Abstract:
We show that $L_1$-norm of linear combinations (with scalar or vector coefficients) of products of i.i.d. nonnegative mean one random variables is comparable to $l_1$-norm of coefficients.

Abstract:
We show that almost all the zeros of any finite linear combination of independent characteristic polynomials of random unitary matrices lie on the unit circle. This result is the random matrix analogue of an earlier result by Bombieri and Hejhal on the distribution of zeros of linear combinations of $L$-functions, thus providing further evidence for the conjectured links between the value distribution of the characteristic polynomial of random unitary matrices and the value distribution of $L$-functions on the critical line.

Abstract:
We give the distribution functions, the expected values, and the moments of linear combinations of lattice polynomials from the uniform distribution. Linear combinations of lattice polynomials, which include weighted sums, linear combinations of order statistics, and lattice polynomials, are actually those continuous functions that reduce to linear functions on each simplex of the standard triangulation of the unit cube. They are mainly used in aggregation theory, combinatorial optimization, and game theory, where they are known as discrete Choquet integrals and Lovasz extensions.

Abstract:
We propose new optimal estimators for the Lipschitz frontier of a set of points. They are defined as kernel estimators being sufficiently regular, covering all the points and whose associated support is of smallest surface. The estimators are written as linear combinations of kernel functions applied to the points of the sample. The coefficients of the linear combination are then computed by solving related linear programming problem. The L_1 error between the estimated and the true frontier function with a known Lipschitz constant is shown to be almost surely converging to zero, and the rate of convergence is proved to be optimal.

Abstract:
We obtain upper bounds on the number of sign changes of linear combinations of derivatives and convolutions of Polya frequency functions using the variation diminishing properties of totally positive functions. These constitute extensions of earlier results of Karlin and Proschan.

Abstract:
The zero distribution of sections of Mittag-Leffler functions of order >1 was studied in 1983 by A. Edrei, E.B. Saff and R.S. Varga. In the present paper, we study the zero distribution of linear combinations of sections and tails of Mittag-Leffler functions of order >1.

Abstract:
We study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with i.i.d. Gaussian coefficients, have only $(2/\pi + o(1))\log{n}$ expected real zeros in terms of the degree $n$. On the other hand, if the basis is given by Legendre (or more generally by Jacobi) polynomials, then random linear combinations have $n/\sqrt{3} + o(n)$ expected real zeros. We prove that the latter asymptotic relation holds universally for a large class of random orthogonal polynomials on the real line, and also give more general local results on the expected number of real zeros.

Abstract:
Linear combinations of independent random variables have been extensively studied in the literature. However, most of the work is based on some specific distribution assumptions. In this paper, a companion of (J. Appl. Probab. 48 (2011) 1179-1188), we unify the study of linear combinations of independent nonnegative random variables under the general setup by using some monotone transforms. The results are further generalized to the case of independent but not necessarily identically distributed nonnegative random variables. The main results complement and generalize the results in the literature including (In Studies in Econometrics, Time Series, and Multivariate Statistics (1983) 465-489 Academic Press; Sankhy\={a} Ser. A 60 (1998) 171-175; Sankhy\={a} Ser. A 63 (2001) 128-132; J. Statist. Plann. Inference 92 (2001) 1-5; Bernoulli 17 (2011) 1044-1053).

Abstract:
Multi-target regression is concerned with the simultaneous prediction of multiple continuous target variables based on the same set of input variables. It arises in several interesting industrial and environmental application domains, such as ecological modelling and energy forecasting. This paper presents an ensemble method for multi-target regression that constructs new target variables via random linear combinations of existing targets. We discuss the connection of our approach with multi-label classification algorithms, in particular RA$k$EL, which originally inspired this work, and a family of recent multi-label classification algorithms that involve output coding. Experimental results on 12 multi-target datasets show that it performs significantly better than a strong baseline that learns a single model for each target using gradient boosting and compares favourably to multi-objective random forest approach, which is a state-of-the-art approach. The experiments further show that our approach improves more when stronger unconditional dependencies exist among the targets.

Abstract:
We consider a transmitter broadcasting random linear combinations (over a field of size $d$) formed from a block of $c$ packets to a collection of $n$ receivers, where the channels between the transmitter and each receiver are independent erasure channels with reception probabilities $\mathbf{q} = (q_1,\ldots,q_n)$. We establish several properties of the random delay until all $n$ receivers have recovered all $c$ packets, denoted $Y_{n:n}^{(c)}$. First, we provide upper and lower bounds, exact expressions, and a recurrence for the moments of $Y_{n:n}^{(c)}$. Second, we study the delay per packet $Y_{n:n}^{(c)}/c$ as a function of $c$, including the asymptotic delay (as $c \to \infty$), and monotonicity properties of the delay per packet (in $c$). Third, we employ extreme value theory to investigate $Y_{n:n}^{(c)}$ as a function of $n$ (as $n \to \infty$). Several results are new, some results are extensions of existing results, and some results are proofs of known results using new (probabilistic) proof techniques.