Positive Half-line Research Articles

The functional equation of the smoothing transform

Given a sequence $T=(T_i)_{i\geq1}$ of nonnegative random variables, a function f on the positive halfline can be transformed to $\mathbb{E}\prod_{i\geq1}f(tT_i)$. We study the fixed points of this transform within the class of decreasing functions. By exploiting the intimate relationship with general branching processes, a full description of the set of solutions is established without the moment conditions that figure in earlier studies. Since the class of functions under consideration contains all Laplace transforms of probability distributions on $[0,\infty)$, the results provide the full description of the set of solutions to the fixed-point equation of the smoothing transform, $X\stackrel{d}{=}\sum_{i\geq1}T_iX_i$, where $\stackrel{d}{=}$ denotes equality of the corresponding laws, and $X_1,X_2,...$ is a sequence of i.i.d. copies of X independent of T. Further, since left-continuous survival functions are covered as well, the results also apply to the fixed-point equation $X\stackrel{d}{=}\inf\{X_i/T_i:i\geq1,T_i>0\}$. Moreover, we investigate the phenomenon of endogeny in the context of the smoothing transform and, thereby, solve an open problem posed by Aldous and Bandyopadhyay.

P-Wavelet frame packets on a half-line using the Walsh–Fourier transform

This paper deals with a construction of wavelet frame packets on the positive half-line ℝ+ using the splitting trick for frames. It is shown that, as long as finitely many splitting steps are applied, the resulting sequence of functions is a frame of L 2(ℝ+). If the matrix M(ξ) associated with the splitting is unitary, then the splitting can be applied infinitely many times to prove the existence of frame with the frame bounds as shown in Theorem 3.3.

Special functions as subordinated semigroups on the real line

We give examples of convolution semigroups on the positive half-line and on the real line. Such semigroups are expressed in terms of special functions which arise in classical differential equations.

Poisson process with different Brownian clocks

In this paper, different types of Poisson processes N subordinated to random time processes X, depending on Brownian motion, are analysed. In particular, the processes X considered here are the elastic Brownian motion , the Brownian sojourn time on the positive half-line the first-passage time (through the level t) of a Brownian motion, with or without drift, and the -Bessel process , for . In all these cases, we obtain the explicit state probability distributions , , , their governing difference-differential equations and some moments. The connections among different models and, in particular, of with birth and death processes are obtained and discussed. The models presented have probability distributions governed by higher order or fractional difference-differential equations. In the case of the Poisson process with Bessel times, equations with time-dependent coefficients appear.

QUANTUM f-DIVERGENCES AND ERROR CORRECTION

Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz' quasi-entropies. Many well-known distinguishability measures of quantum states are given by, or derived from, f-divergences. Special examples include the quantum relative entropy, the Rényi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under substochastic maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results for this class of distinguishability measures. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz' reversibility theorem for a large class of f-divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f-divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Hölder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.

Dyadic wavelet frames on a half-line using the Walsh–Fourier transform

This paper deals with a construction of dyadic wavelet frames on a positive half-line ℝ+ using the Walsh–Fourier transform. We prove necessary and sufficient conditions for the system to be a wavelet frame in L 2(ℝ+). These conditions are found to be better than those of Daubechies [Ten Lecture on Wavelets, NSF-CBMS Regional Conferences in Applied Mathematics, Vol. 61, SIAM, Philadelphia, PA, 1992], Chui and Shi [Inequalities of Littlewood–Paley type for frames and wavelets, SIAM J. Math. Anal. 24 (1993), pp. 263–277], Casazza and Christenson [Weyl–Heisenberg frames for subspaces of L 2(ℝ), Proc. Amer. Math. Soc. 129 (2001), pp. 145–154] and Christenson [Frames, Riesz bases, and discrete Gabor/wavelet expansions, Bull. Amer. Math. Soc. 38 (2001), pp. 273–291; An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003].

