scholarly journals Combinatorial anti-concentration inequalities, with applications

2021 ◽  
pp. 1-22
Author(s):  
JACOB FOX ◽  
MATTHEW KWAN ◽  
LISA SAUERMANN
Keyword(s):  
Random Graphs ◽  
Random Variables ◽  
Concentration Inequalities ◽  
Concentration Inequality ◽  
Concentration Bounds ◽  
Nonnegative Coefficients ◽  
Poisson Type

Abstract We prove several different anti-concentration inequalities for functions of independent Bernoulli-distributed random variables. First, motivated by a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn, we prove some “Poisson-type” anti-concentration theorems that give bounds of the form 1/e + o(1) for the point probabilities of certain polynomials. Second, we prove an anti-concentration inequality for polynomials with nonnegative coefficients which extends the classical Erdős–Littlewood–Offord theorem and improves a theorem of Meka, Nguyen and Vu for polynomials of this type. As an application, we prove some new anti-concentration bounds for subgraph counts in random graphs.

MONOTONIC INDEPENDENCE ON THE WEAKLY MONOTONE FOCK SPACE AND RELATED POISSON TYPE THEOREM

2005 ◽  
Vol 08 (02) ◽  
pp. 259-275 ◽  
Author(s):  
JANUSZ WYSOCZAŃSKI
Keyword(s):  
Limit Theorem ◽  
Fock Space ◽  
Type Theorem ◽  
Random Variables ◽  
Direct Proof ◽  
Binomial Coefficients ◽  
Second Quantization ◽  
The Central Limit Theorem ◽  
Natural Way ◽  
Poisson Type

We show how the construction of t-transformation can be applied to the construction of a sequence of monotonically independent noncommutative random variables. We introduce the weakly monotone Fock space, on which these operators act. This space can be derived in a natural way from the papers by Pusz and Woronowicz on twisted second quantization. It was observed by Bożejko that, by taking μ = 0, for the μ-CAR relations one obtains the Muraki's monotone Fock space, while for the μ-CCR relations one obtains the weakly monotone Fock space. We show that the direct proof of the central limit theorem for these operators provides an interesting recurrence for the highest binomial coefficients. Moreover, we show the Poisson type theorem for these noncommutative random variables.

scholarly journals CLT-related large deviation bounds based on Stein's method

2007 ◽  
Vol 39 (3) ◽  
pp. 731-752 ◽  
Author(s):  
Martin Raič
Keyword(s):  
Random Graphs ◽  
Finite Population ◽  
Large Deviation ◽  
Random Variables ◽  
Stein’S Method ◽  
Stein's Method ◽  
Population Statistics ◽  
Sums Of Random Variables ◽  
Dependence Structures

Large deviation estimates are derived for sums of random variables with certain dependence structures, including finite population statistics and random graphs. The argument is based on Stein's method, but with a novel modification of Stein's equation inspired by the Cramér transform.

scholarly journals Generating stationary random graphs on ℤ with prescribed independent, identically distributed degrees

2006 ◽  
Vol 38 (02) ◽  
pp. 287-298 ◽  
Author(s):  
Maria Deijfen ◽  
Ronald Meester
Keyword(s):  
Probability Distribution ◽  
Random Graph ◽  
Random Graphs ◽  
Random Number ◽  
Random Variables ◽  
Nearest Neighbors ◽  
Expected Length ◽  
Vertex Set ◽  
Finitary Isomorphisms ◽  
Independent Identically Distributed

Let F be a probability distribution with support on the nonnegative integers. We describe two algorithms for generating a stationary random graph, with vertex set ℤ, in which the degrees of the vertices are independent, identically distributed random variables with distribution F. Focus is on an algorithm generating a graph in which, initially, a random number of ‘stubs’ with distribution F is attached to each vertex. Each stub is then randomly assigned a direction (left or right) and the edge configuration obtained by pairing stubs pointing to each other, first exhausting all possible connections between nearest neighbors, then linking second-nearest neighbors, and so on. Under the assumption that F has finite mean, it is shown that this algorithm leads to a well-defined configuration, but that the expected length of the shortest edge attached to a given vertex is infinite. It is also shown that any stationary algorithm for pairing stubs with random, independent directions causes the total length of the edges attached to a given vertex to have infinite mean. Connections to the problem of constructing finitary isomorphisms between Bernoulli shifts are discussed.

