Characterization of Probability Law by Absolute Moments of Its Partial Sums

If Sn = X1 + . . . + Xn, where Xi are independent and identically distributed (i.i.d.) standard normal, then E|Sn| ≡ √2n/π, n ≧ 0. We show that no other symmetric law has exactly these “moments”; the general case remains (stubbornly) open. If X is standard two-sided exponential, then E|Sn| = 2n2-2n(2n/n). We show the latter moments are obtained exactly for all n also for Xi ~ B(2;0.5), the sum of two standard (± 1-valued) Bernoulli’s as well as for many other laws including unsymmetrical ones: Xi ~ G - 1, where G is geometric with mean 1, is one example. Our interest in this delicate nonlinear inverse problem (which was initiated by Klebanov, cf. [12]) of inverting the moments to recover the law was also drawn by the fact that it gives a way to study positive definite functions through the formula E|Sn| = (2/π) ∫0∞Re(1 - φn(1 / u))du, n ≧ 0, expressing E|Sn| in terms of the moments of φ, where φ is the characteristic function of X, φ(u) = Eexp(iuX). We show that if for some b \u3e 0, ψb (u) = φ (btan (u / b)) is a positive definite function then the distributions corresponding to φ and ψb have the same E|Sn| moments for all n. We show that if X is Bernoulli with zero mean and values ±1 then the moments characterize the distribution uniquely even among nonsymmetric laws. In general however we expect that the moments do not characterize the law, and this may well be the only nontrivial case of uniqueness. We extend some of our results to the case of pth moments, p different from an even integer

Executive Summary: Young people, education, employment and ESOL

The report, ‘Young people, education, employment and ESOL’ reviewed 47 studies1 to examine how current ESOL provision2 meets the needs of young people aged 16-25 years who use English as an Additional Language (EAL)3, and who need time and support to develop their English language skills in order to progress in education, training and employment

An historical survey of the development of the vibraphone as an alternative accompanying instrument in jazz

Includes bibliographical references (p. 150-159).Based on commercially available recordings of performances where the vibraphone alone performs the role of harmonic accompanist in the jazz ensemble, this study looks at the historical development of the use of the vibraphone as an alternative harmonic accompanying instrument. Descriptive analyses of key recordings made by leading and influential vibraphonists are given with regard to the use of the vibraphone in the role of an harmonic accompanying instrument

Beam-induced Background Simulations for the CMS Experiment at the LHC

Beam-induced background comes from interactions of the beam and beam halo particles with either the residual gas in the vacuum chamber of accelerator or the collimators that define the beam aperture. Beam-induced processes can potentially be a significant source of background for physics analyses at the LHC. This contribution describes the simulation software environment used for this part of the CMS experiment activity and recent beam-induced background simulation results for the Phase-2 CMS operation design

Analysis of airplane boarding via space-time geometry and random matrix theory

We show that airplane boarding can be asymptotically modeled by 2-dimensional Lorentzian geometry. Boarding time is given by the maximal proper time among curves in the model. Discrepancies between the model and simulation results are closely related to random matrix theory. We then show how such models can be used to explain why some commonly practiced airline boarding policies are ineffective and even detrimental.Comment: 4 page

Limit Distributions of Self-Normalized Sums

If Xi are i.i.d. and have zero mean and arbitrary finite variance the limiting probability distribution of Sn(2) =(∑ni=1 Xi)/(∑nj=1Xj2)1/2 as n→∞ has density f(t) = (2π)−1/2 exp(−t2/2) by the central limit theorem and the law of large numbers. If the tails of Xi are sufficiently smooth and satisfy P(Xi \u3e t) ∼ rt−α and P(Xi \u3c −t) ∼ lt−α as t→∞, where 0 \u3c α \u3c 2, r \u3e 0, l \u3e 0, Sn(2) still has a limiting distribution F even though Xi has infinite variance. The density f of F depends on α as well as on r/l. We also study the limiting distribution of the more general Sn(p) = (∑ni=1Xi)/(∑nj=1 |Xj|p)1/p where Xi are i.i.d. and in the domain of a stable law G with tails as above. In the cases p = 2 (see (4.21)) and p = 1 (see (3.7)) we obtain exact, computable formulas for f(t) = f(t,α,r/l), and give graphs of f for a number of values of α and r/l. For p = 2, we find that f is always symmetric about zero on (−1,1), even though f is symmetric on (−∞,∞) only when r = l

Run Generation Revisited: What Goes Up May or May Not Come Down

In this paper, we revisit the classic problem of run generation. Run generation is the first phase of external-memory sorting, where the objective is to scan through the data, reorder elements using a small buffer of size M , and output runs (contiguously sorted chunks of elements) that are as long as possible. We develop algorithms for minimizing the total number of runs (or equivalently, maximizing the average run length) when the runs are allowed to be sorted or reverse sorted. We study the problem in the online setting, both with and without resource augmentation, and in the offline setting. (1) We analyze alternating-up-down replacement selection (runs alternate between sorted and reverse sorted), which was studied by Knuth as far back as 1963. We show that this simple policy is asymptotically optimal. Specifically, we show that alternating-up-down replacement selection is 2-competitive and no deterministic online algorithm can perform better. (2) We give online algorithms having smaller competitive ratios with resource augmentation. Specifically, we exhibit a deterministic algorithm that, when given a buffer of size 4M , is able to match or beat any optimal algorithm having a buffer of size M . Furthermore, we present a randomized online algorithm which is 7/4-competitive when given a buffer twice that of the optimal. (3) We demonstrate that performance can also be improved with a small amount of foresight. We give an algorithm, which is 3/2-competitive, with foreknowledge of the next 3M elements of the input stream. For the extreme case where all future elements are known, we design a PTAS for computing the optimal strategy a run generation algorithm must follow. (4) Finally, we present algorithms tailored for nearly sorted inputs which are guaranteed to have optimal solutions with sufficiently long runs

Linear regression for numeric symbolic variables: an ordinary least squares approach based on Wasserstein Distance

In this paper we present a linear regression model for modal symbolic data. The observed variables are histogram variables according to the definition given in the framework of Symbolic Data Analysis and the parameters of the model are estimated using the classic Least Squares method. An appropriate metric is introduced in order to measure the error between the observed and the predicted distributions. In particular, the Wasserstein distance is proposed. Some properties of such metric are exploited to predict the response variable as direct linear combination of other independent histogram variables. Measures of goodness of fit are discussed. An application on real data corroborates the proposed method

Adaptive density estimation for stationary processes

We propose an algorithm to estimate the common density $s$ of a stationary process $X_1,...,X_n$. We suppose that the process is either $\beta$ or $\tau$-mixing. We provide a model selection procedure based on a generalization of Mallows' $C_p$ and we prove oracle inequalities for the selected estimator under a few prior assumptions on the collection of models and on the mixing coefficients. We prove that our estimator is adaptive over a class of Besov spaces, namely, we prove that it achieves the same rates of convergence as in the i.i.d framework