Positive Real Line Research Articles

Modeling the cryptocurrency return distribution via Laplace scale mixtures

The search of appropriate models for describing the currency return distribution is one of the main interests not only in finance, but also in the more recent trans-disciplinary econophysics research field. Such a search is recently focusing on cryptocurrencies, due to their proliferation. Although there is no agreement of what theoretical models are the most appropriate, there is a general consensus that the cryptocurrency return distribution is highly-peaked and heavy-tailed, with a large excess kurtosis and the tail distribution decreases as the inverse cubic power-law. With these requirements, the Laplace distribution can be considered as a valid candidate model. However, there are two limitations to be taken into account: 1) the Laplace tail distribution does not decrease as the inverse cubic power-law and 2) the excess kurtosis is fixed at 3. To make the tailedness of the Laplace distribution more flexible, still maintaining its peculiar symmetric shape, we introduce the Laplace scale mixture (LSM) family of distributions. Each member of the family is obtained by dividing the scale parameter of the conditional Laplace distribution by a convenient mixing random variable taking values on all or part of the positive real line and whose distribution depends on a parameter vector θ governing the tail behavior of the resulting LSM. For illustrative purposes, we consider different mixing distributions; they give rise to LSMs having a closed-form probability density function where the Laplace distribution is obtained as a special case under a convenient choice of θ. We describe an EM algorithm to obtain maximum likelihood estimates of the parameters for all the considered LSMs. Interestingly, we show how the influence of observations associated with large scaled absolute distances is reduced (downweighted), with respect to the nested Laplace distribution, in the estimation phase. We fit these models to the returns of 4 cryptocurrencies, considering several classical symmetric distributions for comparison. The analysis shows how the proposed models represent a valid alternative to the considered competitors in terms of AIC, BIC and likelihood-ratio tests, but also in reproducing the larger empirical excess kurtosis and in resembling the empirical inverse cubic power-law decrease of the tail distribution.

Multiple orthogonal polynomials associated with confluent hypergeometric functions

We introduce and analyse a new family of multiple orthogonal polynomials of hypergeometric type with respect to two measures supported on the positive real line which can be described in terms of confluent hypergeometric functions of the second kind. These two measures form a Nikishin system. Our focus is on the multiple orthogonal polynomials for indices on the step line. The sequences of the derivatives of both type I and type II polynomials with respect to these indices are again multiple orthogonal and they correspond to the original sequences with shifted parameters. For the type I polynomials, we provide a Rodrigues-type formula. We characterise the type II polynomials on the step line, also known as d-orthogonal polynomials (where d is the number of measures involved so that here d=2), via their explicit expression as a terminating generalised hypergeometric series, as solutions to a third-order differential equation and via their recurrence relation. The latter involves recurrence coefficients which are unbounded and asymptotically periodic. Based on this information we deduce the asymptotic behaviour of the largest zeros of the type II polynomials. We also discuss limiting relations between these polynomials and the multiple orthogonal polynomials with respect to the modified Bessel weights. Particular choices on the parameters for the 2-orthogonal polynomials under discussion correspond to the cubic components of the already known threefold symmetric Hahn-classical multiple orthogonal polynomials on star-like sets.

ASYMPTOTIC ANALYSIS OF SKOLEM’S EXPONENTIAL FUNCTIONS

AbstractSkolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant $1$ , the identity function ${\mathbf {x}}$ , and such that whenever f and g are in the set, $f+g,fg$ and $f^g$ are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below $2^{2^{\mathbf {x}}}$ . Here we prove that the set of asymptotic classes within any Archimedean class of Skolem functions has order type $\omega $ . As a consequence we obtain, for each positive integer n, an upper bound for the fragment below $2^{n^{\mathbf {x}}}$ . We deduce an epsilon-zero upper bound for the fragment below $2^{{\mathbf {x}}^{\mathbf {x}}}$ , improving the previous epsilon-omega bound by Levitz (1978). A novel feature of our approach is the use of Conway’s surreal number for asymptotic calculations.

Renewal in Hawkes processes with self-excitation and inhibition

AbstractWe investigate the Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of such a point process impacts its future intensity by the addition of a signed reproduction function. The case of a nonnegative reproduction function corresponds to self-excitation, and has been widely investigated in the literature. In particular, there exists a cluster representation of the Hawkes process which allows one to apply known results for Galton–Watson trees. We use renewal techniques to establish limit theorems for Hawkes processes that have reproduction functions which are signed and have bounded support. Notably, we prove exponential concentration inequalities, extending results of Reynaud-Bouret and Roy (2006) previously proven for nonnegative reproduction functions using a cluster representation no longer valid in our case. Importantly, we establish the existence of exponential moments for renewal times of M/G/$\infty$ queues which appear naturally in our problem. These results possess interest independent of the original problem.