scholarly journals Concentration inequalities for sequential dynamical systems of the unit interval

2015 ◽  
Vol 36 (8) ◽  
pp. 2384-2407 ◽  
Author(s):  
ROMAIN AIMINO ◽  
JÉRÔME ROUSSEAU
Keyword(s):  
Dynamical Systems ◽  
Unit Interval ◽  
Concentration Inequalities ◽  
Concentration Inequality ◽  
Sequential Dynamical Systems

We prove a concentration inequality for sequential dynamical systems of the unit interval enjoying an exponential loss of memory in the BV norm and we investigate several of its consequences. In particular, this covers compositions of$\unicode[STIX]{x1D6FD}$-transformations, with all$\unicode[STIX]{x1D6FD}$lying in a neighborhood of a fixed$\unicode[STIX]{x1D6FD}_{\star }>1$, and systems satisfying a covering-type assumption.

A NOISE WITH SPACE PARAMETER AND INVARIANCE OF THE CLASS OF INTENSITIES

2012 ◽  
Vol 15 (04) ◽  
pp. 1250025
Author(s):  
K.-S. LEE ◽  
SI SI
Keyword(s):  
White Noise ◽  
Space Variable ◽  
Dimensional Space ◽  
Random Variables ◽  
Time Dependent ◽  
Two Dimensional ◽  
Parameter Case ◽  
Poisson Type

We are going to discuss a new noise depending on the space variable u. When we discuss random phenomena, we can see significant different characters between the phenomena depending on time and those on space. We can see this fact clearly when we take noises, which we mean systems involving independent idealized random variables, like white noise [Formula: see text] or many other examples. We shall take noises depending specifically on space to discuss different properties from what are found in time-dependent noises. We shall also discuss phenomena where Poisson type variables are involved, however we emphasize that their significant properties are described in terms of the intensity, and we shall therefore consider the phenomena in question with special emphasis on intensity which is often parametrized by the space variables. Interesting results will be seen in the two-dimensional space parameter case, where the group SL(2, R) plays an important role in the representation of the class of intensities.

Local limit theorems for one class of distributions in probabilistic combinatorics

2018 ◽  
Vol 28 (6) ◽  
pp. 405-420
Author(s):  
Aleksandr N. Timashev
Keyword(s):  
Power Series ◽  
Limit Theorems ◽  
Random Variables ◽  
Local Limit ◽  
Random Variable ◽  
Asymptotical Normality ◽  
Probabilistic Combinatorics ◽  
Nonnegative Coefficients ◽  
Positive Coefficients ◽  
Local Limit Theorems

Abstract Let a function f(z) be decomposed into a power series with nonnegative coefficients which converges in a circle of positive radius R. Let the distribution of the random variable ξn, n ∈ {1, 2, …}, be defined by the formula $$\begin{array}{} \displaystyle P\{\xi_n=N\}=\frac{\mathrm{coeff}_{z^n}\left(\frac{\left(f(z)\right)^N}{N!}\right)}{\mathrm{coeff}_{z^n}\left(\exp(f(z))\right)},\,N=0,1,\ldots \end{array} $$ for some ∣z∣ < R (if the denominator is positive). Examples of appearance of such distributions in probabilistic combinatorics are given. Local theorems on asymptotical normality for distributions of ξn are proved in two cases: a) if f(z) = (1 − z)−λ, λ = const ∈ (0, 1] for ∣z∣ < 1, and b) if all positive coefficients of expansion f (z) in a power series are equal to 1 and the set A of their numbers has the form $$\begin{array}{} \displaystyle A = \{m^r \, | \, m \in \mathbb{N} \}, \, \, r = \mathrm {const},\; r \in \{2,3,\ldots\}. \end{array} $$ A hypothetical general local limit normal theorem for random variables ξn is stated. Some examples of validity of the statement of this theorem are given.

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