Related papers: Characteristic function of the positive part of a …
A unifying and generalizing approach to representations of the positive-part and absolute moments $\mathsf{E} X_+^p$ and $\mathsf{E}|X|^p$ of a random variable $X$ for real $p$ in terms of the characteristic function (c.f.) of $X$, as well…
Spitzer's identity describes the position of a reflected random walk over time in terms of a bivariate transform. Among its many applications in probability theory are congestion levels in queues and random walkers in physics. We present a…
We define a characteristic function for probability measures on the signatures of geometric rough paths. We determine sufficient conditions under which a random variable is uniquely determined by its expected signature, thus partially…
In this paper, the classical problem of the probabilistic characterization of a random variable is re-examined. A random variable is usually described by the probability density function (PDF) or by its Fourier transform, namely the…
I develop a function that, for any integer $n \geq 2$, takes a value of 1 if $n$ is prime, 0 if $n$ is composite. I also discuss two applications: First, the characteristic function provides a new expression for the prime counting function.…
The purpose of the present paper is to give unified expressions to the characteristic functions of all elliptical and related distributions. Those distributions including the multivariate elliptical symmetric distributions and some…
This paper introduces max-characteristic functions (max-CFs), which are an offspring of multivariate extreme-value theory. A max-CF characterizes the distribution of a random vector in R^d , whose components are nonnegative and have finite…
Application of the exact statistical inference frequently leads to a non-standard probability distributions of the considered estimators or test statistics. The exact distributions of many estimators and test statistics can be specified by…
We derive necessary and sufficient conditions for a continuous bounded function $f: R\to C$ to be a characteristic function of a probability measure. The Cauchy transform $K_f$ of $f$ is used as analytic continuation of $f$ to the upper and…
The area related to M. Liv\v{s}ic's characteristic matrix functions is too vast to be discussed in one paper and we selected for this article the problems which are close to our scientific interests. We discuss M.Liv\v{s}ic's results…
We consider the distribution of the sum and the maximum of a collection of independent exponentially distributed random variables. The focus is laid on the explicit form of the density functions (pdf) of non-i.i.d. sequences. Those are…
A function of the empirical characteristic function,exists for the stable distribution, which leads to a linear regression and can be used to estimate the parameters. Two approaches are often used, one to find optimal values of t, but these…
There are given characterizations of the exponential distribution by the properties of the independence of linear forms with random coefficients. Related results based on the constancy of regression of one statistic on a linear form are…
We study the continuity properties of trajectories for some random series of functions $\sum a\_kf(\alpha X\_k(\omega))$ where $a\_k$ is a complex sequence, $X\_k$ a sequence of real independent random variables, $f$ is a real valued…
Let $CH(R)$ denote the family of characteristic functions of probability measures (distributions) on the real line $R$. We study the following question: given an integer $n>1$, do there exist two different $f, g\in CH(R)$ such that $…
We consider the role of finite size effects on the value of the effective Hurst exponent H. This problem is motivated by the properties of the high frequency daily stock-prices. For a finite size random walk we derive some exact results…
Sufficient conditions for comparing the convolutions of heterogeneous gamma random variables in terms of the usual stochastic order are established. Such comparisons are characterized by the Schur convexity properties of the cumulative…
Using the spectral resolution of the multiplication operator on the Schwartz class of $L^2(\mathbb{R},\mathbb{C})$, we compute the characteristic function of the cube of a Gaussian random variable.
For any strictly positive martingale $S = \exp(X)$ for which $X$ has a characteristic function, we provide an expansion for the implied volatility. This expansion is explicit in the sense that it involves no integrals, but only polynomials…
We prove that certain quotients of entire functions are characteristic functions. Under some conditions, the probability measure corresponding to a characteristic function of that type has a density which can be expressed as a generalized…