Online Selection of Alternating Subsequences from a Random Sample

We consider sequential selection of an alternating subsequence from a sequence of independent, identically distributed, continuous random variables, and we determine the exact asymptotic behavior of an optimal sequentially selected subsequence. Moreover, we find (in a sense we make precise) that a person who is constrained to make sequential selections does only about 12 percent worse than a person who can make selections with full knowledge of the random sequence

Stationary Gaussian Markov Processes as Limits of Stationary Autoregressive Time Series

We consider the class, ℂp, of all zero mean stationary Gaussian processes, {Yt : t ∈ (—∞, ∞)} with p derivatives, for which the vector valued process {(Yt(0) ,...,Yt(p)) : t ≥ 0} is a p + 1-vector Markov process, where Yt(0) = Y(t). We provide a rigorous description and treatment of these stationary Gaussian processes as limits of stationary AR(p) time series

Some linear models are necessarily parametric

We prove the surprising result that rather general assumptions on the set of admissible signals [xi](t) observed in the presence of additive noise [var epsilon](t) on a closed interval [a, b], imply that the set is finite dimensional, i.e., [xi](t) = [theta]1[xi]1(t) + ... + [theta]m[xi]m(t) for some integer m [greater-or-equal, slanted] 1 and fixed functions [xi]1(t),..., [xi]m(t). Thus, estimating the signal [xi](t) from observations of x(t) = [xi](t) + [var epsilon](t) reduces to estimating the parameters [theta]1,...,[theta]m. This gives a strong argument in favor of parametric linear models.Linear models L2-theory

Tail hypotheses in the signal plus noise model

Based on an infinite sequence of observations Yn=Xn+an, n=1,2,... with independent identically distributed random variables X1,X2,... with known distribution representing noise and constants a1,a2,... representing signal, it is impossible to distinguish with zero error probabilities the class of signals with infinite power [short parallel]a[short parallel]2=a12+a22+... from the noise (a=0).Tail hypothesis Zero error probability

A note on the conditional distribution of X when X - y is given

It is known that the expected conditional variance of the random variable X when X - y is given is strictly positive for at least one value of y if the distribution of X has an absolutely continuous component or has at least two atoms. From this fact, it might be conjectured that this would remain true for any non-degenerate random variable X. However, this is not the case. In this note, we construct a counterexample and show that for every fixed y, with probability one, the conditional distribution of a random variable X with a singularly continuous distribution when X - y is given may be degenerate.Absolutely continuous distributions atoms conditional distributions conditional variances non-degenerate singularly continuous distributions

A mathematical theory of network interference and its applications

n this paper, we introduce a mathematical framework for the characterization of network interference in wireless systems. We consider a network in which the interferers are scattered according to a spatial Poisson process and are operating asynchronously in a wireless environment subject to path loss, shadowing, and multipath fading. We start by determining the statistical distribution of the aggregate network interference. We then investigate four applications of the proposed model: 1) interference in cognitive radio networks; 2) interference in wireless packet networks; 3) spectrum of the aggregate radio-frequency emission of wireless networks; and 4) coexistence between ultrawideband and narrowband systems. Our framework accounts for all the essential physical parameters that affect network interference, such as the wireless propagation effects, the transmission technology, the spatial density of interferers, and the transmitted power of the interferers