Related papers: Distribution of the sum-of-digits function of rand…
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…
The distribution function of a random distance in three dimensions is given and some new three-dimensional d2-tests of randomness are suggested. We show that our test statistics are not correlated with the usual test statistics and are…
The distribution of the sum of r-th power of standard normal random variables is a generalization of the chi-squared distribution. In this paper, we represent the probability density function of the random variable by an one-dimensional…
Sums of of 1-dependent integer-valued random variables are approximated by compound Poisson, negative binomial and Binomial distributions and signed compound Poisson measures. Estimates are obtained for total variation and local metrics.…
In this paper, the statistical properties of Newton s method algorithm output in a specific case have been studied. The relative frequency density of this sample converges to a well-defined function, prompting us to explore its…
Stein's method is used to approximate sums of discrete and locally dependent random variables by a centered and symmetric Binomial distribution. Under appropriate smoothness properties of the summands, the same order of accuracy as in the…
Let $\{X_{i}, i\in J\}$ be a family of locally dependent non-negative integer-valued random variables with finite expectations and variances. We consider the sum $W=\sum_{i\in J}X_i$ and use Stein's method to establish general upper error…
In this paper, we consider the sums of non-negative integer valued $m$-dependent random variables, and its approximation to the power series distribution. We first discuss some relevant results for power series distribution such as Stein…
We consider the continued fraction digits as random variables measured with respect to Lebesgue measure. The logarithmically scaled and normalized fluctuation process of the digit sums converges strongly distributional to a random variable…
This article compares the distributions of integer-valued random variables and Poisson random variables. It considers the total variation and the Wasserstein distance and provides, in particular, explicit bounds on the pointwise difference…
The distribution function of the sum of i.i.d. random variables of the special form is considered. Such sum describes messages posterior probabilities for random coding in binary symmetric channel. Close non-asymptotic lower and upper…
We consider very general "random integers" and (attempt to) prove that many multiplicative and additive functions of such integers have limiting distributions. These integers include, for instance, the curvatures of Apollonian circle…
We prove a general theorem to bound the total variation distance between the distribution of an integer valued random variable of interest and an appropriate discretized normal distribution. We apply the theorem to 2-runs in a sequence of…
In this paper, we analyze the set of all possible aggregate distributions of the sum of standard uniform random variables, a simply stated yet challenging problem in the literature of distributions with given margins. Our main results are…
In this paper, we introduce a new technique to study the distribution in residue classes of sets of integers with digit and sum-of-digits restrictions. From our main theorem, we derive a necessary and sufficient condition for integers with…
Computing the similarity between two probability distributions is a recurring theme across control. We introduce a unified family of distances between the probability distributions of two random variables that is based on the discrepancy…
We consider random multiplicative functions taking the values $\pm 1$. Using Stein's method for normal approximation, we prove a central limit theorem for the sum of such multiplicative functions in appropriate short intervals.
We present a de Bruijn type approximation for quantifying the content of m smooth numbers, derived from samples obtained through a probability measure over the set of integers less than or equal to n, with point mass function at k inversely…
We derive Stein approximation bounds for functionals of uniform random variables, using chaos expansions and the Clark-Ocone representation formula combined with derivation and finite difference operators. This approach covers sums and…
We propose a new general version of Stein's method for univariate distributions. In particular we propose a canonical definition of the Stein operator of a probability distribution {which is based on a linear difference or differential-type…