On Romanovski–Jacobi polynomials and their related approximation results

AbstractThe aim of this article is to present the essential properties of a finite class of orthogonal polynomials related to the probability density function of the F‐distribution over the positive real line. We introduce some basic properties of the Romanovski–Jacobi polynomials, the Romanovski–Jacobi–Gauss type quadrature formulae and the associated interpolation, discrete transforms, spectral differentiation and integration techniques in the physical and frequency spaces, and basic approximation results for the weighted projection operator in the nonuniformly weighted Sobolev space. We discuss the relationship between such kinds of finite orthogonal polynomials and other classes of infinite orthogonal polynomials. Moreover, we derive spectral Galerkin schemes based on a Romanovski–Jacobi expansion in space and time to solve the Cauchy problem for a scalar linear hyperbolic equation in one and two space dimensions posed in the positive real line. Two numerical examples demonstrate the robustness and accuracy of the schemes.

Equivalence of the Chern-Simons state and the Hartle-Hawking and Vilenkin wave functions

We show that the Chern-Simons (CS) state when reduced to minisuperspace is the Fourier dual of the Hartle-Hawking (HH) and Vilenkin (V) wave functions of the Universe. This is to be expected, given that the former and latter solve the same constraint equation, written in terms of conjugate variables (loosely, the expansion factor and the Hubble parameter). A number of subtleties in the mapping, related to the contour of integration of the connection, shed light on the issue of boundary conditions in quantum cosmology. If we insist on a real Hubble parameter, then only the HH wave function can be represented by the CS state, with the Hubble parameter covering the whole real line. For the V (or tunneling) wave function, the Hubble parameter is restricted to the positive real line (which makes sense, since the state only admits outgoing waves), but the contour also covers the whole negative imaginary axis. Hence, the state is not admissible if reality conditions are imposed upon the connection. Modifications of the V state, requiring the addition of source terms to the Hamiltonian constraint, are examined and found to be more palatable. In the dual picture, the HH state predicts a uniform distribution for the Hubble parameter over the whole real line; the modified V state a uniform distribution over the positive real line.

On utility maximization under model uncertainty in discrete‐time markets

AbstractWe study the problem of maximizing terminal utility for an agent facing model uncertainty, in a frictionless discrete‐time market with one safe asset and finitely many risky assets. We show that an optimal investment strategy exists if the utility function, defined either on the positive real line or on the whole real line, is bounded from above. We further find that the boundedness assumption can be dropped, provided that we impose suitable integrability conditions, related to some strengthened form of no‐arbitrage. These results are obtained in an alternative framework for model uncertainty, where all possible dynamics of the stock prices are represented by a collection of stochastic processes on the same filtered probability space, rather than by a family of probability measures.

Regression models for exceedance data: a new approach

The generalized Pareto distribution (GPD) is a family of continuous distributions used to model the tail of the distribution to values higher than a threshold u. Despite the advantages of the GPD representation, its shape and scale parameters do not correspond to the expected value, which complicates the interpretation of regression models specified using the GPD. This study proposes a linear regression model in which the response variable is a GPD, using a new parametrization that is indexed by mean and precision parameters. The main advantage of our new parametrization is the straightforward interpretation of the regression coefficients in terms of the expectation of the positive real line response variable, as is usual in the context of generalized linear models. Furthermore, we propose a model for extreme values, in which the GPD parameters (mean and precision) are defined on the basis of a dynamic linear regression model. The novelty of the study lies in the time variation of the mean and precision parameter of the resulting distribution. The parameter estimation of these new models is performed under the Bayesian paradigm. Simulations are conducted to analyze the performance of our proposed models. Finally, the models are applied to environmental datasets (temperature datasets), illustrating their capabilities in challenging cases in extreme value theory.

Universality for conditional measures of the Bessel point process

The Bessel point process is a rigid point process on the positive real line and its conditional measure on a bounded interval [Formula: see text] is almost surely an orthogonal polynomial ensemble. In this paper, we show that if [Formula: see text] tends to infinity, one almost surely recovers the Bessel point process. In fact, we show this convergence for a deterministic class of probability measures, to which the conditional measure of the Bessel point process almost surely belongs.

