A new large deviation inequality for U-statistics of order 2

Often when analysing randomized algorithms, especially parallel or distributed algorithms, one is called upon to show that some function of many independent choices is tightly concentrated about its expected value. For example, the algorithm might colour the vertices of a given graph with two colours and one would wish to show that, with high probability, very nearly half of all edges are monochromatic.The classic result of Chernoff [3] gives such a large deviation result when the function is a sum of independent indicator random variables. The results of Hoeffding [5] and Azuma [2] give similar results for functions which can be expressed as martingales with a bounded difference property. Roughly speaking, this means that each individual choice has a bounded effect on the value of the function. McDiarmid [9] nicely summarized these results and gave a host of applications. Expressed a little differently, his main result is as follows.

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A large deviation inequality forβ-mixing time series and its applications to the functional kernel regression model

Statistics & Probability Letters ◽

10.1016/j.spl.2017.09.013 ◽

2018 ◽

Vol 133 ◽

pp. 50-58 ◽

Cited By ~ 1

Author(s):

Johannes T.N. Krebs

Keyword(s):

Time Series ◽

Regression Model ◽

Large Deviation ◽

Mixing Time ◽

Kernel Regression ◽

Deviation Inequality ◽

Large Deviation Inequality

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Large Deviations for the Graph Distance in Supercritical Continuum Percolation

Journal of Applied Probability ◽

10.1239/jap/1300198142 ◽

2011 ◽

Vol 48 (1) ◽

pp. 154-172 ◽

Cited By ~ 8

Author(s):

Chang-Long Yao ◽

Ge Chen ◽

Tian-De Guo

Keyword(s):

Large Deviation ◽

Geometric Graph ◽

Graph Distance ◽

Homogeneous Poisson Process ◽

Random Geometric Graph ◽

Asymptotic Shape ◽

Deviation Inequality ◽

Large Deviation Inequality ◽

Palm Measure ◽

Infinite Component

Denote the Palm measure of a homogeneous Poisson process Hλ with two points 0 and x by P0,x. We prove that there exists a constant μ ≥ 1 such that P0,x(D(0, x) / μ||x||2 ∉ (1 − ε, 1 + ε) | 0, x ∈ C∞) exponentially decreases when ||x||2 tends to ∞, where D(0, x) is the graph distance between 0 and x in the infinite component C∞ of the random geometric graph G(Hλ; 1). We derive a large deviation inequality for an asymptotic shape result. Our results have applications in many fields and especially in wireless sensor networks.

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Near-Optimal, Distributed Edge Colouring via the Nibble Method

BRICS Report Series ◽

10.7146/brics.v3i11.19974 ◽

1996 ◽

Vol 3 (11) ◽

Author(s):

Devdatt P. Dubhashi ◽

David A. Grable ◽

Alessandro Panconesi

Keyword(s):

Failure Probability ◽

Positive Constant ◽

Maximum Degree ◽

Large Deviation ◽

Random Variables ◽

Randomized Algorithm ◽

Independent Random Variables ◽

Deviation Inequality ◽

Large Deviation Inequality ◽

Edge Colouring

We give a distributed randomized algorithm to edge colour a network. Let G be a graph with n nodes and maximum degree Delta. Here we prove: If Delta = Omega(log^(1+delta) n) for some delta > 0 and lambda > 0 is fixed, the algorithm almost always colours G with (1 + lambda)Delta colours in time O(log n). If s > 0 is fixed, there exists a positive constant k such that if Delta = omega(log^k n), the algorithm almost always colours G with Delta + Delta / log^s n = (1+o(1))Delta colours in time O(logn + log^s n log log n). By "almost always" we mean that the algorithm may fail, but the failure probability can be made arbitrarily close to 0. The algorithm is based on the nibble method, a probabilistic strategy introduced by Vojtech R¨odl. The analysis makes use of a powerful large deviation inequality for functions of independent random variables.

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Large Deviations for the Graph Distance in Supercritical Continuum Percolation

Journal of Applied Probability ◽

10.1017/s0021900200007695 ◽

2011 ◽

Vol 48 (01) ◽

pp. 154-172 ◽

Cited By ~ 2

Author(s):

Chang-Long Yao ◽

Ge Chen ◽

Tian-De Guo

Keyword(s):

Large Deviation ◽

Geometric Graph ◽

Graph Distance ◽

Homogeneous Poisson Process ◽

Random Geometric Graph ◽

Asymptotic Shape ◽

Deviation Inequality ◽

Large Deviation Inequality ◽

Palm Measure ◽

Infinite Component

Denote the Palm measure of a homogeneous Poisson process H λ with two points 0 and x by P0,x . We prove that there exists a constant μ ≥ 1 such that P0,x (D(0, x) / μ||x||2 ∉ (1 − ε, 1 + ε) | 0, x ∈ C ∞) exponentially decreases when ||x||2 tends to ∞, where D(0, x) is the graph distance between 0 and x in the infinite component C ∞ of the random geometric graph G(H λ; 1). We derive a large deviation inequality for an asymptotic shape result. Our results have applications in many fields and especially in wireless sensor networks.

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Outage Constraint Robust Cooperative Beamforming and Jamming Design for Secure AF Relay Networks

Mathematical Problems in Engineering ◽

10.1155/2021/5919617 ◽

2021 ◽

Vol 2021 ◽

pp. 1-7

Author(s):

Jinxin Zhu ◽

Jun Shao

Keyword(s):

Large Deviation ◽

Type Inequality ◽

Optimization Method ◽

Relay Network ◽

Secrecy Rate ◽

Channel State ◽

Deviation Inequality ◽

Rate Maximization ◽

Large Deviation Inequality ◽

Cooperative Beamforming

This paper investigates a joint robust scheme in a secrecy relay network where distributed relays perform cooperative beamforming and a friendly jammer transmits jamming signal to enhance the information security. Specifically, we consider the outage constraint secrecy rate maximization design with imperfect channel state information. Through semidefinite relaxation and one-dimensional search, we propose a two-layer optimization method to solve the nonconvex problem. In addition, the Bernstein-type inequality and large deviation inequality are utilized to convert the probabilistic constraint. Simulation results demonstrate the performance of the proposed design.

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