EXACT CONVERGENCE RATE AND LEADING TERM IN THE CENTRAL LIMIT THEOREM FOR USTATISTICS


 Leslie Hudson
 2 years ago
 Views:
Transcription
1 Statistica Sinica 6006, EXACT CONVERGENCE RATE AND LEADING TERM IN THE CENTRAL LIMIT THEOREM FOR USTATISTICS Qiying Wang and Neville C Weber The University of Sydney Abstract: The leading term in the normal approimation to the distribution of U statistics of degree is derived. This result is applied to establish the eact rate of convergence in the Central Limit Theorem for Ustatistics and to obtain the oneterm Edgeworth epansion for the distribution function. Analogous results for more general Utype statistics are also considered. Key words and phrases: BerryEsséen theorem, characterisation of rate of convergence, Edgeworth epansion, optimal moments, nonlattice condition, Ustatistics, Lstatistics.. Introduction and Main Results Let X, X,..., X n be a sequence of independent and identically distributed i.i.d. random variables. Let h, y be a realvalued Borel measurable function, symmetric in its arguments with EhX, X = 0. For n, a Ustatistic of degree with kernel h, y is defined by n U n = hx i, X j. i<j n Write g = Eh, X and φ, y = h, y g gy. The statistic U n may be represented as U n = n n j= n gx j + i<j n φx i, X j := U n + U n. See, for eample, Lee 990, p.5. Throughout we assume that Eg X =. This assumption implies that nun / is a standard sum of nondegenerate iid random variables and its distribution may be approimated by a standard normal distribution Φ. Indeed, the classical result see Hall 98, p., for eample shows that sup P nu n Φ + n δn + n. 3
2 40 QIYING WANG AND NEVILLE C WEBER Here and below we define δ n = Eg X I gx n + n Eg 3 X I gx n +n Eg 4 X I gx n, and we say that two sequences of positive numbers a n } and b n } satisfy a n b n if 0 < lim inf n a n /b n lim sup n a n /b n <. Note that 3 yields concise results about the rate of convergence in the Central Limit Theorem. In past decades, there has been considerable interest in stating the accuracy of the normal approimation to the distribution of nu n / in a manner that is similar to 3. In increasing generality, the upper bound for the accuracy of the normal approimation has been established in a number of papers. We mention only Bickel 974, Chan and Wierman 977, Callaert and Janssen 978, Borovskikh 996, 00, Alberink and Bentkus 00, 00 and Wang 00. The result given by Borovskikh 00 also see Alberink and Bentkus 00, which is closest to the upper bound in 3, states that if E hx, X 5/3 <, then sup P nu n Φ [ Eg X I gx n + n E gx 3 I gx n A ] + On, 4 where A is an absolute positive constant. In contrast to rich results on the upper bound, there are only a few papers concerned with the lower bound for the accuracy of the normal approimation to the distribution of nu n /. Maesono 988, 99 obtained a lower bound of order On / under the condition Eh 4 X, X <. Only assuming the eistence of Eh X, X, Wang 99 derived a result for the distribution of nu n / that is similar to 3. In a slightly different problem Bentkus, Götze and Zitikis 994 proved that the best bound of order On / in 4 cannot be obtained under E hx, X 5/3 ɛ < for any ɛ > 0. In the present paper we give the leading term in a normal approimation to the distribution of nu n /. Using the leading term we derive the eact convergence rate in the Central Limit Theorem for Ustatistics, up to terms of order On /, under E hx, X 5/3 <. As mentioned above, to get the terms of order On /, the latter moment condition is the best possible. We also show that, if in addition E gx 3 <, the leading term transforms into the conventional first term in an Edgeworth epansion of the distribution of Ustatistics. Our main result is the following.
3 EXACT CONVERGENCE RATE AND LEADING TERM 4 Theorem.. If E hx, X 5/3 <, then sup P nu n Φ L n + L n = oδ n + On, 5 where δ n is defined as in 3, [ L n = n EΦ gx } ] Φ n Φ, L n = Φ3 n E } gx gx φx, X I φx,x n 3. If in addition gx is nonlattice, then the righthand side of 5 may be replaced by oδ n + n /. As is wellknown, δ n 0, as n, and sup L n δ n, 6 see, for eample, Chapter of Hall 98. We show in Section 3 that sup L n = oδ n + On. 7 Together, 5 7 give concise results about the rate of convergence in the Central Limit Theorem for Ustatistics. Indeed, if E hx, X 5/3 <, then sup P nu n Φ + n δ n + n. 8 Note that 8 refines 4 even for the upper bound. One application of 8 is to characterise the rate of convergence. The following theorem gives eamples. Generalizations of the eamples are readily derived, refer to Theorems.9 and.0 of Hall 98 for more details. Theorem.. Assume E hx, X 5/3 <. If 0 r < /, then n= n r sup P nu n if and only if E gx r+ <. If 0 < r < /, then sup P nu n if and only if Eg X I gx = o r. Φ < 9 Φ = On r 0
