) trace (A 2 ) is positive. (D) All entries of A 2 cannot
More informationConcentration of Measures by Bounded Couplings
Concentration of Measures by Bounded Couplings Subhankar Ghosh, Larry Goldstein and Ümit Işlak University of Southern California [arxiv:0906.3886] [arxiv:1304.5001] May 2013 Concentration of Measure Distributional
More informationHeavy-tailed time series
Heavy-tailed time series Spatio-temporal extremal dependence 1 Thomas Mikosch University of Copenhagen www.math.ku.dk/ mikosch/ 1 Rouen, June 2015 1 2 Contents 1. Heavy tails and extremal dependence -
More informationn! (k 1)!(n k)! = F (X) U(0, 1). (x, y) = n(n 1) ( F (y) F (x) ) n 2
Order statistics Ex. 4.1 (*. Let independent variables X 1,..., X n have U(0, 1 distribution. Show that for every x (0, 1, we have P ( X (1 < x 1 and P ( X (n > x 1 as n. Ex. 4.2 (**. By using induction
More informationConcentration inequalities for non-lipschitz functions
Concentration inequalities for non-lipschitz functions University of Warsaw Berkeley, October 1, 2013 joint work with Radosław Adamczak (University of Warsaw) Gaussian concentration (Sudakov-Tsirelson,
More informationInference For High Dimensional M-estimates. Fixed Design Results
: Fixed Design Results Lihua Lei Advisors: Peter J. Bickel, Michael I. Jordan joint work with Peter J. Bickel and Noureddine El Karoui Dec. 8, 2016 1/57 Table of Contents 1 Background 2 Main Results and
More informationThomas Mikosch and Daniel Straumann: Stable Limits of Martingale Transforms with Application to the Estimation of Garch Parameters
MaPhySto The Danish National Research Foundation: Network in Mathematical Physics and Stochastics Research Report no. 11 March 2004 Thomas Mikosch and Daniel Straumann: Stable Limits of Martingale Transforms
More information8.1 Concentration inequality for Gaussian random matrix (cont d)
MGMT 69: Topics in High-dimensional Data Analysis Falll 26 Lecture 8: Spectral clustering and Laplacian matrices Lecturer: Jiaming Xu Scribe: Hyun-Ju Oh and Taotao He, October 4, 26 Outline Concentration
More informationConcentration inequalities: basics and some new challenges
Concentration inequalities: basics and some new challenges M. Ledoux University of Toulouse, France & Institut Universitaire de France Measure concentration geometric functional analysis, probability theory,
More informationSusceptible-Infective-Removed Epidemics and Erdős-Rényi random
Susceptible-Infective-Removed Epidemics and Erdős-Rényi random graphs MSR-Inria Joint Centre October 13, 2015 SIR epidemics: the Reed-Frost model Individuals i [n] when infected, attempt to infect all
More informationOn the Bennett-Hoeffding inequality
On the Bennett-Hoeffding inequality of Iosif 1,2,3 1 Department of Mathematical Sciences Michigan Technological University 2 Supported by NSF grant DMS-0805946 3 Paper available at http://arxiv.org/abs/0902.4058
More informationInference for High Dimensional Robust Regression
Department of Statistics UC Berkeley Stanford-Berkeley Joint Colloquium, 2015 Table of Contents 1 Background 2 Main Results 3 OLS: A Motivating Example Table of Contents 1 Background 2 Main Results 3 OLS:
More informationStable limits for sums of dependent infinite variance random variables
Probab. Theory Relat. Fields DOI 10.1007/s00440-010-0276-9 Stable limits for sums of dependent infinite variance random variables Katarzyna Bartkiewicz Adam Jakubowski Thomas Mikosch Olivier Wintenberger
More informationA Functional Central Limit Theorem for an ARMA(p, q) Process with Markov Switching
Communications for Statistical Applications and Methods 2013, Vol 20, No 4, 339 345 DOI: http://dxdoiorg/105351/csam2013204339 A Functional Central Limit Theorem for an ARMAp, q) Process with Markov Switching
More informationEigenvalue variance bounds for Wigner and covariance random matrices
Eigenvalue variance bounds for Wigner and covariance random matrices S. Dallaporta University of Toulouse, France Abstract. This work is concerned with finite range bounds on the variance of individual
More informationLifetime Dependence Modelling using a Generalized Multivariate Pareto Distribution
Lifetime Dependence Modelling using a Generalized Multivariate Pareto Distribution Daniel Alai Zinoviy Landsman Centre of Excellence in Population Ageing Research (CEPAR) School of Mathematics, Statistics
More informationOperational Risk and Pareto Lévy Copulas
Operational Risk and Pareto Lévy Copulas Claudia Klüppelberg Technische Universität München email: cklu@ma.tum.de http://www-m4.ma.tum.de References: - Böcker, K. and Klüppelberg, C. (25) Operational VaR
More informationn! (k 1)!(n k)! = F (X) U(0, 1). (x, y) = n(n 1) ( F (y) F (x) ) n 2
Order statistics Ex. 4. (*. Let independent variables X,..., X n have U(0, distribution. Show that for every x (0,, we have P ( X ( < x and P ( X (n > x as n. Ex. 4.2 (**. By using induction or otherwise,
More informationLARGE DEVIATIONS FOR SOLUTIONS TO STOCHASTIC RECURRENCE EQUATIONS UNDER KESTEN S CONDITION D. BURACZEWSKI, E. DAMEK, T. MIKOSCH AND J.
