TESTING FOR THE BUFFERED AUTOREGRESSIVE PROCESSES (SUPPLEMENTARY MATERIAL)
|
|
- Albert Shields
- 5 years ago
- Views:
Transcription
1 TESTING FOR THE BUFFERED AUTOREGRESSIVE PROCESSES SUPPLEMENTARY MATERIAL) By Ke Zhu, Philip L.H. Yu ad Wai Keug Li Chiese Academy of Scieces ad Uiversity of Hog Kog APPENDIX: PROOFS I this appedix, we first give the proofs of Lemmas Deote C as a geeric costat which may vary from place to place i the rest of this paper. The proofs of Lemmas rely o the followig three basic lemmas: Lemma A.. Suppose that y t is strictly statioary, ergodic ad absolutely regular with mixig coefficiets βm) = Om A ) for some A > v/v ) ad r > v > ; ad there exists a A 0 > such that 2A 0 rv/r v) < A. The, for ay γ = r L, r U ) Γ, we have r v)/2a j 0 rv E Ir L < y t i r U ) <. j= i= Proof. First, deote ξ i = Ir L < y t i r U ). The, ξ i is strictly statioary, ergodic ad α-mixig with mixig coefficiets αm) = Om A ). Next, take ι [2A0 rv/r v) + ]/A + ), ), ad let p = j ι ad s = j/j ι, where x is the largest iteger ot greater tha x. Whe j j 0 is large eough, we ca always fid {ξ kp+ } s k=0, a subsequece of {ξ i} j i=. Furthermore, let Fm = σξ i, m i ). The, ξ kp+ F kp+2 kp+. Note that E [ξ kp+ ] < P a y t b) ρ 0, ). Hece, by Propositio 2.6 i Fa ad Yao 2003, p.72), we have that for j j 0, j E ξ i E i= { = [ s E ] ξ kp+ k=0 [ s k=0 ξ kp+ ] 6s )αp) + ρ s s k=0 C j/j ι j ι A + ρ j/jι. E [ξ kp+ ] } + s k=0 E [ξ kp+ ]
2 2 Therefore, sice r v)/2a 0 rv > 0, by usig the iequality x + y) k Cx k + y k ) for ay x, y, k > 0, it follows that A.) r v)/2a j 0 rv r v)/2a j 0 rv E ξ i j 0 ) + E ξ i j= i= j=j 0 i= [ j 0 ) + C j/j ι j ι A] r v)/2a 0 rv j=j 0 + C ρ j/jι r v)/2a 0rv. j=j 0 Sice ι > [2A 0 rv/r v) + ]/A + ), we have ιa + ι )r v)/2a 0 rv >, ad hece j= j ιa+ι )r v)/2a 0rv <, which implies that A.2) [ j/j ι ] r v)/2a0 rv j=j 0 j ι A [ ] r v)/2a0 rv j j= j ι j ι ) A <. O the other had, sice ρ j/jι r v)/2a 0 rv ) /j <, by Cauchy s root test, we have A.3) ρ j/jι r v)/2a 0rv < ρ j/jι r v)/2a 0 rv <. j=j 0 j= Now, the coclusio follows directly from A.)-A.3). This completes the proof. Lemma A.2. Suppose that the coditios i Lemma A. hold, ad y t has a bouded ad cotiuous desity fuctio. The, there exists a B 0 > such that for ay γ, γ 2 Γ, we have R t γ ) R t γ 2 ) 2rv/r v) C γ γ 2 r v)/2b0rv. Proof. Let γ = r L, r U ) ad γ 2 = r 2L, r 2U ). Sice R t γ) = Iy t d r L ) + R t γ)ir L < y t d r U ), we have R t γ ) R t γ 2 ) = t γ, γ 2 ) + Ir L < y t d r U ) [R t γ ) R t γ 2 )], where t γ, γ 2 ) = Ir 2L < y t d r L ) + R t γ 2 ) [Ir L < y t d r U ) Ir 2L < y t d r 2U )].
3 Thus, by iteratio we ca show that R t γ ) R t γ 2 ) TESTING FOR BAR PROCESSES 3 A.4) j = t γ, γ 2 ) + t j γ, γ 2 ) Ir L < y t i d r U ). j= i= Next, for brevity, we assume that r 2L r L r 2U r U, because the proofs for other cases are similar. Note that for ay j 0, R t j γ 2 ) ad Ir L < y t j d r U ) Ir 2L < y t j d r 2U ) = Ir 2U < y t j d r U ) Ir 2L < y t j d r L ). Let fx) be the desity fuctio of y t. Sice x fx) < ad t j γ, γ 2 ) 2, by Hölder s iequality ad Taylor s expasio, it follows that for ay s, A.5) E t j γ, γ 2 ) s 2 s E t j γ, γ 2 ) 2 s [2 x C γ γ 2. fx) r L r 2L + fx) r U r 2U x ] Let A 0 > be specified i Lemma A., ad choose B 0 such that /A 0 + /B 0 =. By Hölder s iequality ad A.5), we ca show that j 2rv/r v) E t jγ, γ 2 ) Ir L < y t i d r U ) i= { E[ t j γ, γ 2 )] } 2B /B 0rv/r v) 0 j E Ir L < y t i d r U ) i= /A 0 A.6) 2 [2B0rv/r v)] {E t j γ, γ 2 ) } /B 0 j E Ir L < y t i d r U ) i= /A 0 j C γ γ 2 /B 0 E Ir L < y t i d r U ) i= /A 0. By A.4)-A.6), Mikowski s iequality, Lemma A. ad the compactess of Γ, we