4 4 QIYING WANG AND NEVILLE C WEBER It is also interesting to note that the effect of L n in 5 on the rate of convergence appears only when δ n = On /. This can be easily seen from 7 and the following corollary, which provides the main result of Jing and Wang 003, about an Edgeworth epansion of the distribution of Ustatistics under optimal conditions. Corollary. below also is an alternative to Theorem 5 of Borovskikh 998 with m =, where the Edgeworth epansion is obtained under lim sup t Ee itgx < instead of the condition that the distribution of gx is nonlattice. Borovskikh 998 also uses a weaker moment condition on the kernel h, y. Corollary.. Assume that E hx, X 5/3 <, E gx 3 <, and the distribution of gx is nonlattice. Then, as n, sup P nu n F n = on, where F n = Φ Φ 3 /6 n Eg 3 X + 3EgX gx hx, X }. The proof of all results will be given in Section 3. To conclude this section we mention that the rate of convergence in the Central Limit Theorem for U statistics depends on the moment conditions for both hx, X and gx. If only E hx, X p <, where 4/3 < p < 5/3, the term of order On / in 8 has to be replaced by a term of lower order. This follows from Theorem. in the net section, which gives an etension of Theorem. to Utype statistics. Throughout the paper we denote constants by A, A, A,..., which may be different at each occurrence.. Etensions to Utype Statistics and Lstatistics Let α and β, y be some realvalued Borel measurable functions of and y. Furthermore, let V n V n X,..., X n be realvalued functions of X,..., X n }. Define a Utype statistic by T n = n n αx j + n 3 βx i, X j + V n. j= i j In this section we derive the leading term in a normal approimation to the distribution of T n under mild conditions, which gives an etension of Theorem.. Theorem.. Assume that a EαX = 0 and Eα X = ; b E[βX, X X i ] = 0, i =,, and E βx, X p < for 4/3 < p 5/3;
5 EXACT CONVERGENCE RATE AND LEADING TERM 43 c P V n C 0 n / C n / for some constants C 0 > 0 and C > 0. Then, as n, sup P T n Φ L n + L n = oδ n + n 4 3p + On, 3 where δ n = Eα X I αx n + n Eα 3 X I αx n +n Eα 4 X I αx n [, L n = n EΦ αx ] } Φ n Φ, L n = Φ3 ]} n E αx αx [βx, X I 3 + βx, X I. β n β n 3 If the condition c is replaced by c P V n on / } on /, and in addition αx is nonlattice, then the righthand side of 3 may be replaced by oδ n + n 4 3p/. Note that the Utype statistic T n defined by is quite general. We net consider an application to Lstatistics. Let X,..., X n be i.i.d. real random variables with distribution function F. Define F n to be the empirical distribution, i.e., F n = n n j= IX i }, where I } is the indicator function. Let Jt be a realvalued function on [0, ] and T G = JG dg. The statistic T F n is called an Lstatistic see Chapter 8 of Serfling 980. Write σ σ J, F = J F s J F t F mins, t} [ F mas, t}] dsdt, and define the distribution function of the standardized Lstatistic T F n by H n = P nσ T F n T F. As is wellknown, H n converges to Φ uniformly in provided E X <, σ > 0, and some smoothness conditions on Jt hold, see Serfling 980 and Helmers, Janssen and Serfling 990 for eample. The upper bounds for the rate of convergence to normality were investigated by Helmers 977 van Zwet 984, Helmers, Janssen and Serfling 990, Wang, Jing and Zhao 000 and Wang 00. As a consequence of Theorem., the following theorem derives the eact convergence rate twosided bound in the Central Limit Theorem for Lstatistics, up to terms of order On /, under mild conditions. Theorem.. Assume that
6 44 QIYING WANG AND NEVILLE C WEBER a Js Jt K s t, 0 < s < t <, for some K > 0; b EX < and σ > 0. Then, as n, sup H n Φ + n δ n + n, 4 where αx = σ JF tix t F t dt, and δ n is defined as in Theorem.. 3. Proofs Proof of Theorem.. The result is an immediate consequence of Theorem.. Proof of Theorem.. Without loss of generality we assume that β, y is symmetric. Otherwise it is enough to replace βx i, Y j by βx i, Y j + βx j, Y i. The proof is along the lines of Jing and Wang 003. Write βx i, X j = βx i, X j I βxi,x j n 3, α X j = E βxi, X j X j, α n X j = α X j I n α X j n, n Tn = n αx j + α X j + n 3 βxi, X j α X i α X j +V n. j= Noting EβX, X = 0, it is easily seen that α X j = E βx i, X j I βxi,x j n 3/ j X E βx i, X j I βxi,x j n 3/ X j, 5 i<j and, as in.4.5 of Jing and Wang 003, sup P T n P Tn np α X n + n P n 4 3p E βx, X p I βx,x n 3 = o n 4 3p βx, X n 3. 6 We further let, m 0 = [0 log n]+/b, where b > 0 is a constant to be chosen
7 EXACT CONVERGENCE RATE AND LEADING TERM 45 later, η j = αx j + α X j Eα X j, [ γ ij = βxi, X j α X i α X j + E βx ] i, X j, n S n = n η j, m = n 3 n,m = n k=m j= n j=m+ γ mj for m n, k if 0 < m < n, and n,m = 0 if m n. It follows immediately that Tn = S n + n,m0 + Ṽn + neα X n n / E βx, X, where Ṽn = V n + m 0 m= m. Note that for any fied m < k n and q, E n,m n,k q 8n 3q + k me γ q ; 7 see Theorem..3 in Koroljuk and Borovskich 994. It follows from 7 with q = p and E γ p < see below that for 4/3 p 5/3, P m 0 m= n m An p log +p n E γ p = o log n n 4 3p. 8 In terms of the condition c or c, 8 and the fact that neα X n n / E βx, X 3 ne βx, X I β n 3/ = on4 3p/, routine calculations show that, to prove 3, it suffices to prove I n := sup P S n + n,m0 Φ L n + L n = oδ n + n 4 3p + On 9 and, if in addition αx is nonlattice, then the righthand side of 9 may be replaced by oδ n + n 4 3p/. We first establish five lemmas before the proof of 9. The proofs of these lemmas will be omitted. The details can be found in Wang and Weber 004, on which the present paper is based.
