Finite Expectation Research Articles

Multiobjective Stopping Problem for Discrete-Time Markov Processes: Convex Analytic Approach

The purpose of this paper is to study an optimal stopping problem with constraints for a Markov chain with general state space by using the convex analytic approach. The costs are assumed to be nonnegative. Our model is not assumed to be transient or absorbing and the stopping time does not necessarily have a finite expectation. As a consequence, the occupation measure is not necessarily finite, which poses some difficulties in the analysis of the associated linear program. Under a very weak hypothesis, it is shown that the linear problem admits an optimal solution, guaranteeing the existence of an optimal stopping strategy for the optimal stopping problem with constraints.

Tail behavior of random sums of negatively associated increments

Let { X , X k , k ⩾ 1 } be a sequence of negatively associated random variables with a common distribution function and finite expectation and let τ be a nonnegative integer-valued random variable independent of { X k , k ⩾ 1 } . In this paper we give unified form for the asymptotic behavior of the random sums S τ = ∑ k = 1 τ X k in the case of lim x → + ∞ P ( X > x ) P ( τ > x ) = C ∈ [ 0 , + ∞ ] . The results extend the earlier results of Aleškevičiené et al. (2008) [2].

On the AC Spectrum of One-dimensional Random Schrödinger Operators with Matrix-valued Potentials

We consider discrete one-dimensional random Schroedinger operators with decaying matrix-valued, independent potentials. We show that if the l^2-norm of this potential has finite expectation value with respect to the product measure then almost surely the Schroedinger operator has an interval of purely absolutely continuous (ac) spectrum. We apply this result to Schroedinger operators on a strip. This work provides a new proof and generalizes a result obtained by Delyon, Simon, and Souillard.

Limit theorems for additive functionals of a Markov chain

Consider a Markov chain $\{X_n\}_{n\ge 0}$ with an ergodic probability measure $\pi$. Let $\Psi$ a function on the state space of the chain, with $\alpha$-tails with respect to $\pi$, $\alpha\in (0,2)$. We find sufficient conditions on the probability transition to prove convergence in law of $N^{1/\alpha}\sum_n^N \Psi(X_n)$ to a $\alpha$-stable law. ``Martingale approximation'' approach and ``coupling'' approach give two different sets of conditions. We extend these results to continuous time Markov jump processes $X_t$, whose skeleton chain satisfies our assumptions. If waiting time between jumps has finite expectation, we prove convergence of $N^{-1/\alpha}\int_0^{Nt} V(X_s) ds$ to a stable process. In the case of waiting times with infinite average, we prove convergence to a Mittag-Leffler process.

On the origin of divergences in the coincidence probabilities in cavity photodetection experiments

The theory of photon correlation is an established part of quantum electronics. However, recently reported divergences in the theory of time correlated detection of photons show that important details of cavity photon statistics are still incompletely understood. The quantum jump superoperators of the SD photon counting model given by Srinivas and Davies (1981 J. Mod. Opt. 28 981–96 ) do not fulfil the assumption of the bounded interaction rate. This has raised doubts about the consistency of the SD photon counting model and especially about the existence of coincidence probability density (CPD) functions (Dodonov et al 2005 J. Opt. B: Quantum Semiclass. Opt. 7 99–108). In this work, we start from the first principles of the quantum trajectory theory and show how the different coincidence probability densities and coincidence probabilities (CPs) have to be calculated. CPDs derived by us are well defined, and CPs are finite and correctly normalized for all fields with finite photon number expectation value. Furthermore, we show that the SD model reproduces photon bunching and antibunching phenomena when consistent derivation for the second-order coherence degree is used.

On a renewal function when the second moment is infinite

Let { S k } be a random walk with independent, identically distributed real-valued increments { X i } , having a nonarithmetic distribution, finite expectation μ > 0 and infinite moment E max ( 0 , X 1 ) 2 . A refinement of the elementary renewal theorem is given in the following form: E min { k : S k > t } − t / μ ∼ ρ ( t ) as t → ∞ , where ρ ( t ) is a specific function such that ρ ( t ) → ∞ as t → ∞ .

Characterization of distributions with the length-bias scaling property

This paper characterizes the density functions of absolutely continuous positive random variables with finite expectation whose respective distribution functions satisfy the so-called length-bias scaling property.

On Wald Optimal Stopping Problem for Geometric Brownian Motions

This note concerns a problem of optimally stopping a nondegenerate, two-dimensional, geometric Brownian motion Q t = (x t ,y t ), with the goal of maximizing where the supremum is taken over the class of all stopping times 𝒳 Q , with finite expectation, H:ℝ+2 → ℝ is a measurable function satisfying a certain growth condition, and c > 0 is a positive constant. It is proved that, under certain conditions, the maximal value Φ(.,.) is a logarithmic function, and the optimal stopping time τ* < ∞ admits the form where ψ( · ) ∈ C 2(0,∞), positive non-decreasing solution of a certain second-order nonlinear ordinary differential equation. The present result extends and supplements a class of Wald-type optimal stopping problems in Graversen and Peskir's paper (1997), which treats the case of a one-dimensional Brownian motion.