Standard ideals in weighted algebras of Korenblyum and Wiener types

We introduce two types of Banach algebras of analytic functions on the unit disc which can be seen as weighted versions of closed primary ideals of the Korenblyum and (analytic) Wiener algebras, respectively. Such types of algebras arise in connection with convolution Banach algebras of Sobolev type, on the positive half-line, and their discrete analogues defined in terms of higher order differences. We show that all closed ideals are standard for algebras in the first class, and that closed ideals with countable hull are standard in algebras of the second class.

Uniform estimates for the finite-time ruin probability in the dependent renewal risk model

This paper investigates the finite-time ruin probability in the dependent renewal risk model, where the claim sizes are independent and identically distributed random variables with strongly subexponential tails, and the interarrival times are negatively dependent. We establish an asymptotic estimate, which holds uniformly for the time horizon varying in the positive half line.

ON BIORTHOGONAL WAVELETS RELATED TO THE WALSH FUNCTIONS

In this paper, we describe an algorithm for computing biorthogonal compactly supported dyadic wavelets related to the Walsh functions on the positive half-line ℝ+. It is noted that a similar technique can be applied in very general situations, e.g., in the case of Cantor and Vilenkin groups. Using the feedback-based approach, some numerical experiments comparing orthogonal and biorthogonal dyadic wavelets with the Haar, Daubechies, and biorthogonal 9/7 wavelets are prepared.

Occupation time distributions for the telegraph process

For the one-dimensional telegraph process, we obtain explicitly the distribution of the occupation time of the positive half-line. The long-term limiting distribution is then derived when the initial location of the process is in the range of subnormal or normal deviations from the origin; in the former case, the limit is given by the arcsine law. These limit theorems are also extended to the case of more general occupation-type functionals.

Extension of Rotfel’d Theorem

Let f ( t ) be a non-negative concave function on the positive half-line. Given an arbitrary partitioned positive semi-definite matrix, we show that f A X X ∗ B ≤ f ( A ) + f ( B ) for all symmetric (i.e. unitarily invariant) norms. This characterization of concave functions extends a famous trace inequality of Rotfel’d, Tr f ( A + B ) ≤ Tr f ( A ) + Tr f ( B ) and contains several classical matrix inequalities.

General Freidlin–Wentzell Large Deviations and positive diffusions

We prove Freidlin–Wentzell Large Deviation estimates under rather minimal assumptions. This allows one to derive Wentzell–Freidlin Large Deviation estimates for diffusions on the positive half line with coefficients that are neither bounded nor Lipschitz continuous. This applies to models of interest in Finance, i.e. the CIR and the CEV models, which are positive diffusion processes whose diffusion coefficient is only Hölder continuous.

Solution bases for the fourth-order Bessel-type and Laguerre-type differential equations

The fourth-order Bessel-type and Laguerre-type linear ordinary differential equations are prototypes of structured linear differential equations of higher even-order, which naturally extend the second-order Bessel and Laguerre equations defined on the positive half-line of the real field R. Whilst the Laguerre-type equation arose from a search for all orthogonal polynomial generated by a linear differential equation, the present authors derived the Bessel-type equations and functions in 1994 by applying a generalized limit process to the Laguerre-type case. Due to their close relationship, the Laguerre- and Bessel-type functions of the same order share many important properties as, for example, orthogonality or a generalized hypergeometric representation. In this article, we first survey the most recent achievements in our study of the fourth-order Bessel equation which led to explicit representations of four linear independent solutions. Our main purpose then is to show how these techniques carry over to establish a solution basis for the fourth-order Laguerre-type differential equation.

Upper and lower solutions for BVPs on the half-line with variable coefficient and derivative depending nonlinearity

This paper is concerned with a second-order nonlinear boundary value problem with a derivative depending nonlinearity and posed on the positive half-line. The derivative operator is time dependent. Upon a priori estimates and under a Nagumo growth condition, the Schauder’s fixed point theorem combined with the method of upper and lower solutions on unbounded domains are used to prove existence of solutions. A uniqueness theorem is also obtained and some examples of application illustrate the obtained results.