8 46 QIYING WANG AND NEVILLE C WEBER Lemma 3.. Write ˆαX = α X Eα X. We have E ˆαX λ E α X λ = o E γ q 6 βx, X q = o Eη = o δ n + n 4 3p n λ+ 3p, for λ, 0 E γ p 6 βx, X p <, n 3q p, for p < q, Lemma 3.. We have Eγ e itη +η n = t n E + o Eγ e itη +η n A min. 3 αx αx βx }, X δ n + n 4 3p n t + t 3, 4 }. 5 t n 4p p, n Eγ t + t 3 Net define, ft = Ee itη / n, gt = Ee itαx / n, and g n t = e t / + ngt + t /. Lemma 3.3. There eists a constant c 0 > 0 such that for all t c 0 n / and all sufficiently large n, ft e t 8n, gt e t 4n, 6 f n t e t A δ n + on 4 3p t + t 4 e t 6, 7 f n t g n t = o δ n + n 4 3p t + t 8 e t 6. 8 If in addition αx is nonlattice, then there eist constants b > 0 and ɛ n such that for c 0 t / n ɛ n, ft e b and gt e b. 9 Lemma 3.4. For any t c 0 n, where c0 is defined as in Lemma 3.3, E n,m0 e itsn + t e t n E = o δ n + n 4 3p αx αx [ βx, X + βx, X ]} t + t 6 e t To introduce the net lemma we first define some notation. As in.3 and
9 EXACT CONVERGENCE RATE AND LEADING TERM 47.6 of Jing and Wang 003, we have Z n t := Ee itsn+ n,m 0 Ee itsn ite n,m0 e itsn [ ] = Z n t + it Z n t + Z n t, where, l m,k = n 3/ n j=k+ γ mj, jm is the largest integer such that mjm < n and Z n t = n Z n n t = m=m 0 Ee itsn+ n,m+ e it m it m, jm m=m 0 j= Z n n t = jm m=m 0 j= t c 0 n El m,jm e itsn e it n,jm+ e it n,j+m+, E l m,jm l m,j+m e its n e it n,j+m+. Lemma 3.5. For 4/3 < p 5/3, we have t c 0 n t Z nt dt = o δ n + n 4 3p, 3 t + Z n t dt = o δ n + n 4 3p, 3 Z n where c 0 is defined as in Lemma 3.3. We are now ready to prove 9. We continue to use the notation defined in Lemmas Furthermore write ϕ n t = t B n e t /, where B n = ]} n E αx αx [βx, X I + βx β n 3, X I. β n 3 Using Lemmas we have J n := Ee itsn+ n,m 0 g n t itϕ n t dt t c 0 n t t c 0 n t Z nt dt + t c 0 n t f n t g n t dt + E n,m0 e itsn ϕ n t dt t c 0 n = oδ n + n 4 3p. 33
10 48 QIYING WANG AND NEVILLE C WEBER Note that eit d Φ + L n L n = g n t + itϕ n t. It follows from Esseen s smoothing lemma and 33 that I n Ee itsn+ n,m 0 g n t itϕ n t dt + t c 0 n t = oδ n + n 4 3p + On. 34 This proves the first part of 9. If αx is nonlatice, it follows from the fact that n,m0 only depends on X m0 +,..., X n, and 9, that for any ɛ n, J n := Ee itsn+ n,m 0 g n t itϕ n t dt c 0 n t ɛn n t c 0 n t ɛn n t ft m 0 dt + c 0 n t ɛn n t g nt + itϕ n t dt = oδ n + n 4 3p. 35 Using 33, 35 and Esséen s smoothing lemma again, we obtain for 4/3 p 5/3, I n Ee itsn+ n,m 0 g n t itϕ n t dt + A t ɛ n n t ɛ n n J n + J n + on = oδ n + n 4 3p. This implies the second part of 9 and hence the proof of 9. The proof of Theorem. is now complete. Proof of Theorem.. Write η j t = IX j t} F t, αx j = σ JF tη j tdt, βx i, X j = Kσ As in 9 of Wang 00, we have n n αx j n 3 βx i, X j V n j= i j A n η i tη j tdt. n T Fn T F σ n n αx j + n 3 βx i, X j + V n, 36 j= i j
11 EXACT CONVERGENCE RATE AND LEADING TERM 49 where V n = n 3/ n j= ZX j with ZX j = Kσ ηj tdt. It is readily seen that EαX = 0, Eα X =, EβX i, X j X i = 0, i j, and similar to the proof of Lemma A in Serfling 980, p.88, αx j + βx i, X j + ZX j Aσ X j + E X. 37 In terms of these facts, 4 follows easily from Theorem.. We omit the details. Proof of Corollary.. Equation follows easily from Theorem., the classical result sup L n + Φ3 6 n Eg3 X = on, and by Hölder s inequality, that L n Φ3 n E gx gx φx, X } A n E gx gx φx, X I φx,x n 3 sup A n E gx E φx, X I φx,x n 3 } = on. Proof of 7. It suffices to show that E gx gx φx, X I φx,x n } 3 n = oδ n + On. 38 Write φx, X = φx, X I φx,x n 3/. It is readily seen that E gx gx φx }, X = E gx I gx n gx φx }, X +E gx I gx < n gx I gx φx } n, X +E gx I gx < ngx I gx < φx } n, X := I 6n + I 7n + I 8n. 39 By Hölder s inequality, and similar to, we have I 6n + I 7n EgX I n gx n E φx, X n [ ] δ n on = o δ n + n. 40
12 40 QIYING WANG AND NEVILLE C WEBER Similarly, by noting E φx, X 5/3 <, I 8n / n n E gx 5 4 I gx < 5 n E φx, X An E gx 5 4 I gx < 5 n. 4 In terms of 4, if E gx 5/ I gx < n <, then I 8n n = On. 4 We show that if E gx 5/ I gx < n =, then ne gx 5 I gx < n A E gx 4 I gx < n, 43 and hence it follows from 4, that I 8n / n = o n E gx 5 I gx < n = o n E gx 4 I gx < n = oδ n. 44 Combining 39 40, 4 and 44, we obtain the proof of 38. We net prove 43. Write l τ = E gx τ I gx <. Note that l 4 is a nondecreasing function and l 4 A. It follows from Proposition.. of Bingham, Goldie and Teugels 987 that lim sup l 4 /l 4 <. Now 43 follows easily from question 34 on page 89 of Feller 97 also see Feller 969 or Theorem.6.6 of Bingham, Goldie and Teugels 987. The proof of 7 is now complete. Acknowledgements The authors would like to thank two referees and an associate editor for their detailed reading of this paper and valuable comments. The research is partially supported by ARC discovery grant DP0457 and Sesqui NSSS grant 004. References Alberink, I. B. and Bentkus, V. 00. BerryEsseen bounds for vonmises and Ustatistics. Lithuanian Math. J. 4, 6. Alberink, I. B. and Bentkus, V. 00. Lyapunov type bounds for Ustatistics. Theory Probab. Appl. 46, Bentkus, V., Götze, F. and Zitikis, M Lower estimates of the convergence rate for Ustatistics. Ann. Probab., Bickel, P. J Edgeworth epansions in nonparametric statistics. Ann. Statist., 0.
