Positive Half-line Research Articles

A central limit theorem for the optimal selection process for monotone subsequences of maximum expected length

This article provides a refinement of the main results for the monotone subsequence selection problem, previously obtained by Bruss and Delbaen (Stoch. Proc. Appl. 96 (2001) 313). Let ( N s ) s ⩾ 0 be a Poisson process with intensity 1 defined on the positive half-line. Let T 1 , T 2 , … be the corresponding occurrence times, and let ( X k ) k = 1 , 2 , … be a sequence of i.i.d. uniform random variables on [0,1], independent of the T j 's. We observe the ( T k , X k ) sequentially. Call X k the observed value at time T k . For a given horizon t , consider the objective to select in sequential order, without recall on preceding observations, a subsequence of monotone increasing values of maximal expected length. Let L t t be the random number of selected values under the optimal strategy. Extending the objective of our first paper the main goal of the present paper is to understand the whole process ( L u t ) 0 ⩽ u ⩽ t , where the random variable L u t denotes the number of the selected values up to time u under the t -optimal strategy. We show that this process obeys, under suitable normalization, a Central Limit Theorem. In particular, we show that this holds in a more complete sense than one would expect. The problem of interdependence of this process with two other processes studied before is overcome by the simultaneous study of three associated martingales. This analysis is based on refined martingale methods, and a non-negligible level of technical sophistication seems unavoidable. But then, the results are rewarding. We not only get the “right” functional Central Limit Theorem for t tending to infinity but also the (singular) covariance matrix of the three-dimensional process summarizing the interacting processes. We feel there is no other way to understand these interactions, and believe that this adds value to our approach.

Vicious walkers in a potential

We consider N vicious walkers moving in one dimension in a one-body potential v(x). Using the backward Fokker-Planck equation we derive exact results for the asymptotic form of the survival probability Q(x,t) of vicious walkers initially located at (x_1,...,x_N) = x, when v(x) is an arbitrary attractive potential. Explicit results are given for a square-well potential with absorbing or reflecting boundary conditions at the walls, and for a harmonic potential with an absorbing or reflecting boundary at the origin and the walkers starting on the positive half line. By mapping the problem of N vicious walkers in zero potential onto the harmonic potential problem, we rederive the results of Fisher [J. Stat. Phys. 34, 667 (1984)] and Krattenthaler et al. [J. Phys. A 33}, 8835 (2000)] respectively for vicious walkers on an infinite line and on a semi-infinite line with an absorbing wall at the origin. This mapping also gives a new result for vicious walkers on a semi-infinite line with a reflecting boundary at the origin: Q(x,t) \sim t^{-N(N-1)/2}.

A homing problem for diffusion processes with control-dependent variance

Controlled one-dimensional diffusion processes, with infinitesimal variance (instead of the infinitesimal mean) depending on the control variable, are considered in an interval located on the positive half-line. The process is controlled until it reaches either end of the interval. The aim is to minimize the expected value of a cost criterion with quadratic control costs on the way and a final cost equal to zero (resp. a large constant) if the process exits the interval through its left (resp. right) end point. Explicit expressions are obtained both for the optimal value of the control variable and the value function when the infinitesimal parameters of the processes are proportional to a power of the state variable.

Random planar lattices and integrated superBrownian excursion

In this paper, a surprising connection is described between a specific brand of random lattices, namely planar quadrangulations, and Aldous’ Integrated SuperBrownian Excursion (ISE). As a consequence, the radius r n of a random quadrangulation with n faces is shown to converge, up to scaling, to the width r=R−L of the support of the one-dimensional ISE, or precisely: More generally the distribution of distances to a random vertex in a random quadrangulation is described in its scaled limit by the random measure ISE shifted to set the minimum of its support in zero. The first combinatorial ingredient is an encoding of quadrangulations by trees embedded in the positive half-line, reminiscent of Cori and Vauquelin’s well labelled trees. The second step relates these trees to embedded (discrete) trees in the sense of Aldous, via the conjugation of tree principle, an analogue for trees of Vervaat’s construction of the Brownian excursion from the bridge. From probability theory, we need a new result of independent interest: the weak convergence of the encoding of a random embedded plane tree by two contour walks to the Brownian snake description of ISE. Our results suggest the existence of a Continuum Random Map describing in term of ISE the scaled limit of the dynamical triangulations considered in two-dimensional pure quantum gravity.