4 4 have R t γ ) R t γ 2 ) 2rv/r v) C γ γ 2 r v)/2rv + C γ γ 2 r v)/2b 0rv j E Ir L < y t i d r U ) j= This completes the proof. i= C γ γ 2 r v)/2b 0rv. r v)/2a 0 rv Lemma A.3. Suppose that the coditios i Lemma A.2 hold ad Ey 4 t <. The, X γx Σ γ 0 a.s. as. Proof. For brevity, we oly prove the uiform covergece for t= φ t γ), the last compoet of X γx, where φ t γ) = yt pr 2 t γ). First, for fix ε > 0, we partitio Γ by {B,, B Kε }, where B k = {r L, r U ); ω k < r L ω k+, ν k < r U ν k+ } Γ. Here, {ω k } ad {ν k } are chose such that A.7) ω k+ ω k ) r v)/2b 0rv < C ε ad ν k+ ν k ) r v)/2b 0rv < C ε, where B 0 > is specified as i Lemma A.2, ad C > 0 will be selected later. Next, we set f u t ε) = y 2 t pr t ω k+, ν k+ ) ad f l tε) = y 2 t pr t ω k, ν k ). By costructio, sice R t γ) is a odecreasig fuctio with respect to r L ad r U, for ay γ Γ, there is some k such that γ B k ad f l tε) φ t γ) f u t ε). Furthermore, sice rv/2rv r + v) <, we have yt p 2 < y 2 2rv/2rv r+v) t p <. 2
5 TESTING FOR BAR PROCESSES 5 Thus, by Hölder s iequality, Lemma A.2 ad A.7), we have E [ ft u ε) ftε) ] l yt p 2 R t ω k+, ν k+ ) R t ω k, ν k ) 2rv/2rv r+v) 2rv/r v) C [ ω k+ ω k ) r v)/2b0rv + ν k+ ν k ) ] r v)/2b 0rv 2CC ε. By settig C = 2C), we have E [ f u t ε) f l tε) ] ε. Thus, the coclusio holds accordig to Theorem 2 i Pollard 984, p.8). This completes the proof. Proof of Lemma 2.. First, sice K γγ is positive defiite by Assumptio 2., we kow that both Σ ad Σ γ are positive defiite. By usig the same argumet as for Lemma 2.iv) i Cha 990), it is ot hard to show that for every γ Γ, Σ γ Σ γ Σ Σ γ is positive defiite. Secod, by the ergodic theorem, it is easy to see that A.8) Third, by Lemma A.3 we have A.9) X γx X X Σ γ 0 ad as. Note that if H 0 holds, we have Σ a.s. as. X γx γ T γ = { X γ X γxx X) X } ε = X γ X)X X), I ) Z γε. Σ γ 0 a.s. The, i) ad ii) follow readily from A.8)-A.9). This completes the proof. Proof of Lemma 2.2. Deote G γ) Z γε = N x t γ)ε t. It is straightforward to show that the fiite dimesioal distributio of {G γ)} coverges to that of {σg γ }. By Pollard 990, Sec.0), we oly eed to verify the stochastic equicotiuity of {G γ)}. To establish it, we use Theorem, Applicatio
6 6 4 i Doukha, Massart, ad Rio 995, p.405); see also Adrews 993) ad Hase 996). First, the evelop fuctio is γ x t γ)ε t = x t ε t, where x t = γ x t γ). By Hölder s iequality ad Assumptio 2., we kow that the evelop fuctio is L 2v bouded. Next, for ay γ, γ 2 Γ, by Assumptios , Lemma A.2 ad Hölder s iequality, we have x t γ )ε t x t γ 2 )ε t 2v = h t γ )ε t h t γ 2 )ε t 2v x t ε t 2r R t γ ) R t γ 2 ) 2rv/r v) C x t 4r ε t 4r γ γ 2 r v)/2b 0rv C γ γ 2 r v)/2b 0rv for some B 0 >, where the last iequality holds sice x t 4r ε t 4r <. Now, followig the argumet i Hase 996, p.426), we kow that G γ) is stochastically equicotiuous. This completes the proof. Next, we give Lemmas A.4-A.6, i which Lemma A.4 is crucial for provig Lemma A.5, ad Lemmas A.5 ad A.6 are eeded to prove Corollary 2. ad Theorem 3., respectively. Lemma A.4. Suppose that y t is strictly statioary ad ergodic. The, i) 0 = O p ); ii) furthermore, if E y t 2 < ad E ε t 2 <, for ay a = o), we have a 0 x t x A.0) tr t γ) = O p) ad A.) a 0 h t γ)ε t = O p). Proof. First, by the ergodic theory, we have that M M Ia y t d b) = P a y t d b) κ > 0 as M. Thus, η > 0, there exists a iteger Mη) > 0 such that P M Ia y t d b) < κ < η. M 2 a.s.
7 By the defiitio of 0, it follows that A.2) TESTING FOR BAR PROCESSES 7 M P 0 > M) = P Ia y t d b) = 0 = P M Ia y t d b) = 0 M P M Ia y t d b) < κ M 2 < η, i.e., i) holds. Furthermore, by takig M = M 2, from A.2) ad Markov s iequality, it follows that η > 0, A.3) 0 P a x t x tr t γ) > M 0 = P a x t x tr t γ) > M, 0 M P max k a x t x tr t γ) > M p k M M k P a x t 2 > M k=p a M = O k k=p a M 2 ) M E x t 2 M = O a ) < η as is large eough. Thus, we kow that equatio A.0) holds. Next, by Hölder s iequality ad a similar argumet as for A.3), it is ot hard to show that η > 0, P a 0 h t γ)ε t > M O a ) < η as is large eough, i.e., A.) holds. This completes the proof.