Spectral analysis of subordinate Brownian motions on the half-line

We study one-dimensional Levy processes with Levy-Khintchine exponent psi(xi^2), where psi is a complete Bernstein function. These processes are subordinate Brownian motions corresponding to subordinators, whose Levy measure has completely monotone density; or, equivalently, symmetric Levy processes whose Levy measure has completely monotone density on the positive half-line. Examples include symmetric stable processes and relativistic processes. The main result is a formula for the generalized eigenfunctions of transition operators of the process killed after exiting the half-line. A generalized eigenfunction expansion of the transition operators is derived. As an application, a formula for the distribution of the first passage time (or the supremum functional) is obtained.

Beta ensembles, stochastic Airy spectrum, and a diffusion

We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigenvalues of the random Schrödinger operator − d 2 d x 2 + x + 2 β b x ′ -\frac {d^2}{dx^2} + x + \frac {2}{\sqrt {\beta }} b_x^{\prime } restricted to the positive half-line, where b x ′ b_x^{\prime } is white noise. In doing so we extend the definition of the Tracy-Widom( β \beta ) distributions to all β > 0 \beta >0 and also analyze their tails. Last, in a parallel development, we provide a second characterization of these laws in terms of a one-dimensional diffusion. The proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converges to that of their continuum operator limit. In particular, we show how Tracy-Widom laws arise from a functional central limit theorem.

Randomness, evenness, and Rényi’s index

This paper links together the notion of entropy and the notion of inequality indices—the former is applied in Statistical Physics to measure randomness, and the latter is applied in Economics to measure evenness. We explore the profound similarities between these diametric notions, construct a mathematical transformation between them, and show how randomness can be used to measure evenness–and vice versa. In particular, we devise and study Rényi’s index—a randomness-based measure of evenness with special properties. Rényi’s index is established as an effectual gauge of statistical heterogeneity in the context of general probability laws defined on the positive half-line.

First-passage exponents of multiple random walks

We investigate first-passage statistics of an ensemble of N noninteracting random walks on a line. Starting from a configuration in which all particles are located in the positive half-line, we study Sn(t), the probability that the nth rightmost particle remains in the positive half-line up to time t. This quantity decays algebraically, , in the long-time limit. Interestingly, there is a family of nontrivial first-passage exponents, β1 < β2 < ⋅⋅⋅ < βN − 1; the only exception is the two-particle case where β1 = 1/3. In the N → ∞ limit, however, the exponents attain a scaling form, βn(N) → β(z) with . We also demonstrate that the smallest exponent decays exponentially with N. We deduce these results from first-passage kinetics of a random walk in an N-dimensional cone and confirm them using numerical simulations. Additionally, we investigate the family of exponents that characterizes leadership statistics of multiple random walks and find that in this case, the cone provides an excellent approximation.

Spectral theory of semibounded Sturm–Liouville operators with local interactions on a discrete set

We study the Hamiltonians HX,α,q with δ-type point interactions at the centers xk on the positive half line in terms of energy forms. We establish analogs of some classical results on operators Hq=−d2/dx2+q with locally integrable potentials q∊Lloc1(R+). In particular, we prove that the Hamiltonian HX,α,q is self-adjoint if it is lower semibounded. This result completes the previous results of Brasche [“Perturbation of Schrödinger Hamiltonians by measures—selfadjointness and semiboundedness,” J. Math. Phys. 26, 621 (1985)] on lower semiboundedness. Also we prove the analog of Molchanov’s discreteness criteria, Birman’s result on stability of a continuous spectrum, and investigate discreteness of a negative spectrum. In the recent paper [Kostenko, A. and Malamud, M., “1–D Schrödinger operators with local point interactions on a discrete set,” J. Differ. Equations 249, 253 (2010)], it was shown that the spectral properties of HX,α≔HX,α,0 correlate with the corresponding spectral properties of a certain class of Jacobi matrices. We apply the above mentioned results to the study of spectral properties of these Jacobi matrices.

Nyman type theorem in convolution Sobolev algebras

We characterize the dense ideals of certain convolution Sobolev algebras, on the positive half-line, as those ideals I which satisfy the Nyman conditions Z(I)=∅ and γ(I)=0. Here Z(I) is the hull of I and γ(I):=inf {inf supp (f):f∈I}.