Finite temperature expectation values of boundary operators

A conjecture is presented for the thermal one-point function of boundary operators in integrable boundary quantum field theories in terms of form factors. It is expected to have applications in studying boundary critical phenomena and boundary flows, which are relevant in the context of condensed matter and string theory. The conjectured formula is verified by a low-temperature expansion developed using finite size techniques, which can also be used to evaluate higher point functions both in the bulk and on the boundary.

Invariant random graphs with iid degrees in a general geography

Let D be a non-negative integer-valued random variable and let G = (V, E) be an infinite transitive finite-degree graph. Continuing the work of Deijfen and Meester (Adv Appl Probab 38:287-298) and Deijfen and Jonasson (Electron Comm Probab 11:336-346), we seek an Aut(G)-invariant random graph model with V as vertex set, iid degrees distributed as D and finite mean connections (i.e., the sum of the edge lengths in the graph metric of G of a given vertex has finite expectation). It is shown that if G has either polynomial growth or rapid growth, then such a random graph model exists if and only if double strok E sign [D,R(D)] 0. All known transitive graphs have either polynomial or rapid growth. It is believed that no other growth rates are possible. When G has rapid growth, the result holds also when the degrees form an arbitrary invariant process. A counter-example shows that this is not the case when G grows polynomially. For this case, we provide other, quite sharp, conditions under which the stronger statement does and does not hold respectively. Our work simplifies and generalizes the results for G,=Z in [4] and proves, e.g., that with G,=Zd, there exists an invariant model with finite mean connections if and only if E D [(d+1) d] < ∞ . When G has exponential growth, e.g., when G is a regular tree, the condition becomes double strok E sign[D log D] < ∞.

From Arthur Cayley via Felix Klein, Sophus Lie, Wilhelm Killing, Elie Cartan, Emmy Noether and superstrings to Cantorian space–time

In this work we present a historical overview of mathematical discoveries which lead to fundamental developments in super string theory, super gravity and finally to E-infinity Cantorian space–time theory. Cantorian space–time is a hierarchical fractal-like semi manifold with formally infinity many dimensions but a finite expectation number for these dimensions. The idea of hierarchy and self-similarity in science was first entertain by Right in the 18th century, later on the idea was repeated by Swedenborg and Charlier. Interestingly, the work of Mohamed El Naschie and his two contra parts Ord and Nottale was done independently without any knowledge of the above starting from non- linear dynamics and fractals.

Near BPS Wilson loop in β-deformed theories

We propose a definition of the Wilson loop operator in the = 1 β-deformed supersymmetric Yang-Mills theory. Although the operator is not BPS, it has a finite expectation value at least up to order (g2N)2. This does not happen generally for a generic non-BPS Wilson loop whose expectation value is UV divergent. For this reason we call this a near-BPS Wilson loop. We derive the general form of the boundary condition satisfied by the dual string worldsheet and find that it is deformed. Finiteness of the expectation value of the Wilson loop fixes the boundary condition to be one which is characterized by the vielbein of the deformed supergravity metric. The Wilson loop operators provide natural candidates as dual descriptions to some of the existing D-brane configurations in the Lunin-Maldacena background. We also construct the string dual configuration for a near-1/4 BPS circular Wilson loop operator. The string lies on a deformed three-sphere instead of a two-sphere as in the undeformed case. The expectation value of the Wilson loop operator is computed using the AdS/CFT correspondence and is found to be independent of the deformation. We conjecture that the exact expectation value of the Wilson loop is given by the same matrix model as in the undeformed case.

Prophet Inequalities for I.I.D. Random Variables with Random Arrival Times

Suppose X 1, X 2,… are independent and identically distributed nonnegative random variables with finite expectation, and for each k, X k is observed at the kth arrival time S k of a Poisson process with a unit rate independent of the sequence {X k }. For t > 0, comparisons are made between the expected maximum M(t) := E[max k≥1 X k I(S k ≤ t)] and the optimal stopping value V(t) := supτ∈𝒯E[X τ I(S τ ≤ t)], where 𝒯 is the set of all ℕ-valued random variables τ such that {τ = i} is measurable with respect to the σ-algebra generated by (X 1, S 1),…, (X i , S i ). For instance, it is shown that M(t)/V(t) ≤ 1 + α0, where α0 ≐ 0.34149 satisfies , and this bound is asymptotically sharp as t → ∞. Another result is that M(t)/V(t) < 2 − (1 − e −t )/t, and this bound is asymptotically sharp as t ↓ 0. Upper bounds for the difference M(t) − V(t) are also given, under the additional assumption that the X k 's are bounded.