13 EXACT CONVERGENCE RATE AND LEADING TERM 4 Bingham, N. H., Goldie, C. M. and Teugels, J. L Regular Variation. Cambridge University Press. Borovskikh, Yu. V Ustatistics in Banach Space. VSP, Utrecht. Borovskikh, Yu. V Sharp estimates of the rate of convergence for Ustatistics. Report 98. School of Mathematics and Statistics, University of Sydney. Borovskikh, Yu. V. 00. On a normal approimation of Ustatistics. Theory Probab. Appl. 45, Callaert, H. and Janssen, P The BerryEsseen theorem for Ustatistics. Ann. Statist. 6, Chan, Y. K. and Wierman, J On the BerryEsseen theorem for Ustatistics. Ann. Probab. 5, Feller, W Onesided analogues of Karamata s regular variation. L Enseignement Mathématique 5, 07. Feller, W. 97. An Introduction to Probability Theory and Its Applications II, second edition. John Wiley, New York. Hall, P. 98. Rates of Convergence in the Central Limit Theorem. Research Notes in Mathematics, N6. Pitman Advanced Publishing Program. Helmers, R The order of the normal approimation for linear combinations of order statistics with smooth weight functions. Ann. Probab. 5, Helmers, R., Janssen, P. and Serfling, R BerryEsséen and bootstrap results for generalized Lstatistics. Scand. J. Statist. 7, Jing, B.Y. and Wang, Q Edgeworth epansion for Ustatistics under minimal conditions. Ann. Statist. 3, Koroljuk, V. S. and Borovskich, Yu. V Theory of Ustatistics. Kluwer Academic Publishers, Dordrecht. Lee, A. J Ustatistics  Theory and Practice. Marcel Dekker, New York. Maesono, Y A lower bound for the normal approimations of Ustatistics, Metrika 35, Maesono, Y. 99. On the normal approimations of Ustatistics of degree two. J. Statist. Plann. Inference 7, Serfling, R. J Approimation Theorems of Mathematical Statistics. John Wiley New York. Wang, Q. 99. Twosided bounds on the rate of convergence to normal distribution of U statistics. J. Systems Sci. Math. Sci., Wang, Q. 00. Nonuniform BerryEsséen bound for Ustatistics. Statist. Sinica, Wang, Q., Jing, B.Y. and Zhao, L The BerryEsséen bound for studentized statistics. Ann. Probab. 8, Wang, Q. and Weber, N. C Eact convergence rate and leading term in the Central Limit Theorem for Ustatistics. School of Mathematics and Statistics, University of Sydney, Research Report 33, 33.html. van Zwet, W. R A BerryEsséen bound for symmetric statistics. Z. Wahrsch. Verw. Gebiete 66,
14 4 QIYING WANG AND NEVILLE C WEBER School of Mathematics and Statistics, The University of Sydney, NSW 006, Australia. School of Mathematics and Statistics, The University of Sydney, NSW 006, Australia. Received April 004; accepted October 004
Normal Approximation for Nonlinear Statistics Using a Concentration Inequality Approach
Normal Approximation for Nonlinear Statistics Using a Concentration Inequality Approach Louis H.Y. Chen 1 National University of Singapore and QiMan Shao 2 Hong Kong University of Science and Technology,
More informationCramér type moderate deviations for trimmed Lstatistics
arxiv:1608.05015v1 [math.pr] 17 Aug 2016 Cramér type moderate deviations for trimmed Lstatistics Nadezhda Gribkova St.Petersburg State University, Mathematics and Mechanics Faculty, 199034, Universitetskaya
More informationPRECISE ASYMPTOTIC IN THE LAW OF THE ITERATED LOGARITHM AND COMPLETE CONVERGENCE FOR USTATISTICS
PRECISE ASYMPTOTIC IN THE LAW OF THE ITERATED LOGARITHM AND COMPLETE CONVERGENCE FOR USTATISTICS REZA HASHEMI and MOLUD ABDOLAHI Department of Statistics, Faculty of Science, Razi University, 67149, Kermanshah,
More informationSelfnormalized CramérType Large Deviations for Independent Random Variables
Selfnormalized CramérType Large Deviations for Independent Random Variables QiMan Shao National University of Singapore and University of Oregon qmshao@darkwing.uoregon.edu 1. Introduction Let X, X
More informationRELATIVE ERRORS IN CENTRAL LIMIT THEOREMS FOR STUDENT S t STATISTIC, WITH APPLICATIONS
Statistica Sinica 19 (2009, 343354 RELATIVE ERRORS IN CENTRAL LIMIT THEOREMS FOR STUDENT S t STATISTIC, WITH APPLICATIONS Qiying Wang and Peter Hall University of Sydney and University of Melbourne Abstract:
More informationOn probabilities of large and moderate deviations for Lstatistics: a survey of some recent developments
UDC 519.2 On probabilities of large and moderate deviations for Lstatistics: a survey of some recent developments N. V. Gribkova Department of Probability Theory and Mathematical Statistics, St.Petersburg
More informationEdgeworth expansions for a sample sum from a finite set of independent random variables
E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Vol. 12 2007), Paper no. 52, pages 1402 1417. Journal URL http://www.math.washington.edu/~ejpecp/ Edgeworth expansions for a sample sum from
More informationIntroduction to Selfnormalized Limit Theory
Introduction to Selfnormalized Limit Theory QiMan Shao The Chinese University of Hong Kong Email: qmshao@cuhk.edu.hk Outline What is the selfnormalization? Why? Classical limit theorems Selfnormalized
More informationConvergence rates in weighted L 1 spaces of kernel density estimators for linear processes
Alea 4, 117 129 (2008) Convergence rates in weighted L 1 spaces of kernel density estimators for linear processes Anton Schick and Wolfgang Wefelmeyer Anton Schick, Department of Mathematical Sciences,
More informationBounds of the normal approximation to randomsum Wilcoxon statistics
R ESEARCH ARTICLE doi: 0.306/scienceasia53874.04.40.8 ScienceAsia 40 04: 8 9 Bounds of the normal approximation to randomsum Wilcoxon statistics Mongkhon Tuntapthai Nattakarn Chaidee Department of Mathematics
More informationA UNIFIED APPROACH TO EDGEWORTH EXPANSIONS FOR A GENERAL CLASS OF STATISTICS
Statistica Siica 20 2010, 613636 A UNIFIED APPROACH TO EDGEWORTH EXPANSIONS FOR A GENERAL CLASS OF STATISTICS BigYi Jig ad Qiyig Wag Hog Kog Uiversity of Sciece ad Techology ad Uiversity of Sydey Abstract:
More informationOn large deviations of sums of independent random variables
On large deviations of sums of independent random variables Zhishui Hu 12, Valentin V. Petrov 23 and John Robinson 2 1 Department of Statistics and Finance, University of Science and Technology of China,
More informationPointwise convergence rates and central limit theorems for kernel density estimators in linear processes
Pointwise convergence rates and central limit theorems for kernel density estimators in linear processes Anton Schick Binghamton University Wolfgang Wefelmeyer Universität zu Köln Abstract Convergence
More informationCENTRAL LIMIT THEOREM AND DIOPHANTINE APPROXIMATIONS. Sergey G. Bobkov. December 24, 2016
CENTRAL LIMIT THEOREM AND DIOPHANTINE APPROXIMATIONS Sergey G. Bobkov December 24, 206 Abstract Let F n denote the distribution function of the normalized sum Z n = X + +X n /σ n of i.i.d. random variables
More informationORTHOGONAL DECOMPOSITION OF FINITE POPULATION STATISTICS AND ITS APPLICATIONS TO DISTRIBUTIONAL ASYMPTOTICS. and Informatics, 2 Bielefeld University
ORTHOGONAL DECOMPOSITION OF FINITE POPULATION STATISTICS AND ITS APPLICATIONS TO DISTRIBUTIONAL ASYMPTOTICS M. Bloznelis 1,3 and F. Götze 2 1 Vilnius University and Institute of Mathematics and Informatics,
More informationBahadur representations for bootstrap quantiles 1
Bahadur representations for bootstrap quantiles 1 Yijun Zuo Department of Statistics and Probability, Michigan State University East Lansing, MI 48824, USA zuo@msu.edu 1 Research partially supported by
More informationOn the Convergence of the Summation Formulas Constructed by Using a Symbolic Operator Approach
On the Convergence of the Summation Formulas Constructed by Using a Symbolic Operator Approach TianXiao He 1, Leetsch C. Hsu 2, and Peter J.S. Shiue 3 1 Department of Mathematics and Computer Science
More informationMOMENT CONVERGENCE RATES OF LIL FOR NEGATIVELY ASSOCIATED SEQUENCES
J. Korean Math. Soc. 47 1, No., pp. 63 75 DOI 1.4134/JKMS.1.47..63 MOMENT CONVERGENCE RATES OF LIL FOR NEGATIVELY ASSOCIATED SEQUENCES KeAng Fu LiHua Hu Abstract. Let X n ; n 1 be a strictly stationary
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 29, 2008 This talk is based on a joint work with Professor Shihong
More informationAdditive functionals of infinitevariance moving averages. Wei Biao Wu The University of Chicago TECHNICAL REPORT NO. 535
Additive functionals of infinitevariance moving averages Wei Biao Wu The University of Chicago TECHNICAL REPORT NO. 535 Departments of Statistics The University of Chicago Chicago, Illinois 60637 June
More informationAN EDGEWORTH EXPANSION FOR SYMMETRIC FINITE POPULATION STATISTICS. M. Bloznelis 1, F. Götze
AN EDGEWORTH EXPANSION FOR SYMMETRIC FINITE POPULATION STATISTICS M. Bloznelis 1, F. Götze Vilnius University and the Institute of Mathematics and Informatics, Bielefeld University 2000 (Revised 2001 Abstract.
