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LittlewoodOfford inequalities for random variables
 11] Nathan Linial, Avner Magen, and Michael
, 1994
"... The concentration of a realvalued random variable X is c(X) sup P(t < X < + 1). Given bounds on the concentrations of n independent random variables, how large can the concentration of their sum be? The main aim of this paper is to give a best possible upper bound for the concentration of the ..."
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Cited by 4 (0 self)
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of the sum of n independent random variables, each of concentration at most 1/k, where k is an integer. Other bounds on the concentration are also discussed, as well as the case of vectorvalued random variables. Key words. LittlewoodOfford problem, concentration, normed spaces AMS subject classifications.
BILINEAR AND QUADRATIC VARIANTS ON THE LITTLEWOODOFFORD PROBLEM
, 2009
"... If f(x1,..., xn) is a polynomial dependent on a large number of independent Bernoulli random variables, what can be said about the maximum concentration of f on any single value? For linear polynomials, this reduces to one version of the classical LittlewoodOfford problem: Given nonzero constants ..."
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Cited by 11 (0 self)
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If f(x1,..., xn) is a polynomial dependent on a large number of independent Bernoulli random variables, what can be said about the maximum concentration of f on any single value? For linear polynomials, this reduces to one version of the classical LittlewoodOfford problem: Given nonzero
The littlewoodofford problem and invertibility of random matrices
 Adv. Math
"... Abstract. We prove two basic conjectures on the distribution of the smallest singular value of random n×n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n −1/2, which is optimal for Gaussian matrices. Moreover, we give a opti ..."
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Cited by 104 (18 self)
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optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in the LittlewoodOfford problem: for i.i.d. random variables Xk and real numbers ak, determine the probability p that the sum � k akXk lies near some number v. For arbitrary coefficients ak
The LittlewoodOfford problem in high dimensions and a conjecture of Frankl and Füredi
 Combinatorica
"... ar ..."
A SHARP INVERSE LITTLEWOODOFFORD THEOREM
, 2009
"... Let ηi, i = 1,...,n be iid Bernoulli random variables. Given a multiset v of n numbers v1,..., vn, the concentration probability P1(v) of v is defined as P1(v): = sup x P(v1η1 +...vnηn = x). A classical result of LittlewoodOfford and Erdős from the 1940s asserts that if the vi are nonzero, then t ..."
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Cited by 12 (3 self)
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Let ηi, i = 1,...,n be iid Bernoulli random variables. Given a multiset v of n numbers v1,..., vn, the concentration probability P1(v) of v is defined as P1(v): = sup x P(v1η1 +...vnηn = x). A classical result of LittlewoodOfford and Erdős from the 1940s asserts that if the vi are non
From the LittlewoodOfford problem to the Circular Law: Universality of the spectral distribution of random matrices
 BULL. AMER. MATH. SOC
, 2009
"... The famous circular law asserts that if Mn is an n×n matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution of the normalized matrix 1 √ Mn converges both in probability and n almost surely to the uniform distribution on the unit disk {z ∈ C: z  ≤1 ..."
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Cited by 53 (7 self)
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in the establishment of the circular law at this level of generality, in particular recent advances in understanding the LittlewoodOfford problem and its inverse.
Some optimal inapproximability results
, 2002
"... We prove optimal, up to an arbitrary ffl? 0, inapproximability results for MaxEkSat for k * 3, maximizing the number of satisfied linear equations in an overdetermined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for ..."
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Cited by 782 (12 self)
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We prove optimal, up to an arbitrary ffl? 0, inapproximability results for MaxEkSat for k * 3, maximizing the number of satisfied linear equations in an overdetermined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds
New results in linear filtering and prediction theory
 Trans. ASME, Ser. D, J. Basic Eng
, 1961
"... A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this "variance equation " completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or nonstationary sta ..."
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Cited by 585 (0 self)
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statistics. The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations. Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results
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