8 8 Lemma A.5. If Assumptios hold, the it follows that uder H 0 or H, i) Xγ X ) γ X = o p ), ii) X γx γ X X ) γ γ = o p ), iii) T γ T γ = op ), where X γ ad T γ are defied i the same way as X γ ad T γ, respectively, with R t γ) beig replaced by R t γ). Proof. i) Note that Xγ X ) γ X = 0 x t x tr t γ). Hece, we kow that i) holds by takig a = /2 i equatio A.0). ii) By a similar argumet as for i), we ca show that ii) holds. iii) Note that whe λ 0 = φ 0, h / ), we have T γ T γ = Xγ X γ ) ε Xγ X γ ) XX X) X ε Xγ X γ ) XX X) X X γ0 h X γxx X) X X γ0 X γ0 ) h + X γ X γ0 X γ X ) γ0 h I γ) I 2 γ) I 3 γ) I 4 γ) + I 5 γ) say. First, sice I γ) = 0 h /4 /4 t γ)ε t, it follows that γ I γ) = o p ) by takig a = /4 i equatio A.). Next, sice I 2 γ) = 0 x t x tr t γ) X X ) X ε, we have that γ I 2 γ) = o p ) from i). Similarly, we ca show that γ I i γ) = o p ) for i = 3, 4, 5. Hece, uder H 0 i.e., h 0) or H, we kow that iii) holds. This completes the proof.
9 TESTING FOR BAR PROCESSES 9 Lemma A.6. If Assumptios hold, the it follows that uder H 0 or H, λ γ) λ 0 = O p ). Proof. First, for ay γ Γ, by Taylor s expasio we have [ ε 2 t λ γ), γ) ε 2 t λ 0, γ) ] A.4) = λ γ) λ 0 ) + λ γ) λ 0 ) 2ε t λ 0, γ)x t γ) x t γ)x t γ) λ γ) λ 0 ). Next, whe λ 0 = φ 0, h / ), we ca show that A.5) N ε t λ 0, γ)x t γ) = Z γε + N x t γ)[x t γ 0 ) x t γ)] λ 0 = Z γε + G γ). x t x t[r t γ 0 ) R t γ)] x t x t[r t γ)r t γ 0 ) R t γ)] h Let λ mi γ) > 0 be the miimum eigevalue of K γγ. The, by equatios A.4)- A.5), η > 0, there exists a Mη) > 0 such that ) P λ γ) λ 0 > M = P λ γ) λ 0 > M, for some γ Γ) [ ε 2 t λ γ), γ) ε 2 t λ 0, γ) ] 0 P λ γ) λ 0 > M, 2 λ γ) λ 0 G γ) + λ γ) λ 0 2 [λ mi γ) + o p )] 0 for some γ Γ ) P M < λ γ) λ 0 2[λ mi γ) + o p )] G γ) for some γ Γ) P G γ) > M[λ mi γ) + o p )]/2 for some γ Γ) η,
10 0 where the last iequality holds because G γ) = O p ) by Lemma 2.2 ad Lemma A.3. Hece, uder H 0 i.e., h 0) or H, our coclusio holds. This completes the proof. Proof of Corollary 2.. The coclusio follows directly from Theorems ad Lemma A.5. Proof of Theorem 3.. We use the method i the proof of Theorem 2 i Hase 996). Let W deote the set of samples ω for which A.6) A.7) lim lim x t γ) ε 2 t < a.s., x t γ)x t δ) ε 2 t σ 2 K γδ γ,δ Γ 0 a.s. Sice x t γ) 2 x t ad E x t ε 2 t ergodic theorem we have < due to Assumptio 2., by the lim 2 x t γ) ε 2 t lim x t ε 2 t < a.s., i.e., A.6) holds. Furthermore, by Assumptios ad a similar argumet as for Lemma A.3, it is ot hard to see that lim x t γ)x t δ) ε 2 t σ 2 K γδ 0 a.s., γ,δ Γ i.e., A.7) holds. Thus, P W ) =. Take ay ω W. For the remaider of the proof, all operatios are coditioally o ω, ad hece all of the radomess appears i the i.i.d. N0, ) variables {v t }. Defie Zγ) = N x t γ)ε t v t. By usig the same argumet as i Hase 996, p ), we have A.8) Z γ) σg γ a.s. as. Note that Ẑγ) Z γ) x t γ)x t γ) v t λ γ) λ 0 ).
11 TESTING FOR BAR PROCESSES Usig the same argumet as for A.8) see, e.g., Hase 996, p.427)), we have A.9) x t γ)x t γ) v t 0 a.s. as. Now, by Lemma A.6 ad A.9), it follows that uder H 0 or H, A.20) Ẑ γ) Z γ) 0 i probability as. Thus, by A.8) ad A.20), we kow that uder H 0 or H, A.2) Ẑ γ) σg γ i probability as. Next, we cosider the fuctioal L : x ) D 2p+2 Γ) σ xγ) Ω 2 γ xγ), where D 2p+2 Γ) deotes the fuctio spaces of all fuctios, mappig R 2 Γ) ito R 2p+2, that are right cotiuous ad have right-had limits. Clearly, L ) is a cotiuous fuctioal; see e.g., Cha 990, p.89). By the cotiuous mappig theory ad A.2), it follows that uder H 0 or H, A.22) LẐγ)) LσG γ ) i probability as. Furthermore, sice σ 2 σ 2 a.s. ad X γ), I) [X 2 γ)] X γ), I) Ω γ uiformly i γ by Lemma A.3, we have that A.23) LR ˆ γ) = LẐγ)) + o p ). Fially, the coclusio follows from A.22)-A.23). This completes the proof. Proof of Corollary 3.. Coditioal o the sample {y 0,, y N }, let ˆF,J ad ˆF be the coditioal empirical c.d.f. ad c.d.f. of LR ˆ, respectively. The, P ) LR c J,α = E [ P )] LR c J,α y 0,, y N = E [ P ˆF,J LR ) α y 0,, y N )].