More informationSOME CONVERSE LIMIT THEOREMS FOR EXCHANGEABLE BOOTSTRAPS
SOME CONVERSE LIMIT THEOREMS OR EXCHANGEABLE BOOTSTRAPS Jon A. Wellner University of Washington The bootstrap GlivenkoCantelli and bootstrap Donsker theorems of Giné and Zinn (990) contain both necessary
More informationA Central Limit Theorem for the Sum of Generalized Linear and Quadratic Forms. by R. L. Eubank and Suojin Wang Texas A&M University
A Central Limit Theorem for the Sum of Generalized Linear and Quadratic Forms by R. L. Eubank and Suojin Wang Texas A&M University ABSTRACT. A central limit theorem is established for the sum of stochastically
More informationHanYing Liang, DongXia Zhang, and JongIl Baek
J. Korean Math. Soc. 41 (2004), No. 5, pp. 883 894 CONVERGENCE OF WEIGHTED SUMS FOR DEPENDENT RANDOM VARIABLES HanYing Liang, DongXia Zhang, and JongIl Baek Abstract. We discuss in this paper the strong
More informationPrecise Asymptotics of Generalized Stochastic Order Statistics for Extreme Value Distributions
Applied Mathematical Sciences, Vol. 5, 2011, no. 22, 10891102 Precise Asymptotics of Generalied Stochastic Order Statistics for Extreme Value Distributions Rea Hashemi and Molood Abdollahi Department
More informationA Note on the Strong Law of Large Numbers
JIRSS (2005) Vol. 4, No. 2, pp 107111 A Note on the Strong Law of Large Numbers V. Fakoor, H. A. Azarnoosh Department of Statistics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Iran.
More informationMath 104: Homework 7 solutions
Math 04: Homework 7 solutions. (a) The derivative of f () = is f () = 2 which is unbounded as 0. Since f () is continuous on [0, ], it is uniformly continous on this interval by Theorem 9.2. Hence for
More informationThis condition was introduced by Chandra [1]. Ordonez Cabrera [5] extended the notion of Cesàro uniform integrability
Bull. Korean Math. Soc. 4 (2004), No. 2, pp. 275 282 WEAK LAWS FOR WEIGHTED SUMS OF RANDOM VARIABLES Soo Hak Sung Abstract. Let {a ni, u n i, n } be an array of constants. Let {X ni, u n i, n } be {a ni
More informationCramérType Moderate Deviation Theorems for TwoSample Studentized (Selfnormalized) UStatistics. WenXin Zhou
CramérType Moderate Deviation Theorems for TwoSample Studentized (Selfnormalized) UStatistics WenXin Zhou Department of Mathematics and Statistics University of Melbourne Joint work with Prof. QiMan
More informationOn large deviations for combinatorial sums
arxiv:1901.0444v1 [math.pr] 14 Jan 019 On large deviations for combinatorial sums Andrei N. Frolov Dept. of Mathematics and Mechanics St. Petersburg State University St. Petersburg, Russia Email address:
More informationSoo Hak Sung and Andrei I. Volodin
Bull Korean Math Soc 38 (200), No 4, pp 763 772 ON CONVERGENCE OF SERIES OF INDEENDENT RANDOM VARIABLES Soo Hak Sung and Andrei I Volodin Abstract The rate of convergence for an almost surely convergent
More informationEMPIRICAL EDGEWORTH EXPANSION FOR FINITE POPULATION STATISTICS. I. M. Bloznelis. April Introduction
EMPIRICAL EDGEWORTH EXPANSION FOR FINITE POPULATION STATISTICS. I M. Bloznelis April 2000 Abstract. For symmetric asymptotically linear statistics based on simple random samples, we construct the oneterm
More informationAlmost sure limit theorems for Ustatistics
Almost sure limit theorems for Ustatistics Hajo Holzmann, Susanne Koch and Alesey Min 3 Institut für Mathematische Stochasti GeorgAugustUniversität Göttingen Maschmühlenweg 8 0 37073 Göttingen Germany
More informationCHAPTER 3: LARGE SAMPLE THEORY
CHAPTER 3 LARGE SAMPLE THEORY 1 CHAPTER 3: LARGE SAMPLE THEORY CHAPTER 3 LARGE SAMPLE THEORY 2 Introduction CHAPTER 3 LARGE SAMPLE THEORY 3 Why large sample theory studying small sample property is usually
More informationA generalization of Strassen s functional LIL
A generalization of Strassen s functional LIL Uwe Einmahl Departement Wiskunde Vrije Universiteit Brussel Pleinlaan 2 B1050 Brussel, Belgium Email: ueinmahl@vub.ac.be Abstract Let X 1, X 2,... be a sequence
More informationOn lower limits and equivalences for distribution tails of randomly stopped sums 1
On lower limits and equivalences for distribution tails of randomly stopped sums 1 D. Denisov, 2 S. Foss, 3 and D. Korshunov 4 Eurandom, HeriotWatt University and Sobolev Institute of Mathematics Abstract
More informationFOURIER TRANSFORMS OF STATIONARY PROCESSES 1. k=1
FOURIER TRANSFORMS OF STATIONARY PROCESSES WEI BIAO WU September 8, 003 Abstract. We consider the asymptotic behavior of Fourier transforms of stationary and ergodic sequences. Under sufficiently mild
More informationCramér large deviation expansions for martingales under Bernstein s condition
Cramér large deviation expansions for martingales under Bernstein s condition Xiequan Fan, Ion Grama, Quansheng Liu To cite this version: Xiequan Fan, Ion Grama, Quansheng Liu. Cramér large deviation expansions
More informationCOMPOSITION SEMIGROUPS AND RANDOM STABILITY. By John Bunge Cornell University
The Annals of Probability 1996, Vol. 24, No. 3, 1476 1489 COMPOSITION SEMIGROUPS AND RANDOM STABILITY By John Bunge Cornell University A random variable X is Ndivisible if it can be decomposed into a
More informationHausdorff moment problem: Reconstruction of probability density functions
Statistics and Probability Letters 78 (8) 869 877 www.elsevier.com/locate/stapro Hausdorff moment problem: Reconstruction of probability density functions Robert M. Mnatsakanov Department of Statistics,