Nonlinear Coherent States Associated with a Measure on the Positive Real Half Line

We construct a class of generalized nonlinear coherent states by means of a newly obtained class of 2D complex orthogonal polynomials. The associated Bargmann-type transform is discussed. A polynomials realization of the basis of the quantum states Hilbert space is also obtained. Here, the entire structure owes its existence to a certain measure on the positive real half line, of finite total mass, together with all its moments. We illustrate this method with the measure $$r^\beta e^{-r}dr$$, where $$\beta $$ is a non-negative constant, which leads to a new generalization of the true-polyanalytic Bargmann transform.

Stationary distribution convergence of the offered waiting processes for $$GI/GI/1+GI$$ queues in heavy traffic

A result of Ward and Glynn (Queueing Syst 50(4):371–400, 2005) asserts that the sequence of scaled offered waiting time processes of the $$GI/GI/1+GI$$ queue converges weakly to a reflected Ornstein–Uhlenbeck process (ROU) in the positive real line, as the traffic intensity approaches one. As a consequence, the stationary distribution of a ROU process, which is a truncated normal, should approximate the scaled stationary distribution of the offered waiting time in a $$GI/GI/1+GI$$ queue; however, no such result has been proved. We prove the aforementioned convergence, and the convergence of the moments, in heavy traffic, thus resolving a question left open in 2005. In comparison with Kingman’s classical result (Kingman in Proc Camb Philos Soc 57:902–904, 1961) showing that an exponential distribution approximates the scaled stationary offered waiting time distribution in a GI / GI / 1 queue in heavy traffic, our result confirms that the addition of customer abandonment has a non-trivial effect on the queue’s stationary behavior.

On the improved estimation of a function of the scale parameter of an exponential distribution based on doubly censored sample

In the present article, we have studied the estimation of entropy, that is, a function of scale parameter of an exponential distribution based on doubly censored sample when the location parameter is restricted to positive real line. The estimation problem is studied under a general class of bowl-shaped non monotone location invariant loss functions. It is established that the best affine equivariant estimator (BAEE) is inadmissible by deriving an improved estimator. This estimator is non-smooth. Further, we have obtained a smooth improved estimator. A class of estimators is considered and sufficient conditions are derived under which these estimators improve upon the BAEE. In particular, using these results we have obtained the improved estimators for the squared error and the linex loss functions. Finally, we have compared the risk performance of the proposed estimators numerically. One data analysis has been performed for illustrative purposes.

Resurgent extrapolation: rebuilding a function from asymptotic data. Painlevé I

Extrapolation is a generic problem in physics and mathematics: how to use asymptotic data in one parametric regime to learn about the behavior of a function in another parametric regime. For example: extending weak coupling expansions to strong coupling, or high temperature expansions to low temperature, or vice versa. Such extrapolations are particularly interesting in systems possessing dualities. Here we study numerical procedures for performing such an extrapolation, combining ideas from resurgent asymptotics with well-known techniques of Borel summation, Padé approximants and conformal mapping. We illustrate the method with the concrete example of the Painlevé I equation, which has applications in many branches of physics and mathematics. Starting solely with a finite number of coefficients from asymptotic data at infinity on the positive real line, we obtain a high precision extrapolation of the function throughout the complex plane, even across the phase transition into the pole region. The precision far exceeds that of state-of-the-art numerical integration methods along the real axis. The methods used are both elementary and general, not relying on Painlevé integrability properties, and so are applicable to a wide class of extrapolation problems.

Expansions of the Real Field by Canonical Products

AbstractWe consider expansions of o-minimal structures on the real field by collections of restrictions to the positive real line of the canonical Weierstrass products associated with sequences such as$(-n^{s})_{n>0}$(for$s>0$) and$(-s^{n})_{n>0}$(for$s>1$), and also expansions by associated functions such as logarithmic derivatives. There are only three possible outcomes known so far: (i) the expansion is o-minimal (that is, definable sets have only finitely many connected components); (ii) every Borel subset of each$\mathbb{R}^{n}$is definable; (iii) the expansion is interdefinable with a structure of the form$(\mathfrak{R}^{\prime },\unicode[STIX]{x1D6FC}^{\mathbb{Z}})$where$\unicode[STIX]{x1D6FC}>1$,$\unicode[STIX]{x1D6FC}^{\mathbb{Z}}$is the set of all integer powers of$\unicode[STIX]{x1D6FC}$, and$\mathfrak{R}^{\prime }$is o-minimal and defines no irrational power functions.