12 2 By the Gliveko-Catelli Theorem ad Theorem 3., it follows that uder H 0 or H, A.24) lim lim P ) LR c J,α J = lim E [ P )] ˆF LR ) α y 0,, y N = lim E [P F 0 LR ) α y 0,, y N )] = lim P F 0 LR ) α), where F 0 is the c.d.f. of G γω γ G γ. Thus, by A.24) ad Theorem 2., uder H 0 we have lim lim P ) ) LR c J,α = P G γω γ G γ F0 α) J = α, i.e., i) holds. Furthermore, by A.24) ad Theorem 2.2, uder H we have lim lim lim P ) LR c J h,α = lim P B h F0 α) ) =, J h where B h { G γ Ω γ G γ + h µ γγ0 h } ad the last equatio holds sice B h i probability as h. Thus, ii) holds. This completes the proof. REFERENCES [] Adrews, D.W.K. 993) A itroductio to ecoometric applicatios of fuctioal limit theory for depedet radom variables. Ecoometric Reviews 2, [2] Cha, K.S. 990) Testig for threshold autoregressio. Aals of Statistics 8, [3] Doukha, P., Massart, P., ad Rio, E. 995) Ivariace priciples for absolutely regular empirical processes. Aales de I Istitut H. Poicare [4] Fa, J. ad Yao, Q. 2003) Noliear time series: Noparametric ad parametric methods. Spriger, New York. [5] Hase, B.E. 996) Iferece whe a uisace parameter is ot idetified uder the ull hypothesis. Ecoometrica 64, [6] Pollard, D. 984) Covergece of Stochastic Processes. New York: Spriger. [7] Pollard, D. 990) Empirical processes: Theory ad applicatios. NSF-CMBS Regioal Coferece Series i Probability ad Statistics, Vol 2. Hayward, CA: Istitute of Mathematical Statistics. Chiese Academy of Scieces Istitute of Applied Mathematics HaiDia District, Zhogguacu Bei Jig, Chia zkxaa@ust.hk Departmet of Statistics ad Actuarial Sciece Uiversity of Hog Kog Pokfulam Road, Hog Kog plhyu@hku.hk hrtlwk@hku.hk
Convergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More information1 Convergence in Probability and the Weak Law of Large Numbers
36-752 Advaced Probability Overview Sprig 2018 8. Covergece Cocepts: i Probability, i L p ad Almost Surely Istructor: Alessadro Rialdo Associated readig: Sec 2.4, 2.5, ad 4.11 of Ash ad Doléas-Dade; Sec
More informationSolution to Chapter 2 Analytical Exercises
Nov. 25, 23, Revised Dec. 27, 23 Hayashi Ecoometrics Solutio to Chapter 2 Aalytical Exercises. For ay ε >, So, plim z =. O the other had, which meas that lim E(z =. 2. As show i the hit, Prob( z > ε =
More informationIntegrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number
MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios
More informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationBerry-Esseen bounds for self-normalized martingales
Berry-Essee bouds for self-ormalized martigales Xiequa Fa a, Qi-Ma Shao b a Ceter for Applied Mathematics, Tiaji Uiversity, Tiaji 30007, Chia b Departmet of Statistics, The Chiese Uiversity of Hog Kog,
More informationEmpirical Processes: Glivenko Cantelli Theorems
Empirical Processes: Gliveko Catelli Theorems Mouliath Baerjee Jue 6, 200 Gliveko Catelli classes of fuctios The reader is referred to Chapter.6 of Weller s Torgo otes, Chapter??? of VDVW ad Chapter 8.3
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationMAS111 Convergence and Continuity
MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece
More informationThe Bilateral Laplace Transform of the Positive Even Functions and a Proof of Riemann Hypothesis
The Bilateral Laplace Trasform of the Positive Eve Fuctios ad a Proof of Riema Hypothesis Seog Wo Cha Ph.D. swcha@dgu.edu Abstract We show that some iterestig properties of the bilateral Laplace trasform
More informationKolmogorov-Smirnov type Tests for Local Gaussianity in High-Frequency Data
Proceedigs 59th ISI World Statistics Cogress, 5-30 August 013, Hog Kog (Sessio STS046) p.09 Kolmogorov-Smirov type Tests for Local Gaussiaity i High-Frequecy Data George Tauche, Duke Uiversity Viktor Todorov,
More informationThe log-behavior of n p(n) and n p(n)/n
Ramauja J. 44 017, 81-99 The log-behavior of p ad p/ William Y.C. Che 1 ad Ke Y. Zheg 1 Ceter for Applied Mathematics Tiaji Uiversity Tiaji 0007, P. R. Chia Ceter for Combiatorics, LPMC Nakai Uivercity
More informationSupplementary Material for Fast Stochastic AUC Maximization with O(1/n)-Convergence Rate
Supplemetary Material for Fast Stochastic AUC Maximizatio with O/-Covergece Rate Migrui Liu Xiaoxua Zhag Zaiyi Che Xiaoyu Wag 3 iabao Yag echical Lemmas ized versio of Hoeffdig s iequality, ote that We
More informationSelf-normalized deviation inequalities with application to t-statistic
Self-ormalized deviatio iequalities with applicatio to t-statistic Xiequa Fa Ceter for Applied Mathematics, Tiaji Uiversity, 30007 Tiaji, Chia Abstract Let ξ i i 1 be a sequece of idepedet ad symmetric
More informationBIRKHOFF ERGODIC THEOREM
BIRKHOFF ERGODIC THEOREM Abstract. We will give a proof of the poitwise ergodic theorem, which was first proved by Birkhoff. May improvemets have bee made sice Birkhoff s orgial proof. The versio we give
More informationNotes 27 : Brownian motion: path properties
Notes 27 : Browia motio: path properties Math 733-734: Theory of Probability Lecturer: Sebastie Roch Refereces:[Dur10, Sectio 8.1], [MP10, Sectio 1.1, 1.2, 1.3]. Recall: DEF 27.1 (Covariace) Let X = (X
More informationUniversity of Colorado Denver Dept. Math. & Stat. Sciences Applied Analysis Preliminary Exam 13 January 2012, 10:00 am 2:00 pm. Good luck!