More informationTESTING FOR EQUAL DISTRIBUTIONS IN HIGH DIMENSION
TESTING FOR EQUAL DISTRIBUTIONS IN HIGH DIMENSION Gábor J. Székely Bowling Green State University Maria L. Rizzo Ohio University October 30, 2004 Abstract We propose a new nonparametric test for equality
More informationThe Lindeberg central limit theorem
The Lindeberg central limit theorem Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto May 29, 205 Convergence in distribution We denote by P d the collection of Borel probability
More informationA BERRY ESSEEN BOUND FOR STUDENT S STATISTIC IN THE NONI.I.D. CASE. 1. Introduction and Results
A BERRY ESSEE BOUD FOR STUDET S STATISTIC I THE OI.I.D. CASE V. Bentkus 1 M. Bloznelis 1 F. Götze 1 Abstract. We establish a Berry Esséen bound for Student s statistic for independent (nonidentically)
More informationAsymptotic behavior for sums of nonidentically distributed random variables
Appl. Math. J. Chinese Univ. 2019, 34(1: 4554 Asymptotic behavior for sums of nonidentically distributed random variables YU Changjun 1 CHENG Dongya 2,3 Abstract. For any given positive integer m,
More informationA CHARACTERIZATION OF STRICT LOCAL MINIMIZERS OF ORDER ONE FOR STATIC MINMAX PROBLEMS IN THE PARAMETRIC CONSTRAINT CASE
Journal of Applied Analysis Vol. 6, No. 1 (2000), pp. 139 148 A CHARACTERIZATION OF STRICT LOCAL MINIMIZERS OF ORDER ONE FOR STATIC MINMAX PROBLEMS IN THE PARAMETRIC CONSTRAINT CASE A. W. A. TAHA Received
More informationRefining the Central Limit Theorem Approximation via Extreme Value Theory
Refining the Central Limit Theorem Approximation via Extreme Value Theory Ulrich K. Müller Economics Department Princeton University February 2018 Abstract We suggest approximating the distribution of
More informationRANDOM WALKS WHOSE CONCAVE MAJORANTS OFTEN HAVE FEW FACES
RANDOM WALKS WHOSE CONCAVE MAJORANTS OFTEN HAVE FEW FACES ZHIHUA QIAO and J. MICHAEL STEELE Abstract. We construct a continuous distribution G such that the number of faces in the smallest concave majorant
More informationRandom Walks Conditioned to Stay Positive
1 Random Walks Conditioned to Stay Positive Bob Keener Let S n be a random walk formed by summing i.i.d. integer valued random variables X i, i 1: S n = X 1 + + X n. If the drift EX i is negative, then
More informationEdgeworth Expansion for Studentized Statistics
Edgeworth Expasio for Studetized Statistics BigYi JING Hog Kog Uiversity of Sciece ad Techology Qiyig WANG Australia Natioal Uiversity ABSTRACT Edgeworth expasios are developed for a geeral class of studetized
More informationCOMPROMISE PLANS WITH CLEAR TWOFACTOR INTERACTIONS
Statistica Sinica 15(2005), 709715 COMPROMISE PLANS WITH CLEAR TWOFACTOR INTERACTIONS Weiming Ke 1, Boxin Tang 1,2 and Huaiqing Wu 3 1 University of Memphis, 2 Simon Fraser University and 3 Iowa State
More informationMean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability
J. Math. Anal. Appl. 305 2005) 644 658 www.elsevier.com/locate/jmaa Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability
More informationLecture 3: Expected Value. These integrals are taken over all of Ω. If we wish to integrate over a measurable subset A Ω, we will write
Lecture 3: Expected Value 1.) Definitions. If X 0 is a random variable on (Ω, F, P), then we define its expected value to be EX = XdP. Notice that this quantity may be. For general X, we say that EX exists
More informationNonuniform Berry Esseen Bounds for Weighted UStatistics and Generalized LStatistics
Coun Math Stat 0 :5 67 DOI 0.007/s40040009 Nonunifor Berry Esseen Bounds for Weighted UStatistics and Generalized LStatistics Haojun Hu QiMan Shao Received: 9 August 0 / Accepted: Septeber 0 / Published
More informationA Symbolic Operator Approach to Several Summation Formulas for Power Series
A Symbolic Operator Approach to Several Summation Formulas for Power Series T. X. He, L. C. Hsu 2, P. J.S. Shiue 3, and D. C. Torney 4 Department of Mathematics and Computer Science Illinois Wesleyan
More informationA regeneration proof of the central limit theorem for uniformly ergodic Markov chains
A regeneration proof of the central limit theorem for uniformly ergodic Markov chains By AJAY JASRA Department of Mathematics, Imperial College London, SW7 2AZ, London, UK and CHAO YANG Department of Mathematics,
More informationA TWO PARAMETERS AMBROSETTI PRODI PROBLEM*
PORTUGALIAE MATHEMATICA Vol. 53 Fasc. 3 1996 A TWO PARAMETERS AMBROSETTI PRODI PROBLEM* C. De Coster** and P. Habets 1 Introduction The study of the Ambrosetti Prodi problem has started with the paper
More informationAlmost Sure Convergence of the General Jamison Weighted Sum of BValued Random Variables
Acta Mathematica Sinica, English Series Feb., 24, Vol.2, No., pp. 8 92 Almost Sure Convergence of the General Jamison Weighted Sum of BValued Random Variables Chun SU Tie Jun TONG Department of Statistics
More informationAlmost Sure Central Limit Theorem for SelfNormalized Partial Sums of Negatively Associated Random Variables
Filomat 3:5 (207), 43 422 DOI 0.2298/FIL70543W Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Almost Sure Central Limit Theorem
More informationQED. Queen s Economics Department Working Paper No. 1244
