[{"issue":"5","date_updated":"2019-04-26T07:22:03Z","page":"1582 - 1591","extern":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"journal_article","date_published":"2015-03-14T00:00:00Z","quality_controlled":"1","oa":1,"intvolume":" 35","year":"2015","publist_id":"5675","volume":35,"language":[{"iso":"eng"}],"citation":{"chicago":"Sadel, Christian. “A Herman-Avila-Bochi Formula for Higher-Dimensional Pseudo-Unitary and Hermitian-Symplectic-Cocycles.” *Ergodic Theory and Dynamical Systems* 35, no. 5 (2015): 1582–91. https://doi.org/10.1017/etds.2013.103.","short":"C. Sadel, Ergodic Theory and Dynamical Systems 35 (2015) 1582–1591.","ista":"Sadel C. 2015. A Herman-Avila-Bochi formula for higher-dimensional pseudo-unitary and Hermitian-symplectic-cocycles. Ergodic Theory and Dynamical Systems. 35(5), 1582–1591.","mla":"Sadel, Christian. “A Herman-Avila-Bochi Formula for Higher-Dimensional Pseudo-Unitary and Hermitian-Symplectic-Cocycles.” *Ergodic Theory and Dynamical Systems*, vol. 35, no. 5, Cambridge University Press, 2015, pp. 1582–91, doi:10.1017/etds.2013.103.","ieee":"C. Sadel, “A Herman-Avila-Bochi formula for higher-dimensional pseudo-unitary and Hermitian-symplectic-cocycles,” *Ergodic Theory and Dynamical Systems*, vol. 35, no. 5, pp. 1582–1591, 2015.","ama":"Sadel C. A Herman-Avila-Bochi formula for higher-dimensional pseudo-unitary and Hermitian-symplectic-cocycles. *Ergodic Theory and Dynamical Systems*. 2015;35(5):1582-1591. doi:10.1017/etds.2013.103","apa":"Sadel, C. (2015). A Herman-Avila-Bochi formula for higher-dimensional pseudo-unitary and Hermitian-symplectic-cocycles. *Ergodic Theory and Dynamical Systems*, *35*(5), 1582–1591. https://doi.org/10.1017/etds.2013.103"},"publisher":"Cambridge University Press","status":"public","doi":"10.1017/etds.2013.103","month":"03","publication":"Ergodic Theory and Dynamical Systems","title":"A Herman-Avila-Bochi formula for higher-dimensional pseudo-unitary and Hermitian-symplectic-cocycles","date_created":"2018-12-11T11:52:24Z","publication_status":"published","abstract":[{"lang":"eng","text":"A Herman-Avila-Bochi type formula is obtained for the average sum of the top d Lyapunov exponents over a one-parameter family of double-struck G-cocycles, where double-struck G is the group that leaves a certain, non-degenerate Hermitian form of signature (c, d) invariant. The generic example of such a group is the pseudo-unitary group U(c, d) or, in the case c = d, the Hermitian-symplectic group HSp(2d) which naturally appears for cocycles related to Schrödinger operators. In the case d = 1, the formula for HSp(2d) cocycles reduces to the Herman-Avila-Bochi formula for SL(2, ℝ) cocycles."}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1307.8414"}],"author":[{"full_name":"Sadel, Christian","id":"4760E9F8-F248-11E8-B48F-1D18A9856A87","first_name":"Christian","orcid":"0000-0001-8255-3968","last_name":"Sadel"}],"day":"14","oa_version":"Preprint","_id":"1503"},{"publist_id":"5674","intvolume":" 43","year":"2015","quality_controlled":"1","oa":1,"date_published":"2015-12-01T00:00:00Z","type":"journal_article","issue":"6","date_updated":"2019-01-24T13:03:01Z","extern":"1","page":"2588 - 2623","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"1504","author":[{"last_name":"Bao","orcid":"0000-0003-3036-1475","first_name":"Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","full_name":"Bao, Zhigang"},{"last_name":"Lin","full_name":"Lin, Liang","first_name":"Liang"},{"first_name":"Guangming","full_name":"Pan, Guangming","last_name":"Pan"},{"last_name":"Zhou","full_name":"Zhou, Wang","first_name":"Wang"}],"abstract":[{"text":"Let