Smooth estimation of a monotone density

We investigate the interplay of smoothness and monotonicity assumptions when estimating a density from a sample of observations. The nonparametric maximum likelihood estimator of a decreasing density on the positive half line attains a rate of convergence of [Formula: See Text] at a fixed point t if the density has a negative derivative at t. The same rate is obtained by a kernel estimator of bandwidth [Formula: See Text], but the limit distributions are different. If the density is both differentiable at t and known to be monotone, then a third estimator is obtained by isotonization of a kernel estimator. We show that this again attains the rate of convergence [Formula: See Text], and compare the limit distributions of the three types of estimators. It is shown that both isotonization and smoothing lead to a more concentrated limit distribution and we study the dependence on the proportionality constant in the bandwidth. We also show that isotonization does not change the limit behaviour of a kernel estimator with a bandwidth larger than [Formula: See Text], in the case that the density is known to have more than one derivative.

On the coexistence of absolutely continuous and singular continuous components of the spectral measure for some Sturm–Liouville operators with square summable potential

We prove that there exist some Sturm–Liouville operators with square summable potentials such that the singular continuous component of the spectral measure lies on the positive half-line. The Hausdorff dimension of the support of this singular measure can be arbitrary number from 0 to 1.

On the continuous analog of Rakhmanov's Theorem for orthogonal polynomials

We obtain the continuous analogs of Rakhmanov's Theorem for polynomials orthogonal on the unit circle. Sturm–Liouville operators and Krein systems are considered. For a Sturm–Liouville operator with bounded potential q, we prove the following statement. If the essential spectrum and absolutely continuous component of the spectral measure fill the whole positive half-line, then q decays at infinity in the certain integral sense.

The double gamma function and related Pick functions

Denoting by G the Barnes double gamma function we find the Stieltjes representation of the related Pick functionz↦logG(z+1)z2Logz.As a corollary we obtain that this function has a completely monotone derivative on the positive half line.

Entropy and information in the interest rate term structure

Associated with every positive interest term structure there is a probability density function over the positive half-line. This fact can be used to turn the problem of term structure analysis into a problem in the comparison of probability distributions, an area well developed in statistics, known as information geometry. The information-theoretic and geometric aspects of term structures thus arising are here illustrated. In particular, we introduce a new term structure calibration methodology based on maximization of entropy, and also present some new families of interest rate models arising naturally in this context.

Small perturbation of diffusions in inhomogeneous media

For the system of d-dim stochastic differential equations, dX ε(t)=b(X ε(t)) dt+εσ(X ε(t)) dW(t),t∈[0,1],X ε(0)=x 0∈R d, where b( x) and σ( x) are smooth except possibly along the hyperplane {( x 1,…, x d ); x 1=0}, we shall demonstrate that the natural setup of its large deviation principle is to consider the probability ε 2 logP(‖X ε−ϕ‖<δ, ‖u ε−ψ‖<δ, ‖ℓ ε−η‖<δ)∼−I(ϕ,ψ,η) of the triplet ( X ε , u ε ,ℓ ε ) simultaneously. Here, u ε is the occupation time of X ε 1(·) in the positive half line and ℓ ε (·) is the local time of X ε 1(·) at 0. The explicit form of the rate function I(·,·,·) is obtained. The usual Wentzell–Friedlin theory concerns only probabilities of the form ε 2log P(‖ X ε − ϕ‖< δ) and its limit is a consequence of the contraction principle of our result.

Limit Clusters in the Inviscid Burgers Turbulence with Certain Random Initial Velocities

We study the infinite time shock limits given certain Markovian initial velocities to the inviscid Burgers turbulence. Specifically, we consider the one-sided case where initial velocities are zero on the negative half-line and follow a time-homogeneous nice Markov process X on the positive half-line. Finite shock limits occur if the Markov process is transient tending to infinity. They form a Poisson point process if X is spectrally negative. We give an explicit description when X is furthermore spatially homogeneous (a Levy process) or a self-similar process on (0, ∞). We also consider the two-sided case where we suppose an independent dual process in the negative spatial direction. Both spatial homogeneity and an exponential Levy condition lead to stationary shock limits.

