Related papers: On the Distribution Function of the Complexity of …
The notion of probability density for a random function is not as straightforward as in finite-dimensional cases. While a probability density function generally does not exist for functional data, we show that it is possible to develop the…
We consider the problem of reconstructing a function given its values on a set of points with finite density. We prove that with probability one, the values of an almost periodic function on a random array of points (with finite density)…
We present estimators for entropy and other functions of a discrete probability distribution when the data is a finite sample drawn from that probability distribution. In particular, for the case when the probability distribution is a joint…
The wrapped normal distribution arises when a the density of a one-dimensional normal distribution is wrapped around the circle infinitely many times. At first look, evaluation of its probability density function appears tedious as an…
Determining the randomized (or distributional) communication complexity of disjointness is a central problem in communication complexity, having roots in the foundational work of Babai, Frankl, and Simon in the 1980s and culminating in the…
The first section starts with the basic definitions following mainly the notations of the book written by E. Kushilevitz and N. Nisan. At the end of the first section I examine tree-balancing. In the second section I summarize the…
We derive a simple and precise approximation to probability density functions in sampling distributions based on the Fourier cosine series. After clarifying the required conditions, we illustrate the approximation on two examples: the…
Exact formulas are derived for the probability density functions of the sum and difference of two independent non-central gamma distributed random variables, with both series and integral representations of the density presented. These…
We study the complexity of sampling, rounding, and integrating arbitrary logconcave functions. Our new approach provides the first complexity improvements in nearly two decades for general logconcave functions for all three problems, and…
As a part of our works on effective properties of probability distributions, we deal with the corresponding characteristic functions. A sequence of probability distributions is computable if and only if the corresponding sequence of…
In the probability theory \emph{selfdecomposable, or class $L_0$ distributions} play an important role as they are limiting distributions of normalized partial sums of sequences of independent, not necessarily identically distributed,…
We determine the distributions of lengths of runs in random sequences of elements from a totally ordered set (total order) or partially ordered set (partial order). In particular, we produce novel formulae for the expected value, variance,…
We extend previously proposed measures of complexity, emergence, and self-organization to continuous distributions using differential entropy. This allows us to calculate the complexity of phenomena for which distributions are known. We…
V.I. Arnold has recently defined the complexity of a sequence of $n$ zeros and ones with the help of the operator of finite differences. In this paper we describe the results obtained for almost most complicated sequences of elements of a…
A measure for the complexity of a differentiable function f(x) on an interval is introduced. It is based on approximations of the function by piecewise constant functions. The measure takes into account the quality of the approximation and…
The property of perfectness plays an important role in the theory of Bayesian networks. First, the existence of perfect distributions for arbitrary sets of variables and directed acyclic graphs implies that various methods for reading…
Dispersion is a fundamental concept in statistics, yet standard approaches - especially via stochastic orders - face limitations in the discrete setting. In particular, the classical dispersive order, well-established for continuous…
We introduce a complexity measure for symbolic sequences. Starting from a segmentation procedure of the sequence, we define its complexity as the entropy of the distribution of lengths of the domains of relatively uniform composition in…
Probability distribution theory helps in studying the impact of various dimensions in life while the Mittag-Leffler function and bicomplex are used in electromagnetism, quantum mechanics, and signal theory. Considering the importance of…
We present the construction of a theory of distributions (generalized functions) with a ``thick submanifold'', that is, a new theory of thick distributions on $\mathbb{R}^n$ whose domain contains a smooth submanifold on which the test…