Related papers: The asymptotic distribution of the remainder in a …

Let $X=\sum_{k=1}^\infty X_k \beta^{-k}$ be the base-$\beta$ expansion of a continuous random variable $X$ on the unit interval where $\beta$ is the positive solution to $\beta^n = 1 + \beta + \cdots + \beta^{n-1}$ for an integer $n\ge 2$…

We provide an asymptotic expansion for $\sum_{k=1}^n \left\{\sqrt{k}\right\}$. In the same spirit, we discuss the case of n-th root and it relation to special values of Riemman's zeta function.

In this work in progress, we study the asymptotic behaviour of the $p$-quantile of the Beta distribution, i.e. the quantity $q$ defined implicitly by $\int_0^q t^{a - 1} (1 - t)^{b - 1} \text{d} t = p B (a, b)$, as a function of the first…

We obtain an explicit formula for the variance of the number of $k$-peaks in a uniformly random permutation. This is then used to obtain an asymptotic formula for the variance of the length of longest $k$-alternating subsequence in random…

We establish the asymptotic expansion in $\beta$ matrix models with a confining, off-critical potential, in the regime where the support of the equilibrium measure is a union of segments. We first address the case where the filling…

Let $\beta>1$. For $x \in [0,\infty)$, we have so-called the $\beta$-expansion of $x$ in base $\beta$ as follows: $$x= \sum_{j \leq k} x_{j}\beta^{j} = x_{k}\beta^{k}+ \cdots + x_{1}\beta+x_{0}+x_{-1}\beta^{-1} + x_{-2}\beta^{-2} + \cdots$$…

A sequence of real numbers $\{x_{n}\}_{n\in \mathbb{N}}$ is said to be $\alpha \beta$-statistically convergent of order $\gamma$ (where $0<\gamma\leq 1$) to a real number $x$ \cite{a} if for every $\delta>0,$ $$\underset{n\rightarrow…

In [Temme N.M., Special functions. An introduction to the classical functions of mathematical physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996, Section 11.3.3.1] a uniform asymptotic expansion for the…

This note displays an interesting phenomenon for percentiles of independent but non-identical random variables. Let $X_1,\cdots,X_n$ be independent random variables obeying non-identical continuous distributions and $X^{(1)}\geq \cdots\geq…

Let \{X_1, X_2, ...\} be a sequence of independent and identically distributed positive random variables of Pareto-type with index \alpha>0 and let \{N(t); t\geq 0\} be a counting process independent of the X_i's. For any fixed t\geq 0,…

Let $X_{nr}$ be the $r$th largest of a random sample of size $n$ from a distribution $F (x) = 1 - \sum_{i = 0}^\infty c_i x^{-\alpha - i \beta}$ for $\alpha > 0$ and $\beta > 0$. An inversion theorem is proved and used to derive an…

Let X be a generalised symmetrised Dirichlet random vector in R^k, and let u_n be thresholds such that P{X> u_n} tends to 0 as n goes infinity. In this paper we derive an exact asymptotic expansion of P{X> u_n} assuming that the associated…

Let $\be\in(1,2)$. Each $x\in I_\be:=[0,\frac{1}{\be-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty a_k\be^{-k}, \] where $a_k\in\{0,1\}$ for all $k$ (a $\be$-expansion of $x$). It was shown in \cite{S} that a.e. $x\in I_\be$…

An infinite convergent sum of independent and identically distributed random variables discounted by a multiplicative random walk is called perpetuity, because of a possible actuarial application. We give three disjoint groups of sufficient…

This paper studies the asymptotic properties of weighted sums of the form $Z_n=\sum_{i=1}^n a_i X_i$, in which $X_1, X_2, \ldots, X_n$ are i.i.d.~random variables and $a_1, a_2, \ldots, a_n$ correspond to either eigenvalues or singular…

The mean absolute deviation about the mean is an alternative to the standard deviation for measuring dispersion in a sample or in a population. For stationary, ergodic time series with a finite first moment, an asymptotic expansion for the…

This paper develops a general framework for analyzing asymptotics of $V$-statistics. Previous literature on limiting distribution mainly focuses on the cases when $n \to \infty$ with fixed kernel size $k$. Under some regularity conditions,…

We consider base-$\beta$ expansions of Parry's type, where $a_0 \geq a_1 \geq 1$ are integers and $a_0<\beta <a_0+1$ is the positive solution to $\beta^2 = a_0\beta + a_1$ (the golden ratio corresponds to $a_0=a_1=1$). The map $x\mapsto…

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…

Asymptotic expansions of series $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma e^{-(k+a)^\alpha x}$ and $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma / (x(k+a)^\alpha+1)^\mu}$ in powers of $x$ as $x\to+0$ are found, where $\epsilon=1$ or…