Uiversity of Colorado Dever Dept. Math. & Stat. Scieces Applied Aalysis Prelimiary Exam 13 Jauary 01, 10:00 am :00 pm Name: The proctor will let you read the followig coditios before the exam begis, ad
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationChapter 7 Isoperimetric problem
Chapter 7 Isoperimetric problem Recall that the isoperimetric problem (see the itroductio its coectio with ido s proble) is oe of the most classical problem of a shape optimizatio. It ca be formulated
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More information5.1 A mutual information bound based on metric entropy
Chapter 5 Global Fao Method I this chapter, we exted the techiques of Chapter 2.4 o Fao s method the local Fao method) to a more global costructio. I particular, we show that, rather tha costructig a local
More informationAn Introduction to Asymptotic Theory
A Itroductio to Asymptotic Theory Pig Yu School of Ecoomics ad Fiace The Uiversity of Hog Kog Pig Yu (HKU) Asymptotic Theory 1 / 20 Five Weapos i Asymptotic Theory Five Weapos i Asymptotic Theory Pig Yu
More informationNotes 19 : Martingale CLT
Notes 9 : Martigale CLT Math 733-734: Theory of Probability Lecturer: Sebastie Roch Refereces: [Bil95, Chapter 35], [Roc, Chapter 3]. Sice we have ot ecoutered weak covergece i some time, we first recall
More informationAsymptotic distribution of products of sums of independent random variables
Proc. Idia Acad. Sci. Math. Sci. Vol. 3, No., May 03, pp. 83 9. c Idia Academy of Scieces Asymptotic distributio of products of sums of idepedet radom variables YANLING WANG, SUXIA YAO ad HONGXIA DU ollege
More informationRandom Walks on Discrete and Continuous Circles. by Jeffrey S. Rosenthal School of Mathematics, University of Minnesota, Minneapolis, MN, U.S.A.
Radom Walks o Discrete ad Cotiuous Circles by Jeffrey S. Rosethal School of Mathematics, Uiversity of Miesota, Mieapolis, MN, U.S.A. 55455 (Appeared i Joural of Applied Probability 30 (1993), 780 789.)
More informationInformation Theory and Statistics Lecture 4: Lempel-Ziv code
Iformatio Theory ad Statistics Lecture 4: Lempel-Ziv code Łukasz Dębowski ldebowsk@ipipa.waw.pl Ph. D. Programme 203/204 Etropy rate is the limitig compressio rate Theorem For a statioary process (X i)
More informationLecture 8: Convergence of transformations and law of large numbers
Lecture 8: Covergece of trasformatios ad law of large umbers Trasformatio ad covergece Trasformatio is a importat tool i statistics. If X coverges to X i some sese, we ofte eed to check whether g(x ) coverges
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More informationEntropy Rates and Asymptotic Equipartition
Chapter 29 Etropy Rates ad Asymptotic Equipartitio Sectio 29. itroduces the etropy rate the asymptotic etropy per time-step of a stochastic process ad shows that it is well-defied; ad similarly for iformatio,
More informationA RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS
J. Japa Statist. Soc. Vol. 41 No. 1 2011 67 73 A RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS Yoichi Nishiyama* We cosider k-sample ad chage poit problems for idepedet data i a
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationProperties of Fuzzy Length on Fuzzy Set
Ope Access Library Joural 206, Volume 3, e3068 ISSN Olie: 2333-972 ISSN Prit: 2333-9705 Properties of Fuzzy Legth o Fuzzy Set Jehad R Kider, Jaafar Imra Mousa Departmet of Mathematics ad Computer Applicatios,
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More informationKernel density estimator
Jauary, 07 NONPARAMETRIC ERNEL DENSITY ESTIMATION I this lecture, we discuss kerel estimatio of probability desity fuctios PDF Noparametric desity estimatio is oe of the cetral problems i statistics I
More informationS1 Notation and Assumptions
Statistica Siica: Supplemet Robust-BD Estimatio ad Iferece for Varyig-Dimesioal Geeral Liear Models Chumig Zhag Xiao Guo Che Cheg Zhegju Zhag Uiversity of Wiscosi-Madiso Supplemetary Material S Notatio
More informationApproximation theorems for localized szász Mirakjan operators
Joural of Approximatio Theory 152 (2008) 125 134 www.elsevier.com/locate/jat Approximatio theorems for localized szász Miraja operators Lise Xie a,,1, Tigfa Xie b a Departmet of Mathematics, Lishui Uiversity,
More informationEFFECTIVE WLLN, SLLN, AND CLT IN STATISTICAL MODELS
EFFECTIVE WLLN, SLLN, AND CLT IN STATISTICAL MODELS Ryszard Zieliński Ist Math Polish Acad Sc POBox 21, 00-956 Warszawa 10, Polad e-mail: rziel@impagovpl ABSTRACT Weak laws of large umbers (W LLN), strog
More informationLecture Notes for Analysis Class
Lecture Notes for Aalysis Class Topological Spaces A topology for a set X is a collectio T of subsets of X such that: (a) X ad the empty set are i T (b) Uios of elemets of T are i T (c) Fiite itersectios
More informationSTAT Homework 1 - Solutions
STAT-36700 Homework 1 - Solutios Fall 018 September 11, 018 This cotais solutios for Homework 1. Please ote that we have icluded several additioal commets ad approaches to the problems to give you better
More informationSupplementary Material for Fast Stochastic AUC Maximization with O(1/n)-Convergence Rate
Supplemetary Material for Fast Stochastic AUC Maximizatio with O/-Covergece Rate Migrui Liu Xiaoxua Zhag Zaiyi Che Xiaoyu Wag 3 iabao Yag echical Lemmas ized versio of Hoeffdig s iequality, ote that We
More informationHomework 4. x n x X = f(x n x) +
Homework 4 1. Let X ad Y be ormed spaces, T B(X, Y ) ad {x } a sequece i X. If x x weakly, show that T x T x weakly. Solutio: We eed to show that g(t x) g(t x) g Y. It suffices to do this whe g Y = 1.