LARGE DEVIATIONS FOR SOLUTIONS TO STOCHASTIC RECURRENCE EQUATIONS UNDER KESTEN S CONDITION D. BURACZEWSKI, E. DAMEK, T. MIKOSCH AND J. ZIENKIEWICZ Abstract. In this paper we prove large deviations results
More informationLecture 2: Review of Basic Probability Theory
ECE 830 Fall 2010 Statistical Signal Processing instructor: R. Nowak, scribe: R. Nowak Lecture 2: Review of Basic Probability Theory Probabilistic models will be used throughout the course to represent
More informationBasic concepts of probability theory
Basic concepts of probability theory Random variable discrete/continuous random variable Transform Z transform, Laplace transform Distribution Geometric, mixed-geometric, Binomial, Poisson, exponential,
More informationAsymptotic Tail Probabilities of Sums of Dependent Subexponential Random Variables
Asymptotic Tail Probabilities of Sums of Dependent Subexponential Random Variables Jaap Geluk 1 and Qihe Tang 2 1 Department of Mathematics The Petroleum Institute P.O. Box 2533, Abu Dhabi, United Arab
More informationLectures 6 7 : Marchenko-Pastur Law
Fall 2009 MATH 833 Random Matrices B. Valkó Lectures 6 7 : Marchenko-Pastur Law Notes prepared by: A. Ganguly We will now turn our attention to rectangular matrices. Let X = (X 1, X 2,..., X n ) R p n
More informationLecture 32: Asymptotic confidence sets and likelihoods
Lecture 32: Asymptotic confidence sets and likelihoods Asymptotic criterion In some problems, especially in nonparametric problems, it is difficult to find a reasonable confidence set with a given confidence
More informationLesson 4: Stationary stochastic processes
Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@univaq.it Stationary stochastic processes Stationarity is a rather intuitive concept, it means
More informationZeros of Random Analytic Functions
Zeros of Random Analytic Functions Zakhar Kabluchko University of Ulm September 3, 2013 Algebraic equations Consider a polynomial of degree n: P(x) = a n x n + a n 1 x n 1 +... + a 1 x + a 0. Here, a 0,
More informationApplications of Distance Correlation to Time Series
Applications of Distance Correlation to Time Series Richard A. Davis, Columbia University Muneya Matsui, Nanzan University Thomas Mikosch, University of Copenhagen Phyllis Wan, Columbia University May
More information2 Eigenvectors and Eigenvalues in abstract spaces.
MA322 Sathaye Notes on Eigenvalues Spring 27 Introduction In these notes, we start with the definition of eigenvectors in abstract vector spaces and follow with the more common definition of eigenvectors
More informationCS Lecture 19. Exponential Families & Expectation Propagation
CS 6347 Lecture 19 Exponential Families & Expectation Propagation Discrete State Spaces We have been focusing on the case of MRFs over discrete state spaces Probability distributions over discrete spaces
More informationA note on a Marčenko-Pastur type theorem for time series. Jianfeng. Yao
A note on a Marčenko-Pastur type theorem for time series Jianfeng Yao Workshop on High-dimensional statistics The University of Hong Kong, October 2011 Overview 1 High-dimensional data and the sample covariance
More informationJanuary 22, 2013 MEASURES OF SERIAL EXTREMAL DEPENDENCE AND THEIR ESTIMATION
January 22, 2013 MASURS OF SRIAL XTRMAL DPNDNC AND THIR STIMATION RICHARD A. DAVIS, THOMAS MIKOSCH, AND YUWI ZHAO Abstract. The goal of this paper is two-fold: 1. We review classical and recent measures
More informationThe Convergence Rate for the Normal Approximation of Extreme Sums
The Convergence Rate for the Normal Approximation of Extreme Sums Yongcheng Qi University of Minnesota Duluth WCNA 2008, Orlando, July 2-9, 2008 This talk is based on a joint work with Professor Shihong
More informationRandom Correlation Matrices, Top Eigenvalue with Heavy Tails and Financial Applications
Random Correlation Matrices, Top Eigenvalue with Heavy Tails and Financial Applications J.P Bouchaud with: M. Potters, G. Biroli, L. Laloux, M. A. Miceli http://www.cfm.fr Portfolio theory: Basics Portfolio
More informationRandom Matrices and Multivariate Statistical Analysis
Random Matrices and Multivariate Statistical Analysis Iain Johnstone, Statistics, Stanford imj@stanford.edu SEA 06@MIT p.1 Agenda Classical multivariate techniques Principal Component Analysis Canonical
More informationReconstruction from Anisotropic Random Measurements
Reconstruction from Anisotropic Random Measurements Mark Rudelson and Shuheng Zhou The University of Michigan, Ann Arbor Coding, Complexity, and Sparsity Workshop, 013 Ann Arbor, Michigan August 7, 013
More informationConcentration inequalities and the entropy method
Concentration inequalities and the entropy method Gábor Lugosi ICREA and Pompeu Fabra University Barcelona what is concentration? We are interested in bounding random fluctuations of functions of many
More informationSubmitted to the Brazilian Journal of Probability and Statistics
Submitted to the Brazilian Journal of Probability and Statistics Multivariate normal approximation of the maximum likelihood estimator via the delta method Andreas Anastasiou a and Robert E. Gaunt b a
More information