Q = (Q1, . . . , Qn) be a random vector drawn from the uniform distribution on the set of all n! permutations of {1, 2, . . . , n}. Let Z = (Z1, . . . , Zn), where Zj is the mean zero variance one random variable obtained by centralizing and normalizing Qj , j = 1, . . . , n. Assume that Xi , i = 1, . . . ,p are i.i.d. copies of 1/√ p Z and X = Xp,n is the p × n random matrix with Xi as its ith row. Then Sn = XX is called the p × n Spearman's rank correlation matrix which can be regarded as a high dimensional extension of the classical nonparametric statistic Spearman's rank correlation coefficient between two independent random variables. In this paper, we establish a CLT for the linear spectral statistics of this nonparametric random matrix model in the scenario of high dimension, namely, p = p(n) and p/n→c ∈ (0,∞) as n→∞.We propose a novel evaluation scheme to estimate the core quantity in Anderson and Zeitouni's cumulant method in [Ann. Statist. 36 (2008) 2553-2576] to bypass the so-called joint cumulant summability. In addition, we raise a two-step comparison approach to obtain the explicit formulae for the mean and covariance functions in the CLT. Relying on this CLT, we then construct a distribution-free statistic to test complete independence for components of random vectors. Owing to the nonparametric property, we can use this test on generally distributed random variables including the heavy-tailed ones.","lang":"eng"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1312.5119"}],"oa_version":"Published Version","day":"01","publication_status":"published","publication":"Annals of Statistics","month":"12","title":"Spectral statistics of large dimensional spearman s rank correlation matrix and its application","date_created":"2018-12-11T11:52:24Z","doi":"10.1214/15-AOS1353","status":"public","publisher":"Institute of Mathematical Statistics","citation":{"apa":"Bao, Z., Lin, L., Pan, G., & Zhou, W. (2015). Spectral statistics of large dimensional spearman s rank correlation matrix and its application. *Annals of Statistics*, *43*(6), 2588–2623. https://doi.org/10.1214/15-AOS1353","ama":"Bao Z, Lin L, Pan G, Zhou W. Spectral statistics of large dimensional spearman s rank correlation matrix and its application. *Annals of Statistics*. 2015;43(6):2588-2623. doi:10.1214/15-AOS1353","ieee":"Z. Bao, L. Lin, G. Pan, and W. Zhou, “Spectral statistics of large dimensional spearman s rank correlation matrix and its application,” *Annals of Statistics*, vol. 43, no. 6, pp. 2588–2623, 2015.","mla":"Bao, Zhigang, et al. “Spectral Statistics of Large Dimensional Spearman s Rank Correlation Matrix and Its Application.” *Annals of Statistics*, vol. 43, no. 6, Institute of Mathematical Statistics, 2015, pp. 2588–623, doi:10.1214/15-AOS1353.","ista":"Bao Z, Lin L, Pan G, Zhou W. 2015. Spectral statistics of large dimensional spearman s rank correlation matrix and its application. Annals of Statistics. 43(6), 2588–2623.","chicago":"Bao, Zhigang, Liang Lin, Guangming Pan, and Wang Zhou. “Spectral Statistics of Large Dimensional Spearman s Rank Correlation Matrix and Its Application.” *Annals of Statistics* 43, no. 6 (2015): 2588–2623. https://doi.org/10.1214/15-AOS1353.","short":"Z. Bao, L. Lin, G. Pan, W. Zhou, Annals of Statistics 43 (2015) 2588–2623."