Some aspects of reversed hazard rate and past entropy

The entropy of past lifetime is closely related to the reversed hazard rate. Since the reversed hazard rate of a non negative random variable with an absolutely continuous distribution can neither be constant nor monotonic increasing on the positive real line, there are limitations on deriving various properties of past entropy as well. In this work we investigate conditions under which reversed hazard rate can be increasing and can assume different non monotone shapes and use them to derive some new classes of past entropy-based life distributions and characterizations.

Persistence probability of a random polynomial arising from evolutionary game theory

AbstractWe obtain an asymptotic formula for the persistence probability in the positive real line of a random polynomial arising from evolutionary game theory. It corresponds to the probability that a multi-player two-strategy random evolutionary game has no internal equilibria. The key ingredient is to approximate the sequence of random polynomials indexed by their degrees by an appropriate centered stationary Gaussian process.

Smoluchowski flux and lamb-lion problems for random walks and Lévy flights with a constant drift

We consider non-interacting particles (or lions) performing one-dimensional random walks or Lévy flights (with Lévy index ) in the presence of a constant drift c. Initially these random walkers are uniformly distributed over the positive real line with a density . At the origin z = 0 there is an immobile absorbing trap (or a lamb), such that when a particle crosses the origin, it gets absorbed there. Our main focus is on (i) the flux of particles out of the system (the ‘Smoluchowski problem’) and (ii) the survival probability Sc(n) of the trap or lamb (the ‘lamb-lion problem’) until step n. We show that both observables can be expressed in terms of the average maximum of a single random walk or Lévy flight after n steps. This allows us to obtain the precise asymptotic behavior of both and Sc(n) analytically for large n in the two problems, for any value of and . In particular, for c > 0 and , we show that vanishes as , where is a -dependent positive constant, in contrast with the case of standard random walks (i.e. with ) for which . Our analytical results are confirmed by numerical simulations.

The local universality of Muttalib–Borodin biorthogonal ensembles with parameter

The Muttalib–Borodin biorthogonal ensemble is a probability density function for n particles on the positive real line that depends on a parameter and an external field . For we find the large n behavior of the associated correlation kernel with only few restrictions on . The idea is to relate the ensemble to a type II multiple orthogonal polynomial ensemble that can in turn be related to a Riemann–Hilbert problem which we then solve with the Deift–Zhou steepest descent method. The main ingredient is the construction of the local parametrix at the origin, with the help of Meijer G-functions, and its matching condition with a global parametrix. We will present a new iterative technique to obtain the matching condition, which we expect to be applicable in more general situations as well.

On a Monotone Scheme for Nonconvex Nonsmooth Optimization with Applications to Fracture Mechanics

A general class of nonconvex optimization problems is considered, where the penalty is the composition of a linear operator with a nonsmooth nonconvex mapping, which is concave on the positive real line. The necessary optimality condition of a regularized version of the original problem is solved by means of a monotonically convergent scheme. Such problems arise in continuum mechanics, as for instance cohesive fractures, where singular behaviour is usually modelled by nonsmooth nonconvex energies. The proposed algorithm is successfully tested for fracture mechanics problems. Its performance is also compared to two alternative algorithms for nonsmooth nonconvex optimization arising in optimal control and mathematical imaging.

On the transient dynamic antiplane contact problem in the presence of dry friction and slip

This article models the elastodynamic transient contact between two elastically similar half planes under antiplane loading and in the presence of friction. Contact is maintained along the positive real line under the presence of a certain remote contact pressure. An antiplane shear load is applied, which entails interfacial shear traction that opposes the frictional force entailed by the contact pressure. In order to balance the surface tractions, the surface must be allowed to slip. We derive the closed form solution of the interfacial traction due to a general antiplanar displacement distribution using a variant of the Wiener-Hopf technique. We also find closed-form expressions for the interfacial shear traction due to this remote antiplane load. In combination with the frictional force, this leads to an integral equation the solution to which is the distribution of relative slip. We quantify both this and the magnitude of the interfacial shear tractions under diverse loading, showing that transient loading leads to partial reverse slip of the contact surfaces. We show that the reverse slip tends to vanish over time, and that it is ameliorated if the friction coefficient is reduced.