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

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