Fractional fermion number in a(1+1)-dimensional Dirac equation with a scalar Coulomb field

An elementary example of fermion number fractionalization is given. The model considered is a massive Dirac fermion moving on the positive halfline in an external scalar Coulomb field. The theory is symmetric under charge conjugation. We demonstrate the existence of a nondegenerate, normalizable, and self-charge-conjugate zero energy solution in the theory. The vacuum thus has a fermion number $\ifmmode\pm\else\textpm\fi{}1/2$ depending on whether the lowest positive energy state without the external scalar field was filled or empty.

Price systems constructed by optimal dynamic portfolios

The paper studies connections between arbitrage and utility maximization in a discrete-time financial market. The market is incomplete. Thus one has several choices of equivalent martingale measures to price contingent claims. Davis determines a unique price for a contingent claim which is based on an optimal dynamic portfolio by use of a `marginal rate of substitution' argument. Here conditions will be given such that this price is determined by a martingale measure and thus by a consistent price system. The underlying utility function U is defined on the positive half-line. Then dynamic portfolios are admissible if the terminal wealth is positive. In case of the logarithmic utility function, the optimal dynamic portfolio is the numeraire portfolio.

On Sojourn Distributions of Processes Related to Some Higher-Order Heat-Type Equations

It is well known that the sojourn time of Brownian motion B(t), t>0 on the positive half-line, during the interval [0, t] and under the condition B(t)=0, is uniformly distributed, while it has the form of the “corrected arc-sine law” when the condition B(t)>0 is assumed. We find the analogues of these laws for “processes” X(t), t>0 governed by signed measures whose densities are the fundamental solutions of third and fourth-order heat-type equations. Surprisingly, both laws hold for the fourth-order “process.” The uniform law is still valid for the third-order “process” but a different law emerges when the condition X(t)>0 is considered.

Positive Solutions to Second Order Singular Differential Equations Involving the One-Dimensional M-Laplace Operator

Abstract We consider a class of second order quasilinear differential equations with singular ninlinearities. Our main purpose is to investigate in detail the asymptotic behavior of their solutions defined on a positive half-line. The set of all possible positive solutions is classified into five types according to their asymptotic behavior near infinity, and sharp conditions are established for the existence of solutions belonging to each of the classified types.

The Paley-Wiener theorem for Schwartz distributions with support on a half-line

LetD′+ be the space of Schwartz distributions with support on the closed positive half-line [0, +∞). We give a generalization of the Paley-Wiener theorem to the case of the distributions inD′+.

Positive Solutions to Second Order Singular Differential Equations Involving the One-Dimensional M-Laplace Operator

We consider a class of second order quasilinear differential equations with singular ninlinearities. Our main purpose is to investigate in detail the asymptotic behavior of their solutions defined on a positive half-line. The set of all possible positive solutions is classified into five types according to their asymptotic behavior near infinity, and sharp conditions are established for the existence of solutions belonging to each of the classified types.

Point processes in time and stein's method

This article gives an upper bound for a Wasserstein distance between the distribution of a simple point process and that of a Poisson process on the positive half line. The bound is partly expressed in terms of their compensators, and partly in terms of the expected future effect of having a point at a given time. The argument is based on Stein's method, together with a martingale approach. Some examples are provided, which illustrate the computation of the upper bound and demonstrate its accuracy

Positive solutions to a class of second order differential equations with singular nonlinearities

A class of second order quasilinear differential equations with singular nonlinearities is considered. The set of all possible solutions defined on a positive half-line [a,∞) is classified into six types according to their aymptotic behavior as t→∞, and sharp conditions are established for the existence of solutions belonging to each of the classified types.

Differentiation of Operator Functions and Perturbation Bounds

Given a smooth real function f on the positive half line consider the induced map \(\) on the set of positive Hilbert space operators. Let f(k be the kth derivative of the real function f and >E5>Dkf the kth Frechet derivative of the operator map f. We identify large classes of functions for which \(\), for k= 1,2,... . This reduction of a noncommutative problem to a commutative one makes it easy to obtain perturbation bounds for several operator maps. Our techniques serve to illustrate the use of a formalism for “quantum analysis” that is like the one recently developed by M. Suzuki.