More informationResearch Article On the Strong Laws for Weighted Sums of ρ -Mixing Random Variables
Hidawi Publishig Corporatio Joural of Iequalities ad Applicatios Volume 2011, Article ID 157816, 8 pages doi:10.1155/2011/157816 Research Article O the Strog Laws for Weighted Sums of ρ -Mixig Radom Variables
More informationMa 530 Introduction to Power Series
Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power
More informationECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors
ECONOMETRIC THEORY MODULE XIII Lecture - 34 Asymptotic Theory ad Stochastic Regressors Dr. Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Asymptotic theory The asymptotic
More informationTheorem 3. A subset S of a topological space X is compact if and only if every open cover of S by open sets in X has a finite subcover.
Compactess Defiitio 1. A cover or a coverig of a topological space X is a family C of subsets of X whose uio is X. A subcover of a cover C is a subfamily of C which is a cover of X. A ope cover of X is
More information32 estimating the cumulative distribution function
32 estimatig the cumulative distributio fuctio 4.6 types of cofidece itervals/bads Let F be a class of distributio fuctios F ad let θ be some quatity of iterest, such as the mea of F or the whole fuctio
More information2 Markov Chain Monte Carlo Sampling
22 Part I. Markov Chais ad Stochastic Samplig Figure 10: Hard-core colourig of a lattice. 2 Markov Chai Mote Carlo Samplig We ow itroduce Markov chai Mote Carlo (MCMC) samplig, which is a extremely importat
More informationECE 901 Lecture 12: Complexity Regularization and the Squared Loss
ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality
More informationlim za n n = z lim a n n.
Lecture 6 Sequeces ad Series Defiitio 1 By a sequece i a set A, we mea a mappig f : N A. It is customary to deote a sequece f by {s } where, s := f(). A sequece {z } of (complex) umbers is said to be coverget
More informationIt is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial.
Taylor Polyomials ad Taylor Series It is ofte useful to approximate complicated fuctios usig simpler oes We cosider the task of approximatig a fuctio by a polyomial If f is at least -times differetiable
More informationNotes 5 : More on the a.s. convergence of sums
Notes 5 : More o the a.s. covergece of sums Math 733-734: Theory of Probability Lecturer: Sebastie Roch Refereces: Dur0, Sectios.5; Wil9, Sectio 4.7, Shi96, Sectio IV.4, Dur0, Sectio.. Radom series. Three-series
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 3 9/11/2013. Large deviations Theory. Cramér s Theorem
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/5.070J Fall 203 Lecture 3 9//203 Large deviatios Theory. Cramér s Theorem Cotet.. Cramér s Theorem. 2. Rate fuctio ad properties. 3. Chage of measure techique.
More informationLet us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.
Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,
More informationOptimally Sparse SVMs
A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but
More informationCentral Limit Theorem using Characteristic functions
Cetral Limit Theorem usig Characteristic fuctios RogXi Guo MAT 477 Jauary 20, 2014 RogXi Guo (2014 Cetral Limit Theorem usig Characteristic fuctios Jauary 20, 2014 1 / 15 Itroductio study a radom variable
More informationAppendix to Quicksort Asymptotics
Appedix to Quicksort Asymptotics James Alle Fill Departmet of Mathematical Scieces The Johs Hopkis Uiversity jimfill@jhu.edu ad http://www.mts.jhu.edu/~fill/ ad Svate Jaso Departmet of Mathematics Uppsala
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013 Fuctioal Law of Large Numbers. Costructio of the Wieer Measure Cotet. 1. Additioal techical results o weak covergece
More informationOn Weak and Strong Convergence Theorems for a Finite Family of Nonself I-asymptotically Nonexpansive Mappings
Mathematica Moravica Vol. 19-2 2015, 49 64 O Weak ad Strog Covergece Theorems for a Fiite Family of Noself I-asymptotically Noexpasive Mappigs Birol Güdüz ad Sezgi Akbulut Abstract. We prove the weak ad
More informationPeriodic solutions for a class of second-order Hamiltonian systems of prescribed energy
Electroic Joural of Qualitative Theory of Differetial Equatios 215, No. 77, 1 1; doi: 1.14232/ejqtde.215.1.77 http://www.math.u-szeged.hu/ejqtde/ Periodic solutios for a class of secod-order Hamiltoia
More informationAppendix to: Hypothesis Testing for Multiple Mean and Correlation Curves with Functional Data
Appedix to: Hypothesis Testig for Multiple Mea ad Correlatio Curves with Fuctioal Data Ao Yua 1, Hog-Bi Fag 1, Haiou Li 1, Coli O. Wu, Mig T. Ta 1, 1 Departmet of Biostatistics, Bioiformatics ad Biomathematics,
More informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
More informationfor all x ; ;x R. A ifiite sequece fx ; g is said to be ND if every fiite subset X ; ;X is ND. The coditios (.) ad (.3) are equivalet for =, but these
