It is, perhaps, surprising that the location of the unique supremum of a stationary process on an interval can fail to be uniformly distributed over that interval. We show that this distribution is absolutely continuous in the interior of the interval and describe very specific conditions the density has to satisfy. We establish universal upper bounds on the density and demonstrate their optima...

Since the 1930’s many authors have studied the distribution νλ of the random series Yλ = ∑±λn where the signs are chosen independently with probability (1/2, 1/2) and 0 < λ < 1. Solomyak recently proved that for almost every λ ∈ [ 1 2 , 1], the distribution νλ is absolutely continuous with respect to Lebesgue measure. In this paper we prove that νλ is even equivalent to Lebesgue measure for alm...

A general construction for ?nite absolutely continuous invariant measure will be presented. It will be shown that the local bounded distortion of the Radon-Nykodym derivatives of f n () will imply the existence of a ?nite invariant measure for the map f which is absolutely continuous with respect to , a measure on the phase space describing the sets of measure zero. Furthermore we will discuss ...

We assume without loss of generality that E{Xn} = 0. Let rs = i£{XnXn+s}. Then r8 = / I T edF(h), where F(X) is the spectral distribution function of the process. In §3 the spectral distribution function of any process of the form (2.2) is shown to be absolutely continuous. Finally it is shown in §4 that under some additional assumptions on the moment structure of the process the central limit ...

Let {Sn, n ≥ 0} with S0 = 0 be a random walk with negative drift and let τx = min {k > 0 : Sk < −x} , x ≥ 0. Assuming that the distribution of the i.i.d. increments of the random walk is absolutely continuous with subexponential density we describe the asymptotic behavior, as n→∞, of the probabilities P (τx = n) and P(Sn ∈ [y, y+ ∆), τx > n) for fixed x and various ranges of y. The case of latt...

For estimating the entropy of an absolutely continuous multivariate distribution, we propose nonparametric estimators based on the Euclidean distances between the n sample points and their kn-nearest neighbors, where {kn : n = 1, 2, . . .} is a sequence of positive integers varying with n. The proposed estimators are shown to be asymptotically unbiased and consistent.

We give the distribution of Mn, the maximum of a sequence of n observations from a moving average of order 1. Solutions are first given in terms of repeated integrals and then for the case where the underlying independent random variables have an absolutely continuous density. When the correlation is positive , P (Mn < x) = ∞

Empirical processes for stationary, causal sequences are considered. We establish empirical central limit theorems for classes of indicators of left half lines, absolutely continuous functions and piecewise differentiable functions. Sample path properties of empirical distribution functions are also discussed. The results are applied to linear processes and Markov chains.

We give the distribution of Mn, the maximum of a sequence of n observations from a moving average of order 1. Solutions are first given in terms of repeated integrals and then for the case where the underlying independent random variables have an absolutely continuous density. When the correlation is positive , P (Mn ≤ x) = ∞

We characterize the subexponential densities on $(0,\infty)$ for compound Poisson distributions $[0,\infty)$ with absolutely continuous L\'evy measures. As a corollary, we show that class of all probability density functions $\mathbb R_+$ is closed under generalized convolution roots sums. Moreover, give an application to distribution supremum random walk.