},"language":[{"iso":"eng"}],"volume":43},{"oa":1,"quality_controlled":"1","intvolume":" 43","year":"2015","publist_id":"5672","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"382 - 421","date_updated":"2019-01-24T13:03:02Z","issue":"1","type":"journal_article","date_published":"2015-02-01T00:00:00Z","department":[{"_id":"LaEr"}],"title":"Universality for the largest eigenvalue of sample covariance matrices with general population","date_created":"2018-12-11T11:52:25Z","publication":"Annals of Statistics","month":"02","publication_status":"published","day":"01","oa_version":"Preprint","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1304.5690"}],"abstract":[{"lang":"eng","text":"This paper is aimed at deriving the universality of the largest eigenvalue of a class of high-dimensional real or complex sample covariance matrices of the form W N =Σ 1/2XX∗Σ 1/2 . Here, X = (xij )M,N is an M× N random matrix with independent entries xij , 1 ≤ i M,≤ 1 ≤ j ≤ N such that Exij = 0, E|xij |2 = 1/N . On dimensionality, we assume that M = M(N) and N/M → d ε (0, ∞) as N ∞→. For a class of general deterministic positive-definite M × M matrices Σ , under some additional assumptions on the distribution of xij 's, we show that the limiting behavior of the largest eigenvalue of W N is universal, via pursuing a Green function comparison strategy raised in [Probab. Theory Related Fields 154 (2012) 341-407, Adv. Math. 229 (2012) 1435-1515] by Erd″os, Yau and Yin for Wigner matrices and extended by Pillai and Yin [Ann. Appl. Probab. 24 (2014) 935-1001] to sample covariance matrices in the null case (&Epsi = I ). Consequently, in the standard complex case (Ex2 ij = 0), combing this universality property and the results known for Gaussian matrices obtained by El Karoui in [Ann. Probab. 35 (2007) 663-714] (nonsingular case) and Onatski in [Ann. Appl. Probab. 18 (2008) 470-490] (singular case), we show that after an appropriate normalization the largest eigenvalue of W N converges weakly to the type 2 Tracy-Widom distribution TW2 . Moreover, in the real case, we show that whenΣ is spiked with a fixed number of subcritical spikes, the type 1 Tracy-Widom limit TW1 holds for the normalized largest eigenvalue of W N , which extends a result of Féral and Péché in [J. Math. Phys. 50 (2009) 073302] to the scenario of nondiagonal Σ and more generally distributed X . In summary, we establish the Tracy-Widom type universality for the largest eigenvalue of generally distributed sample covariance matrices under quite light assumptions on &Sigma . Applications of these limiting results to statistical signal detection and structure recognition of separable covariance matrices are also discussed."}],"author":[{"id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","first_name":"Zhigang","orcid":"0000-0003-3036-1475","full_name":"Bao, Zhigang","last_name":"Bao"},{"first_name":"Guangming","full_name":"Pan, Guangming","last_name":"Pan"},{"first_name":"Wang","full_name":"Zhou, Wang","last_name":"Zhou"}],"_id":"1505","language":[{"iso":"eng"}],"volume":43,"publisher":"Institute of Mathematical Statistics","citation":{"ama":"Bao Z, Pan G, Zhou W. Universality for the largest eigenvalue of sample covariance matrices with general population. *Annals of Statistics*. 2015;43(1):382-421. doi:10.1214/14-AOS1281","apa":"Bao, Z., Pan, G., & Zhou, W. (2015). Universality for the largest eigenvalue of sample covariance matrices with general population. *Annals of Statistics*, *43*(1), 382–421. https://doi.org/10.1214/14-AOS1281","ista":"Bao Z, Pan G, Zhou W. 2015. Universality for the largest eigenvalue of sample covariance matrices with general population. Annals of Statistics. 