QED Queen s Economics Department Working Paper No. 1244 A necessary moment condition for the fractional functional central limit theorem Søren Johansen University of Copenhagen and CREATES Morten Ørregaard
More informationTHE GEOMETRICAL ERGODICITY OF NONLINEAR AUTOREGRESSIVE MODELS
Statistica Sinica 6(1996), 943956 THE GEOMETRICAL ERGODICITY OF NONLINEAR AUTOREGRESSIVE MODELS H. Z. An and F. C. Huang Academia Sinica Abstract: Consider the nonlinear autoregressive model x t = φ(x
More informationA CLT FOR MULTIDIMENSIONAL MARTINGALE DIFFERENCES IN A LEXICOGRAPHIC ORDER GUY COHEN. Dedicated to the memory of Mikhail Gordin
A CLT FOR MULTIDIMENSIONAL MARTINGALE DIFFERENCES IN A LEXICOGRAPHIC ORDER GUY COHEN Dedicated to the memory of Mikhail Gordin Abstract. We prove a central limit theorem for a squareintegrable ergodic
More informationA NOTE ON THE COMPLETE MOMENT CONVERGENCE FOR ARRAYS OF BVALUED RANDOM VARIABLES
Bull. Korean Math. Soc. 52 (205), No. 3, pp. 825 836 http://dx.doi.org/0.434/bkms.205.52.3.825 A NOTE ON THE COMPLETE MOMENT CONVERGENCE FOR ARRAYS OF BVALUED RANDOM VARIABLES Yongfeng Wu and Mingzhu
More informationTHE LINDEBERGFELLER CENTRAL LIMIT THEOREM VIA ZERO BIAS TRANSFORMATION
THE LINDEBERGFELLER CENTRAL LIMIT THEOREM VIA ZERO BIAS TRANSFORMATION JAINUL VAGHASIA Contents. Introduction. Notations 3. Background in Probability Theory 3.. Expectation and Variance 3.. Convergence
More informationX n D X lim n F n (x) = F (x) for all x C F. lim n F n(u) = F (u) for all u C F. (2)
14:17 11/16/2 TOPIC. Convergence in distribution and related notions. This section studies the notion of the socalled convergence in distribution of real random variables. This is the kind of convergence
More informationLimit Theorems for Exchangeable Random Variables via Martingales
Limit Theorems for Exchangeable Random Variables via Martingales Neville Weber, University of Sydney. May 15, 2006 Probabilistic Symmetries and Their Applications A sequence of random variables {X 1, X
More information1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989),
Real Analysis 2, Math 651, Spring 2005 April 26, 2005 1 Real Analysis 2, Math 651, Spring 2005 Krzysztof Chris Ciesielski 1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer
More informationModel Specification Testing in Nonparametric and Semiparametric Time Series Econometrics. Jiti Gao
Model Specification Testing in Nonparametric and Semiparametric Time Series Econometrics Jiti Gao Department of Statistics School of Mathematics and Statistics The University of Western Australia Crawley
More informationJackknife Euclidean LikelihoodBased Inference for Spearman s Rho
Jackknife Euclidean LikelihoodBased Inference for Spearman s Rho M. de Carvalho and F. J. Marques Abstract We discuss jackknife Euclidean likelihoodbased inference methods, with a special focus on the
More informationA NOTE ON A DISTRIBUTION OF WEIGHTED SUMS OF I.I.D. RAYLEIGH RANDOM VARIABLES
Sankhyā : The Indian Journal of Statistics 1998, Volume 6, Series A, Pt. 2, pp. 171175 A NOTE ON A DISTRIBUTION OF WEIGHTED SUMS OF I.I.D. RAYLEIGH RANDOM VARIABLES By P. HITCZENKO North Carolina State
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 informationMath 321 Final Examination April 1995 Notation used in this exam: N. (1) S N (f,x) = f(t)e int dt e inx.
Math 321 Final Examination April 1995 Notation used in this exam: N 1 π (1) S N (f,x) = f(t)e int dt e inx. 2π n= N π (2) C(X, R) is the space of bounded realvalued functions on the metric space X, equipped
More informationAdvanced Probability Theory (Math541)
Advanced Probability Theory (Math541) Instructor: Kani Chen (Classic)/Modern Probability Theory (19001960) Instructor: Kani Chen (HKUST) Advanced Probability Theory (Math541) 1 / 17 Primitive/Classic
More informationThe Hilbert Transform and Fine Continuity
Irish Math. Soc. Bulletin 58 (2006), 8 9 8 The Hilbert Transform and Fine Continuity J. B. TWOMEY Abstract. It is shown that the Hilbert transform of a function having bounded variation in a finite interval
More informationOn rate of convergence in distribution of asymptotically normal statistics based on samples of random size
Annales Mathematicae et Informaticae 39 212 pp. 17 28 Proceedings of the Conference on Stochastic Models and their Applications Faculty of Informatics, University of Debrecen, Debrecen, Hungary, August
More informationON THE COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF DEPENDENT RANDOM VARIABLES UNDER CONDITION OF WEIGHTED INTEGRABILITY
J. Korean Math. Soc. 45 (2008), No. 4, pp. 1101 1111 ON THE COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF DEPENDENT RANDOM VARIABLES UNDER CONDITION OF WEIGHTED INTEGRABILITY JongIl Baek, MiHwa Ko, and TaeSung
More informationUNIFORM IN BANDWIDTH CONSISTENCY OF KERNEL REGRESSION ESTIMATORS AT A FIXED POINT
UNIFORM IN BANDWIDTH CONSISTENCY OF KERNEL RERESSION ESTIMATORS AT A FIXED POINT JULIA DONY AND UWE EINMAHL Abstract. We consider pointwise consistency properties of kernel regression function type estimators
More informationMiHwa Ko. t=1 Z t is true. j=0
Commun. Korean Math. Soc. 21 (2006), No. 4, pp. 779 786 FUNCTIONAL CENTRAL LIMIT THEOREMS FOR MULTIVARIATE LINEAR PROCESSES GENERATED BY DEPENDENT RANDOM VECTORS MiHwa Ko Abstract. Let X t be an mdimensional
More information= 1 2 x (x 1) + 1 {x} (1 {x}). [t] dt = 1 x (x 1) + O (1), [t] dt = 1 2 x2 + O (x), (where the error is not now zero when x is an integer.
