7,853 research outputs found
Concentration inequalities for order statistics
This note describes non-asymptotic variance and tail bounds for order
statistics of samples of independent identically distributed random variables.
Those bounds are checked to be asymptotically tight when the sampling
distribution belongs to a maximum domain of attraction. If the sampling
distribution has non-decreasing hazard rate (this includes the Gaussian
distribution), we derive an exponential Efron-Stein inequality for order
statistics: an inequality connecting the logarithmic moment generating function
of centered order statistics with exponential moments of Efron-Stein
(jackknife) estimates of variance. We use this general connection to derive
variance and tail bounds for order statistics of Gaussian sample. Those bounds
are not within the scope of the Tsirelson-Ibragimov-Sudakov
Gaussian concentration inequality. Proofs are elementary and combine
R\'enyi's representation of order statistics and the so-called entropy approach
to concentration inequalities popularized by M. Ledoux.Comment: 13 page
Concentration inequalities for random tensors
We show how to extend several basic concentration inequalities for simple
random tensors where all are
independent random vectors in with independent coefficients. The
new results have optimal dependence on the dimension and the degree . As
an application, we show that random tensors are well conditioned: independent copies of the simple random tensor
are far from being linearly dependent with high probability. We prove this fact
for any degree and conjecture that it is true for any
.Comment: A few more typos were correcte
Concentration inequalities for disordered models
We use a generalization of Hoeffding's inequality to show concentration
results for the free energy of disordered pinning models, assuming only that
the disorder has a finite exponential moment. We also prove some concentration
inequalities for directed polymers in random environment, which we use to
establish a large deviations results for the end position of the polymer under
the polymer measure.Comment: Revised versio
Second-Order Matrix Concentration Inequalities
Matrix concentration inequalities give bounds for the spectral-norm deviation
of a random matrix from its expected value. These results have a weak
dimensional dependence that is sometimes, but not always, necessary. This paper
identifies one of the sources of the dimensional term and exploits this insight
to develop sharper matrix concentration inequalities. In particular, this
analysis delivers two refinements of the matrix Khintchine inequality that use
information beyond the matrix variance to reduce or eliminate the dimensional
dependence.Comment: 27 pages. Revision corrects technical errors in several place
Concentration Inequalities from Likelihood Ratio Method
We explore the applications of our previously established likelihood-ratio
method for deriving concentration inequalities for a wide variety of univariate
and multivariate distributions. New concentration inequalities for various
distributions are developed without the idea of minimizing moment generating
functions.Comment: 43 pages, no figur
Concentration Inequalities for Bounded Random Vectors
We derive simple concentration inequalities for bounded random vectors, which
generalize Hoeffding's inequalities for bounded scalar random variables. As
applications, we apply the general results to multinomial and Dirichlet
distributions to obtain multivariate concentration inequalities.Comment: 9 pages, no figure
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