43(1), 382–421.","short":"Z. Bao, G. Pan, W. Zhou, Annals of Statistics 43 (2015) 382–421.","chicago":"Bao, Zhigang, Guangming Pan, and Wang Zhou. “Universality for the Largest Eigenvalue of Sample Covariance Matrices with General Population.” *Annals of Statistics* 43, no. 1 (2015): 382–421. https://doi.org/10.1214/14-AOS1281.","ieee":"Z. Bao, G. Pan, and W. Zhou, “Universality for the largest eigenvalue of sample covariance matrices with general population,” *Annals of Statistics*, vol. 43, no. 1, pp. 382–421, 2015.","mla":"Bao, Zhigang, et al. “Universality for the Largest Eigenvalue of Sample Covariance Matrices with General Population.” *Annals of Statistics*, vol. 43, no. 1, Institute of Mathematical Statistics, 2015, pp. 382–421, doi:10.1214/14-AOS1281."},"status":"public","doi":"10.1214/14-AOS1281","acknowledgement":"B.Z. was supported in part by NSFC Grant 11071213, ZJNSF Grant R6090034 and SRFDP Grant 20100101110001. P.G. was supported in part by the Ministry of Education, Singapore, under Grant ARC 14/11. Z.W. was supported in part by the Ministry of Education, Singapore, under Grant ARC 14/11, and by a Grant R-155-000-131-112 at the National University of Singapore\r\n"},{"doi":"10.3150/14-BEJ615","status":"public","publisher":"Bernoulli Society for Mathematical Statistics and Probability","citation":{"ama":"Bao Z, Pan G, Zhou W. The logarithmic law of random determinant. *Bernoulli*. 2015;21(3):1600-1628. doi:10.3150/14-BEJ615","apa":"Bao, Z., Pan, G., & Zhou, W. (2015). The logarithmic law of random determinant. *Bernoulli*, *21*(3), 1600–1628. https://doi.org/10.3150/14-BEJ615","chicago":"Bao, Zhigang, Guangming Pan, and Wang Zhou. “The Logarithmic Law of Random Determinant.” *Bernoulli* 21, no. 3 (2015): 1600–1628. https://doi.org/10.3150/14-BEJ615.","short":"Z. Bao, G. Pan, W. Zhou, Bernoulli 21 (2015) 1600–1628.","ista":"Bao Z, Pan G, Zhou W. 2015. The logarithmic law of random determinant. Bernoulli. 21(3), 1600–1628.","mla":"Bao, Zhigang, et al. “The Logarithmic Law of Random Determinant.” *Bernoulli*, vol. 21, no. 3, Bernoulli Society for Mathematical Statistics and Probability, 2015, pp. 1600–28, doi:10.3150/14-BEJ615.","ieee":"Z. Bao, G. Pan, and W. Zhou, “The logarithmic law of random determinant,” *Bernoulli*, vol. 21, no. 3, pp. 1600–1628, 2015."},"language":[{"iso":"eng"}],"volume":21,"_id":"1506","oa_version":"Preprint","day":"01","author":[{"full_name":"Bao, Zhigang","orcid":"0000-0003-3036-1475","first_name":"Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","last_name":"Bao"},{"full_name":"Pan, Guangming","first_name":"Guangming","last_name":"Pan"},{"last_name":"Zhou","first_name":"Wang","full_name":"Zhou, Wang"}],"abstract":[{"lang":"eng","text":"Consider the square random matrix An = (aij)n,n, where {aij:= a(n)ij , i, j = 1, . . . , n} is a collection of independent real random variables with means zero and variances one. Under the additional moment condition supn max1≤i,j ≤n Ea4ij <∞, we prove Girko's logarithmic law of det An in the sense that as n→∞ log | detAn| ? (1/2) log(n-1)! d/→√(1/2) log n N(0, 1)."}],"main_file_link":[{"url":"http://arxiv.org/abs/1208.5823","open_access":"1"}],"publication_status":"published","date_created":"2018-12-11T11:52:25Z","title":"The logarithmic law of random determinant","month":"08","publication":"Bernoulli","date_published":"2015-08-01T00:00:00Z","department":[{"_id":"LaEr"}],"type":"journal_article","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2019-01-24T13:03:02Z","issue":"3","page":"1600 - 1628","publist_id":"5671","year":"2015","intvolume":" 21","oa":1,"quality_controlled":"1"},{"year":"2015","oa":1,"external_id":{"arxiv":["1511.03501"]},"article_processing_charge":"No","related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"8156"}]},"department":[{"_id":"UlWa"}],"date_published":"2015-11-15T00:00:00Z","type":"preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"24","date_updated":"2020-07-30T12:50:39Z","_id":"8183","oa_version":"Preprint","day":"15","author":[{"last_name":"Avvakumov","id":"3827DAC8-F248-11E8-B48F-1D18A9856A87","first_name":"Sergey","full_name":"Avvakumov, Sergey"},{"full_name":"Mabillard, Isaac","first_name":"Isaac","id":"32BF9DAA-F248-11E8-B48F-1D18A9856A87","last_name":"Mabillard"},{"last_name":"Skopenkov","full_name":"Skopenkov, A.","first_name":"A."