sub-gaussia techiques i provig some strog it theorems Λ M. Amii A. Bozorgia Departmet of Mathematics, Faculty of Scieces Sista ad Baluchesta Uiversity, Zaheda, Ira Amii@hamoo.usb.ac.ir, Fax:054446565 Departmet
More informationON POINTWISE BINOMIAL APPROXIMATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 57-66 ON POINTWISE BINOMIAL APPROXIMATION BY w-functions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece
More informationPrecise Rates in Complete Moment Convergence for Negatively Associated Sequences
Commuicatios of the Korea Statistical Society 29, Vol. 16, No. 5, 841 849 Precise Rates i Complete Momet Covergece for Negatively Associated Sequeces Dae-Hee Ryu 1,a a Departmet of Computer Sciece, ChugWoo
More informationarxiv: v1 [math.pr] 4 Dec 2013
Squared-Norm Empirical Process i Baach Space arxiv:32005v [mathpr] 4 Dec 203 Vicet Q Vu Departmet of Statistics The Ohio State Uiversity Columbus, OH vqv@statosuedu Abstract Jig Lei Departmet of Statistics
More informationSolutions to home assignments (sketches)
Matematiska Istitutioe Peter Kumli 26th May 2004 TMA401 Fuctioal Aalysis MAN670 Applied Fuctioal Aalysis 4th quarter 2003/2004 All documet cocerig the course ca be foud o the course home page: http://www.math.chalmers.se/math/grudutb/cth/tma401/
More informationEconometrica Supplementary Material
Ecoometrica Supplemetary Material SUPPLEMENT TO NONPARAMETRIC INSTRUMENTAL VARIABLES ESTIMATION OF A QUANTILE REGRESSION MODEL : MATHEMATICAL APPENDIX (Ecoometrica, Vol. 75, No. 4, July 27, 1191 128) BY
More informationJournal of Multivariate Analysis. Superefficient estimation of the marginals by exploiting knowledge on the copula
Joural of Multivariate Aalysis 102 (2011) 1315 1319 Cotets lists available at ScieceDirect Joural of Multivariate Aalysis joural homepage: www.elsevier.com/locate/jmva Superefficiet estimatio of the margials
More informationECE 901 Lecture 14: Maximum Likelihood Estimation and Complexity Regularization
ECE 90 Lecture 4: Maximum Likelihood Estimatio ad Complexity Regularizatio R Nowak 5/7/009 Review : Maximum Likelihood Estimatio We have iid observatios draw from a ukow distributio Y i iid p θ, i,, where
More informationThis section is optional.
4 Momet Geeratig Fuctios* This sectio is optioal. The momet geeratig fuctio g : R R of a radom variable X is defied as g(t) = E[e tx ]. Propositio 1. We have g () (0) = E[X ] for = 1, 2,... Proof. Therefore
More informationThe Wasserstein distances
The Wasserstei distaces March 20, 2011 This documet presets the proof of the mai results we proved o Wasserstei distaces themselves (ad ot o curves i the Wasserstei space). I particular, triagle iequality
More informationLecture 19: Convergence
Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may
More informationSlide Set 13 Linear Model with Endogenous Regressors and the GMM estimator
Slide Set 13 Liear Model with Edogeous Regressors ad the GMM estimator Pietro Coretto pcoretto@uisa.it Ecoometrics Master i Ecoomics ad Fiace (MEF) Uiversità degli Studi di Napoli Federico II Versio: Friday
More informationRademacher Complexity
EECS 598: Statistical Learig Theory, Witer 204 Topic 0 Rademacher Complexity Lecturer: Clayto Scott Scribe: Ya Deg, Kevi Moo Disclaimer: These otes have ot bee subjected to the usual scrutiy reserved for
More information1 Review and Overview
DRAFT a fial versio will be posted shortly CS229T/STATS231: Statistical Learig Theory Lecturer: Tegyu Ma Lecture #3 Scribe: Migda Qiao October 1, 2013 1 Review ad Overview I the first half of this course,
More informationExponential Functions and Taylor Series
MATH 4530: Aalysis Oe Expoetial Fuctios ad Taylor Series James K. Peterso Departmet of Biological Scieces ad Departmet of Mathematical Scieces Clemso Uiversity March 29, 2017 MATH 4530: Aalysis Oe Outlie
More informationHAJEK-RENYI-TYPE INEQUALITY FOR SOME NONMONOTONIC FUNCTIONS OF ASSOCIATED RANDOM VARIABLES
HAJEK-RENYI-TYPE INEQUALITY FOR SOME NONMONOTONIC FUNCTIONS OF ASSOCIATED RANDOM VARIABLES ISHA DEWAN AND B. L. S. PRAKASA RAO Received 1 April 005; Revised 6 October 005; Accepted 11 December 005 Let
More informationModern Discrete Probability Spectral Techniques
Moder Discrete Probability VI - Spectral Techiques Backgroud Sébastie Roch UW Madiso Mathematics December 1, 2014 1 Review 2 3 4 Mixig time I Theorem (Covergece to statioarity) Cosider a fiite state space
More information17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15
17. Joit distributios of extreme order statistics Lehma 5.1; Ferguso 15 I Example 10., we derived the asymptotic distributio of the maximum from a radom sample from a uiform distributio. We did this usig
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS
More informationExponential Functions and Taylor Series
Expoetial Fuctios ad Taylor Series James K. Peterso Departmet of Biological Scieces ad Departmet of Mathematical Scieces Clemso Uiversity March 29, 207 Outlie Revistig the Expoetial Fuctio Taylor Series
