Classical change point analysis aims at (1) detecting abrupt changes in the mean of a possibly nonstationary time series and (2) identifying regions where exhibits piecewise constant behavior. In many applications however, it is more reasonable to assume that gradually smooth way. Those gradual may either be nonrelevant (i.e., small), or relevant for specific problem hand, present paper present...

In statistical physics textbooks, one often encounters systems which consist of macroscopic numbers of identical small parts which do not interact with each other. This is not too unrealistic since there are many physical systems (such as certain spin systems, including nuclear spin systems) which are well approximated by such non-interacting models in some ranges of temperature and time. The q...

In this paper, we explore two conjectures about Rademacher sequences. Let (εi) be a sequence, i.e., sequence of independent {−1, 1}-valued symmetric random variables. Set Sn = a1ε1 + … anεn for (a1, …, an) ∈ ℝn. The rst conjecture says that P (|Sn| ≤ ‖a‖) ≥ \({\mathbb{R}^n}\) all ℝn and n \(\mathbb{N}\). second ℕ. Regarding the first conjecture, present several new equivalent formulations. Thes...

and Applied Analysis 3 To investigate the approximate controllability of system (9), we assume the following conditions. (H4) The functionf:[0, b]×X → X satisfies the following: (a) f(t, ⋅) : X → X is continuous for each t ∈ [0, b] and for each x ∈ X, f(⋅, x) : [0, b] → X is strongly measurable; (b) there is a positive integrable function n ∈ L 1 ([0, b], [0, +∞)) and a continuous nondecreasing...

Let {εi : −∞ < i < ∞} be a strictly stationary sequence of linearly positive quadrant dependent random variables and P∞ i=−∞ |ai| < ∞. In this paper, we prove the precise asymptotics in the law of iterated logarithm for the moment convergence of moving-average process of the form Xk = P∞ i=−∞ ai+kεi, k ≥ 1.

In this model it is supposed that shocks contributing additively to the amount of damage occur at random points of time. Common assumptions are that the sequence of shock amounts and the process N (t), say counting their number are stochastically independent and that the shock amounts ε1 , ε2...(Where εi is the increment of the damage caused by the i-th shock) are independent and identically di...

Let k and n be positive integers, and let d(n;k) be the maximum density in f0;1;2 : : : ;kn 1g of a set containing no arithmetic progression of k terms with first term a = ∑aik and common difference d = ∑εik, where 0 ai k 1, εi = 0 or 1, and εi = 1 ) ai = 0. Setting βk = limn!∞ d(n;k), we show that limk!∞ βk is either 0 or 1. Throughout, we shall use the notation [a;b) = fa;a+ 1;a+ 2; : : : ;b ...

We give conditions for consistency and asymptotic normality of a FGLS estimator under conditions similar to the ones in [White, 1980a]. This is motivated by the estimator suggested in [Wansbeek, 2004]. In particular, we show that this estimator is not always consistent (even when the OLS estimator is) and is generally inefficient. 1. Motivation and Problem Formulation Consider the linear regres...

Given a set P of points in the plane, a set of points Q is a weak ε-net with respect to a family of sets S (e.g., rectangles, disks, or convex sets) if every set of S containing ε|P | points contains a point of Q. In this paper, we determine bounds on εi , the Preprint submitted to Elsevier Science 10 February 2008 smallest epsilon that can be guaranteed for any P when |Q| = i, for small values...

Let vi be vectors in Rd and {εi} independent Rademacher random variables. Then the Littlewood–Offord problem entails finding best upper bound for supx∈RdP(∑εivi=x). Generalizing uniform bounds of Littlewood–Offord, Erdős Kleitman, a recent result Dzindzalieta Juškevičius provides non-uniform that is optimal its dependence on ‖x‖2. In this short note, we provide simple alternative proof their re...