},{"last_name":"Wagner","orcid":"0000-0002-1494-0568","first_name":"Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Uli"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1511.03501"}],"abstract":[{"text":"We study conditions under which a finite simplicial complex $K$ can be mapped to $\\mathbb R^d$ without higher-multiplicity intersections. An almost $r$-embedding is a map $f: K\\to \\mathbb R^d$ such that the images of any $r$\r\npairwise disjoint simplices of $K$ do not have a common point. We show that if $r$ is not a prime power and $d\\geq 2r+1$, then there is a counterexample to the topological Tverberg conjecture, i.e., there is an almost $r$-embedding of\r\nthe $(d+1)(r-1)$-simplex in $\\mathbb R^d$. This improves on previous constructions of counterexamples (for $d\\geq 3r$) based on a series of papers by M. \\\"Ozaydin, M. Gromov, P. Blagojevi\\'c, F. Frick, G. Ziegler, and the second and fourth present authors. The counterexamples are obtained by proving the following algebraic criterion in codimension 2: If $r\\ge3$ and if $K$ is a finite $2(r-1)$-complex then there exists an almost $r$-embedding $K\\to \\mathbb R^{2r}$ if and only if there exists a general position PL map $f:K\\to \\mathbb R^{2r}$ such that the algebraic intersection number of the $f$-images of any $r$ pairwise disjoint simplices of $K$ is zero. This result can be restated in terms of cohomological obstructions or equivariant maps, and extends an analogous codimension 3 criterion by the second and fourth authors. As another application we classify ornaments $f:S^3 \\sqcup S^3\\sqcup S^3\\to \\mathbb R^5$ up to ornament\r\nconcordance. It follows from work of M. Freedman, V. Krushkal and P. Teichner that the analogous criterion for $r=2$ is false. We prove a lemma on singular higher-dimensional Borromean rings, yielding an elementary proof of the counterexample.","lang":"eng"}],"publication_status":"submitted","title":"Eliminating higher-multiplicity intersections, III. Codimension 2","date_created":"2020-07-30T10:45:19Z","month":"11","acknowledgement":"We would like to thank A. Klyachko, V. Krushkal, S. Melikhov, M. Tancer, P. Teichner and anonymous referees\r\nfor helpful discussions.","status":"public","publisher":"arXiv","citation":{"apa":"Avvakumov, S., Mabillard, I., Skopenkov, A., & Wagner, U. (n.d.). Eliminating higher-multiplicity intersections, III. Codimension 2. arXiv.","ama":"Avvakumov S, Mabillard I, Skopenkov A, Wagner U. Eliminating higher-multiplicity intersections, III. Codimension 2.","ieee":"S. Avvakumov, I. Mabillard, A. Skopenkov, and U. Wagner, “Eliminating higher-multiplicity intersections, III. Codimension 2.” arXiv.","mla":"Avvakumov, Sergey, et al. *Eliminating Higher-Multiplicity Intersections, III. Codimension 2*. 1511.03501, arXiv.","ista":"Avvakumov S, Mabillard I, Skopenkov A, Wagner U. Eliminating higher-multiplicity intersections, III. Codimension 2.","short":"S. Avvakumov, I. Mabillard, A. Skopenkov, U. Wagner, (n.d.).","chicago":"Avvakumov, Sergey, Isaac Mabillard, A. Skopenkov, and Uli Wagner. “Eliminating Higher-Multiplicity Intersections, III. Codimension 2.” arXiv, n.d."},"article_number":"1511.03501","language":[{"iso":"eng"}]}]