More informationM17 MAT25-21 HOMEWORK 5 SOLUTIONS
M17 MAT5-1 HOMEWORK 5 SOLUTIONS 1. To Had I Cauchy Codesatio Test. Exercise 1: Applicatio of the Cauchy Codesatio Test Use the Cauchy Codesatio Test to prove that 1 diverges. Solutio 1. Give the series
More informationMAXIMAL INEQUALITIES AND STRONG LAW OF LARGE NUMBERS FOR AANA SEQUENCES
Commu Korea Math Soc 26 20, No, pp 5 6 DOI 0434/CKMS20265 MAXIMAL INEQUALITIES AND STRONG LAW OF LARGE NUMBERS FOR AANA SEQUENCES Wag Xueju, Hu Shuhe, Li Xiaoqi, ad Yag Wezhi Abstract Let {X, } be a sequece
More informationDifferent kinds of Mathematical Induction
Differet ids of Mathematical Iductio () Mathematical Iductio Give A N, [ A (a A a A)] A N () (First) Priciple of Mathematical Iductio Let P() be a propositio (ope setece), if we put A { : N p() is true}
More informationA constructive analysis of convex-valued demand correspondence for weakly uniformly rotund and monotonic preference
MPRA Muich Persoal RePEc Archive A costructive aalysis of covex-valued demad correspodece for weakly uiformly rotud ad mootoic preferece Yasuhito Taaka ad Atsuhiro Satoh. May 04 Olie at http://mpra.ub.ui-mueche.de/55889/
More informationTR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT
TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the
More informationSieve Estimators: Consistency and Rates of Convergence
EECS 598: Statistical Learig Theory, Witer 2014 Topic 6 Sieve Estimators: Cosistecy ad Rates of Covergece Lecturer: Clayto Scott Scribe: Julia Katz-Samuels, Brado Oselio, Pi-Yu Che Disclaimer: These otes
More informationStrong Convergence Theorems According. to a New Iterative Scheme with Errors for. Mapping Nonself I-Asymptotically. Quasi-Nonexpansive Types
It. Joural of Math. Aalysis, Vol. 4, 00, o. 5, 37-45 Strog Covergece Theorems Accordig to a New Iterative Scheme with Errors for Mappig Noself I-Asymptotically Quasi-Noexpasive Types Narogrit Puturog Mathematics
More informationAn alternative proof of a theorem of Aldous. concerning convergence in distribution for martingales.
A alterative proof of a theorem of Aldous cocerig covergece i distributio for martigales. Maurizio Pratelli Dipartimeto di Matematica, Uiversità di Pisa. Via Buoarroti 2. I-56127 Pisa, Italy e-mail: pratelli@dm.uipi.it
More informationFirst Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise
First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I-1 Part I - 203A A radom variable X is distributed with the margial desity: >
More informationProbability 2 - Notes 10. Lemma. If X is a random variable and g(x) 0 for all x in the support of f X, then P(g(X) 1) E[g(X)].
Probability 2 - Notes 0 Some Useful Iequalities. Lemma. If X is a radom variable ad g(x 0 for all x i the support of f X, the P(g(X E[g(X]. Proof. (cotiuous case P(g(X Corollaries x:g(x f X (xdx x:g(x
More informationECE 330:541, Stochastic Signals and Systems Lecture Notes on Limit Theorems from Probability Fall 2002
ECE 330:541, Stochastic Sigals ad Systems Lecture Notes o Limit Theorems from robability Fall 00 I practice, there are two ways we ca costruct a ew sequece of radom variables from a old sequece of radom
More informationGamma Distribution and Gamma Approximation
Gamma Distributio ad Gamma Approimatio Xiaomig Zeg a Fuhua (Frak Cheg b a Xiame Uiversity, Xiame 365, Chia mzeg@jigia.mu.edu.c b Uiversity of Ketucky, Leigto, Ketucky 456-46, USA cheg@cs.uky.edu Abstract
More informationRank tests and regression rank scores tests in measurement error models
Rak tests ad regressio rak scores tests i measuremet error models J. Jurečková ad A.K.Md.E. Saleh Charles Uiversity i Prague ad Carleto Uiversity i Ottawa Abstract The rak ad regressio rak score tests
More informationAn almost sure invariance principle for trimmed sums of random vectors
Proc. Idia Acad. Sci. Math. Sci. Vol. 20, No. 5, November 200, pp. 6 68. Idia Academy of Scieces A almost sure ivariace priciple for trimmed sums of radom vectors KE-ANG FU School of Statistics ad Mathematics,
More informationAnalytic Continuation
Aalytic Cotiuatio The stadard example of this is give by Example Let h (z) = 1 + z + z 2 + z 3 +... kow to coverge oly for z < 1. I fact h (z) = 1/ (1 z) for such z. Yet H (z) = 1/ (1 z) is defied for
More informationA Note on the Kolmogorov-Feller Weak Law of Large Numbers
Joural of Mathematical Research with Applicatios Mar., 015, Vol. 35, No., pp. 3 8 DOI:10.3770/j.iss:095-651.015.0.013 Http://jmre.dlut.edu.c A Note o the Kolmogorov-Feller Weak Law of Large Numbers Yachu
More informationResearch Article Moment Inequality for ϕ-mixing Sequences and Its Applications
Hidawi Publishig Corporatio Joural of Iequalities ad Applicatios Volume 2009, Article ID 379743, 2 pages doi:0.55/2009/379743 Research Article Momet Iequality for ϕ-mixig Sequeces ad Its Applicatios Wag
More information