Quantum gravity unification via transfinite arithmetic and geometrical averaging

In E-Infinity theory, we have not only infinitely many dimensions but also infinitely many fundamental forces. However, due to the hierarchical structure of ε ( ∞ ) spacetime we have a finite expectation number for its dimensionality and likewise a finite expectation number for the corresponding interactions. Starting from the preceding fundamental principles and using the experimental findings as well as the theoretical value of the coupling constants of the electroweak and the strong forces we present an extremely simple averaging procedure for determining the quantum gravity unification coupling constant with and without super symmetry. The work draws heavily on previous results, in particular a paper, by the Slovian Prof. Marek-Crnjac [Marek-Crnjac L. On the unification of all fundamental forces in a fundamentally fuzzy Cantorian ε ( ∞ ) manifold and high energy physics. Chaos, Solitons & Fractals 2004;4:657–68].

Spontaneous nonthermal leptogenesis in high-scale inflation models

We argue that a nonthermal leptogenesis occurs spontaneously, without direct couplings of the inflaton with right-handed neutrinos, in a wide class of high-scale inflation models such as the chaotic and hybrid inflation. It is only a finite vacuum expectation value of the inflaton, or more precisely, a linear term in the K\ahler potential, that is a prerequisite for the spontaneous nonthermal leptogenesis. To exemplify how it works, we show that a chaotic inflation model in supergravity naturally produces a right amount of baryon asymmetry via the spontaneous nonthermal leptogenesis. We also discuss the gravitino production from the inflaton.

Self-consistent variational approach to the minimal left-right symmetric model of electroweak interactions

The problem of mass generation is addressed by a Gaussian variational method for the minimal left-right symmetric model of electroweak interactions. Without any scalar bidoublet, the Gaussian effective potential is shown to have a minimum for a broken symmetry vacuum with a finite expectation value for both the scalar Higgs doublets. The symmetry is broken by the fermionic coupling that destabilizes the symmetric vacuum, yielding a self-consistent fermionic mass. In this framework a light Higgs is only compatible with the existence of a new high energy mass scale below 2 TeV.

Large deviations for random sums of negatively dependent random variables with consistently varying tails

Let { X k , k = 1 , 2 , … } be a sequence of negatively dependent random variables with common distribution F and finite expectation μ . Under the assumption that the tail probability F ¯ ( x ) = 1 - F ( x ) is consistently varying as x tends to infinity, this paper investigates precise large deviations for the random sum S N ( t ) = ∑ n = 1 N ( t ) X n , where { N ( t ) , t ⩾ 0 } is a nonnegative and integer-valued process independent of { X k , k = 1 , 2 , … } .

Maximal avalanches in the Bak-Sneppen model

We study the durations of the avalanches in the maximal avalanche decomposition of the Bak-Sneppen evolution model. We show that all the avalanches in this maximal decomposition have infinite expectation, but only ‘barely’, in the sense that if we made the appropriate threshold a tiny amount smaller (in a certain sense), then the avalanches would have finite expectation. The first of these results is somewhat surprising, since simulations suggest finite expectations.

Maximal avalanches in the Bak-Sneppen model

We study the durations of the avalanches in the maximal avalanche decomposition of the Bak-Sneppen evolution model. We show that all the avalanches in this maximal decomposition have infinite expectation, but only ‘barely’, in the sense that if we made the appropriate threshold a tiny amount smaller (in a certain sense), then the avalanches would have finite expectation. The first of these results is somewhat surprising, since simulations suggest finite expectations.

Functional renormalization group with vacuum expectation values and spontaneous symmetry breaking

We generalize our recently developed super-field functional renormalization group (RG) method involving both Fermi and Bose fields [F. Schuetz, L. Bartosch, and P. Kopietz, Phys. Rev. B 72, 035105 (2005)] to include the possibility that some bosonic components of the field have a finite vacuum expectation value. We derive an exact hierarchy of flow equations for the one-line irreducible vertices and the vacuum expectation value of the field. We apply our method to an interacting Fermi system where the interaction can be decoupled in the zero-sound channel and is then mediated by a collective bosonic field. The vacuum expectation value of the zero-frequency and zero-momentum component of the bosonic field is then closely related to the fermionic density. This can be exploited to calculate the compressibility of the interacting system. By using a cutoff in the bosonic propagator, the RG can be set up such that the self-consistent Hartree approximation is imposed as initial condition for the RG flow and the corrections to this approximation are generated as the remaining degrees of freedom are successively eliminated by the RG procedure.