---
_id: '1503'
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.
author:
- first_name: Christian
full_name: Sadel, Christian
id: 4760E9F8-F248-11E8-B48F-1D18A9856A87
last_name: Sadel
orcid: 0000-0001-8255-3968
citation:
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
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.'
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.
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.
short: C. Sadel, Ergodic Theory and Dynamical Systems 35 (2015) 1582–1591.
date_created: 2018-12-11T11:52:24Z
date_published: 2015-03-14T00:00:00Z
date_updated: 2019-04-26T07:22:03Z
day: '14'
doi: 10.1017/etds.2013.103
extern: '1'
intvolume: ' 35'
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1307.8414
month: '03'
oa: 1
oa_version: Preprint
page: 1582 - 1591
publication: Ergodic Theory and Dynamical Systems
publication_status: published
publisher: Cambridge University Press
publist_id: '5675'
quality_controlled: '1'
status: public
title: A Herman-Avila-Bochi formula for higher-dimensional pseudo-unitary and Hermitian-symplectic-cocycles
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 35
year: '2015'
...
---
_id: '1504'
abstract:
- lang: eng
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.
author:
- first_name: Zhigang
full_name: Bao, Zhigang
id: 442E6A6C-F248-11E8-B48F-1D18A9856A87
last_name: Bao
orcid: 0000-0003-3036-1475
- first_name: Liang
full_name: Lin, Liang
last_name: Lin
- first_name: Guangming
full_name: Pan, Guangming
last_name: Pan
- first_name: Wang
full_name: Zhou, Wang
last_name: Zhou
citation:
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
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
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.'
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.
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.
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.
short: Z. Bao, L. Lin, G. Pan, W. Zhou, Annals of Statistics 43 (2015) 2588–2623.
date_created: 2018-12-11T11:52:24Z
date_published: 2015-12-01T00:00:00Z
date_updated: 2019-01-24T13:03:01Z
day: '01'
doi: 10.1214/15-AOS1353
extern: '1'
intvolume: ' 43'
issue: '6'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1312.5119
month: '12'
oa: 1
oa_version: Published Version
page: 2588 - 2623
publication: Annals of Statistics
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '5674'
quality_controlled: '1'
status: public
title: Spectral statistics of large dimensional spearman s rank correlation matrix
and its application
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 43
year: '2015'
...
---
_id: '1505'
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.
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"
author:
- first_name: Zhigang
full_name: Bao, Zhigang
id: 442E6A6C-F248-11E8-B48F-1D18A9856A87
last_name: Bao
orcid: 0000-0003-3036-1475
- first_name: Guangming
full_name: Pan, Guangming
last_name: Pan
- first_name: Wang
full_name: Zhou, Wang
last_name: Zhou
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
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.
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.
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.
short: Z. Bao, G. Pan, W. Zhou, Annals of Statistics 43 (2015) 382–421.
date_created: 2018-12-11T11:52:25Z
date_published: 2015-02-01T00:00:00Z
date_updated: 2019-01-24T13:03:02Z
day: '01'
department:
- _id: LaEr
doi: 10.1214/14-AOS1281
intvolume: ' 43'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1304.5690
month: '02'
oa: 1
oa_version: Preprint
page: 382 - 421
publication: Annals of Statistics
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '5672'
quality_controlled: '1'
status: public
title: Universality for the largest eigenvalue of sample covariance matrices with
general population
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 43
year: '2015'
...
---
_id: '1506'
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).
author:
- first_name: Zhigang
full_name: Bao, Zhigang
id: 442E6A6C-F248-11E8-B48F-1D18A9856A87
last_name: Bao
orcid: 0000-0003-3036-1475
- first_name: Guangming
full_name: Pan, Guangming
last_name: Pan
- first_name: Wang
full_name: Zhou, Wang
last_name: Zhou
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.'
ieee: Z. Bao, G. Pan, and W. Zhou, “The logarithmic law of random determinant,”
*Bernoulli*, vol. 21, no. 3, pp. 1600–1628, 2015.
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.
short: Z. Bao, G. Pan, W. Zhou, Bernoulli 21 (2015) 1600–1628.
date_created: 2018-12-11T11:52:25Z
date_published: 2015-08-01T00:00:00Z
date_updated: 2019-01-24T13:03:02Z
day: '01'
department:
- _id: LaEr
doi: 10.3150/14-BEJ615
intvolume: ' 21'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1208.5823
month: '08'
oa: 1
oa_version: Preprint
page: 1600 - 1628
publication: Bernoulli
publication_status: published
publisher: Bernoulli Society for Mathematical Statistics and Probability
publist_id: '5671'
quality_controlled: '1'
status: public
title: The logarithmic law of random determinant
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 21
year: '2015'
...
---
_id: '8183'
abstract:
- lang: eng
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."
acknowledgement: "We would like to thank A. Klyachko, V. Krushkal, S. Melikhov, M.
Tancer, P. Teichner and anonymous referees\r\nfor helpful discussions."
article_number: '1511.03501'
article_processing_charge: No
author:
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
- first_name: Isaac
full_name: Mabillard, Isaac
id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87
last_name: Mabillard
- first_name: A.
full_name: Skopenkov, A.
last_name: Skopenkov
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: Avvakumov S, Mabillard I, Skopenkov A, Wagner U. Eliminating higher-multiplicity
intersections, III. Codimension 2.
apa: Avvakumov, S., Mabillard, I., Skopenkov, A., & Wagner, U. (n.d.). Eliminating
higher-multiplicity intersections, III. Codimension 2. arXiv.
chicago: Avvakumov, Sergey, Isaac Mabillard, A. Skopenkov, and Uli Wagner. “Eliminating
Higher-Multiplicity Intersections, III. Codimension 2.” arXiv, n.d.
ieee: S. Avvakumov, I. Mabillard, A. Skopenkov, and U. Wagner, “Eliminating higher-multiplicity
intersections, III. Codimension 2.” arXiv.
ista: Avvakumov S, Mabillard I, Skopenkov A, Wagner U. Eliminating higher-multiplicity
intersections, III. Codimension 2.
mla: Avvakumov, Sergey, et al. *Eliminating Higher-Multiplicity Intersections,
III. Codimension 2*. 1511.03501, arXiv.
short: S. Avvakumov, I. Mabillard, A. Skopenkov, U. Wagner, (n.d.).
date_created: 2020-07-30T10:45:19Z
date_published: 2015-11-15T00:00:00Z
date_updated: 2020-07-30T12:50:39Z
day: '15'
department:
- _id: UlWa
external_id:
arxiv:
- '1511.03501'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1511.03501
month: '11'
oa: 1
oa_version: Preprint
page: '24'
publication_status: submitted
publisher: arXiv
related_material:
record:
- id: '8156'
relation: dissertation_contains
status: public
status: public
title: Eliminating higher-multiplicity intersections, III. Codimension 2
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2015'
...