Zero-free neighborhoods around the unit circle for Kac polynomials

In this paper, we study how the roots of the Kac polynomials $$W_n(z) = \sum _{k=0}^{n-1} \xi _k z^k$$ W n ( z ) = ∑ k = 0 n - 1 ξ k z k concentrate around the unit circle when the coefficients of $$W_n$$ W n are independent and identically distributed nondegenerate real random variables. It is well known that the roots of a Kac polynomial concentrate around the unit circle as $$n\rightarrow \infty $$ n → ∞ if and only if $${\mathbb {E}}[\log ( 1+ |\xi _0|)]<\infty $$ E [ log ( 1 + | ξ 0 | ) ] < ∞ . Under the condition $${\mathbb {E}}[\xi ^2_0]<\infty $$ E [ ξ 0 2 ] < ∞ , we show that there exists an annulus of width $${\text {O}}(n^{-2}(\log n)^{-3})$$ O ( n - 2 ( log n ) - 3 ) around the unit circle which is free of roots with probability $$1-{\text {O}}({(\log n)^{-{1}/{2}}})$$ 1 - O ( ( log n ) - 1 / 2 ) . The proof relies on small ball probability inequalities and the least common denominator used in [17].

NEW EXPONENTIAL PROBABILITY INEQUALITY AND COMPLETE CONVERGENCE FOR F-LNQD RANDOM VARIABLES SEQUENCE WITH APPLICATION TO AR(1)MODEL GENERATED BY F-LNQD ERRORS

The exponential probability inequalities have been important tools in probability and statistics. In this paper, we prove a new tail probability inequality for the distri-butions of sums of conditionally linearly negative quadrant dependent (F-LNQD , in short) random variables, and obtain a result dealing with conditionally complete con-vergence of first-order autoregressive processes with identically distributed (F-LNQD) innovations.

Relaxation schemes for the joint linear chance constraint based on probability inequalities

<p style='text-indent:20px;'>This paper is concerned with the joint chance constraint for a system of linear inequalities. We discuss computationally tractble relaxations of this constraint based on various probability inequalities, including Chebyshev inequality, Petrov exponential inequalities, and others. Under the linear decision rule and additional assumptions about first and second order moments of the random vector, we establish several upper bounds for a single chance constraint. This approach is then extended to handle the joint linear constraint. It is shown that the relaxed constraints are second-order cone representable. Numerical test results are presented and the problem of how to choose proper probability inequalities is discussed.</p>