Problem Sheet,. i) Draw the graphs for [] and {}. ii) Show that for α R, α+ α [t] dt = α and α+ α {t} dt =. Hint Split these integrals at the integer which must lie in any interval of length, such as [α,
More informationA generalization of Cramér large deviations for martingales
A generalization of Cramér large deviations for martingales Xiequan Fan, Ion Grama, Quansheng Liu To cite this version: Xiequan Fan, Ion Grama, Quansheng Liu. A generalization of Cramér large deviations
More informationd(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N
Problem 1. Let f : A R R have the property that for every x A, there exists ɛ > 0 such that f(t) > ɛ if t (x ɛ, x + ɛ) A. If the set A is compact, prove there exists c > 0 such that f(x) > c for all x
More informationRegularity of the density for the stochastic heat equation
Regularity of the density for the stochastic heat equation Carl Mueller 1 Department of Mathematics University of Rochester Rochester, NY 15627 USA email: cmlr@math.rochester.edu David Nualart 2 Department
More informationarxiv:math/ v2 [math.pr] 16 Mar 2007
CHARACTERIZATION OF LIL BEHAVIOR IN BANACH SPACE UWE EINMAHL a, and DELI LI b, a Departement Wiskunde, Vrije Universiteit Brussel, arxiv:math/0608687v2 [math.pr] 16 Mar 2007 Pleinlaan 2, B1050 Brussel,
More informationUniversity of California San Diego and Stanford University and
First International Workshop on Functional and Operatorial Statistics. Toulouse, June 1921, 2008 Ksample Subsampling Dimitris N. olitis andjoseph.romano University of California San Diego and Stanford
More informationTHEOREMS, ETC., FOR MATH 515
THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every
More informationReal Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi
Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.
More informationV. Gupta, A. Aral and M. Ozhavzali. APPROXIMATION BY qszászmirakyanbaskakov OPERATORS
F A S C I C U L I M A T H E M A T I C I Nr 48 212 V. Gupta, A. Aral and M. Ozhavzali APPROXIMATION BY SZÁSZMIRAKYANBASKAKOV OPERATORS Abstract. In the present paper we propose the analogue of well known
More informationCharacterization through Hazard Rate of heavy tailed distributions and some Convolution Closure Properties
Characterization through Hazard Rate of heavy tailed distributions and some Convolution Closure Properties Department of Statistics and Actuarial  Financial Mathematics, University of the Aegean October
More informationAN EXTENSION OF THE HONGPARK VERSION OF THE CHOWROBBINS THEOREM ON SUMS OF NONINTEGRABLE RANDOM VARIABLES
J. Korean Math. Soc. 32 (1995), No. 2, pp. 363 370 AN EXTENSION OF THE HONGPARK VERSION OF THE CHOWROBBINS THEOREM ON SUMS OF NONINTEGRABLE RANDOM VARIABLES ANDRÉ ADLER AND ANDREW ROSALSKY A famous result
More informationStability of Adjointable Mappings in Hilbert
Stability of Adjointable Mappings in Hilbert arxiv:math/0501139v2 [math.fa] 1 Aug 2005 C Modules M. S. Moslehian Abstract The generalized Hyers Ulam Rassias stability of adjointable mappings on Hilbert
More informationEpiconvergence and εsubgradients of Convex Functions
Journal of Convex Analysis Volume 1 (1994), No.1, 87 100 Epiconvergence and εsubgradients of Convex Functions Andrei Verona Department of Mathematics, California State University Los Angeles, CA 90032,
More informationMIXED TYPE OF ADDITIVE AND QUINTIC FUNCTIONAL EQUATIONS. Abasalt Bodaghi, Pasupathi Narasimman, Krishnan Ravi, Behrouz Shojaee
Annales Mathematicae Silesianae 29 (205, 35 50 Prace Naukowe Uniwersytetu Śląskiego nr 3332, Katowice DOI: 0.55/amsil2050004 MIXED TYPE OF ADDITIVE AND QUINTIC FUNCTIONAL EQUATIONS Abasalt Bodaghi, Pasupathi
More informationGeneralized Neyman Pearson optimality of empirical likelihood for testing parameter hypotheses
Ann Inst Stat Math (2009) 61:773 787 DOI 10.1007/s1046300801726 Generalized Neyman Pearson optimality of empirical likelihood for testing parameter hypotheses Taisuke Otsu Received: 1 June 2007 / Revised:
More informationHarmonic sets and the harmonic prime number theorem
Harmonic sets and the harmonic prime number theorem Version: 9th September 2004 Kevin A. Broughan and Rory J. Casey University of Waikato, Hamilton, New Zealand Email: kab@waikato.ac.nz We restrict primes
More informationLecture I: Asymptotics for large GUE random matrices
Lecture I: Asymptotics for large GUE random matrices Steen Thorbjørnsen, University of Aarhus andom Matrices Definition. Let (Ω, F, P) be a probability space and let n be a positive integer. Then a random
More informationMonotonicity and Aging Properties of Random Sums
Monotonicity and Aging Properties of Random Sums Jun Cai and Gordon E. Willmot Department of Statistics and Actuarial Science University of Waterloo Waterloo, Ontario Canada N2L 3G1 Email: jcai@uwaterloo.ca,
More informationFunctions of several variables of finite variation and their differentiability
ANNALES POLONICI MATHEMATICI LX.1 (1994) Functions of several variables of finite variation and their differentiability by Dariusz Idczak ( Lódź) Abstract. Some differentiability properties of functions
More informationPhenomena in high dimensions in geometric analysis, random matrices, and computational geometry Roscoff, France, June 2529, 2012
Phenomena in high dimensions in geometric analysis, random matrices, and computational geometry Roscoff, France, June 2529, 202 BOUNDS AND ASYMPTOTICS FOR FISHER INFORMATION IN THE CENTRAL LIMIT THEOREM
More informationWald for nonstopping times: The rewards of impatient prophets
Electron. Commun. Probab. 19 (2014), no. 78, 1 9. DOI: 10.1214/ECP.v193609 ISSN: 1083589X ELECTRONIC COMMUNICATIONS in PROBABILITY Wald for nonstopping times: The rewards of impatient prophets Alexander
More informationA square bias transformation: properties and applications
A square bias transformation: properties and applications Irina Shevtsova arxiv:11.6775v4 [math.pr] 13 Dec 13 Abstract The properties of the square bias transformation are studied, in particular, the precise
More informationAsymptotic normality of the L k error of the Grenander estimator
Asymptotic normality of the L k error of the Grenander estimator Vladimir N. Kulikov Hendrik P. Lopuhaä 1.7.24 Abstract: We investigate the limit behavior of the L k distance between a decreasing density
More information