Khinchin Formula Research Articles

On positivity preservers with constant coefficients and their generators

In this work we study positivity preservers T:R[x1,…,xn]→R[x1,…,xn] with constant coefficients and define their generators A if they exist, i.e., exp⁡(A)=T. We use the theory of regular Fréchet Lie groups to show the first main result. A positivity preserver with constant coefficients has a generator if and only if it is represented by an infinitely divisible measure (Main Theorem 4.7). In the second main result (Main Theorem 4.11) we use the Lévy–Khinchin formula to fully characterize the generators of positivity preservers with constant coefficients.

Sub-Feller semigroups generated by pseudodifferential operators on symmetric spaces of noncompact type

We consider global pseudodifferential operators on symmetric spaces of noncompact type, defined using spherical functions. The associated symbols have a natural probabilistic form that extend the notion of the characteristic exponent appearing in Gangolli's Lévy–Khinchine formula to a function of two variables. The Hille–Yosida–Ray theorem is used to obtain conditions on such a symbol so that the corresponding pseudodifferential operator has an extension that generates a sub-Feller semigroup, generalising existing results for Euclidean space.

The Speed of Convergence of the Threshold Estimator of Ruin Probability under the Tempered α-Stable Lévy Subordinator

In this paper, a nonparametric estimator of ruin probability is introduced in a spectrally negative Lévy process where the jump component is a tempered α-stable subordinator. Given a discrete record of high-frequency data, a threshold technique is proposed to estimate the mean of the jump size and use the Fourier transform and the Pollaczek–Khinchin formula to construct the estimator of ruin probability. The convergence rate of the integrated squared error for the estimator is studied.

Nonlinear free L\'evy--Khinchine formula and conformal mapping

There are two natural notions of L\'evy processes in free probability: the first one has free increments with homogeneous distributions and the other has homogeneous transition probabilities (P.~Biane, \textit{Math. Z.} {\bf 227}(1998), 143--174). In the two cases one can associate a Nevanlinna function to a free L\'evy process. The Nevanlinna functions appearing in the first notion were characterized by Bercovici and Voiculescu, \textit{Pacific J. Math.} {\bf 153}(1992), 217--248. I give an explicit parametrization for the Nevanlinna functions associated with the second kind of free L\'evy processes. This gives a nonlinear free L\'evy--Khinchine formula.

Olshanski spherical pairs of semigroups type

Let [Formula: see text] be the infinite semigroup, inductive limit of the increasing sequence of the semigroups [Formula: see text], where [Formula: see text] is the unitary group of matrices and [Formula: see text] is the semigroup of positive hermitian matrices. The main purpose of this work is twofold. First, we give a complete classification of spherical functions defined on [Formula: see text], by following a general approach introduced by Olshanski and Vershik.10 Second, we prove an integral representation for functions of positive-type analog to the Bochner–Godement theorem, and a Lévy–Khinchin formula for functions of negative type defined on [Formula: see text].

Generalised Pollaczek–Khinchin formula for the Polya/G/1 queue

A generalised Pollaczek–Khinchin formula for the Polya/G/1 queue, with a Polya peaked arrival process, general distributed service times, and infinite number of waiting positions, is obtained. It is shown that the peakedness of the number of arrivals and the variance of the service time lead to a significant increase in the service delay and queue length.

A multidimensional ruin problem and an associated notion of duality

ABSTRACTWe consider a dimensional general Cramer–Lundberg type network, with initial capital operating under a risk diversifying treaty; this is described in terms of the appropriate Skorokhod problem (SP) in determined by a reflection matrix R. Ruin of the insurance network is defined as the marginal deficit of each company being positive (and hence surplus of each company being 0) at the same time. The corresponding ruin problem is studied using the associated regulated random walk {(Y(a)n, Z(a)n)} with {Z(a)n} denoting the reflected part, and {Y(a)n} the pushing part. A dual process is introduced through time reversal at sample path level over finite time horizon; the stochastic analogue is again a regulated random walk in starting at 0; let {Wn} denote the reflected part of the dual regulated random walk. It is shown that ruin for the insurance network corresponds to {Wn} hitting the open upper orthant with vertex at R− 1a before hitting the boundary of even at the sample path level. Under natural hypotheses, we show that the ruin probability is for some n ⩾ 1) boundary hitting time of storage process) A notion of dimensional ladder height distribution is defined, and a Pollaczek–Khinchine formula is derived. An expression for the ladder height distribution is given. (One-dimensional ladder height distributions are known for about 50 years.) Some examples are presented.

A Lévy–Khinchin formula for the space of infinite dimensional square complex matrices

Using a generalized Bochner type representation for Olshanski spherical pairs, we establish a Lévy–Khinchin formula for the continuous functions of negative type on the space V∞=M(∞,C) of infinite dimensional square complex matrices relatively to the action of the product group K∞=U(∞)×U(∞). The space V∞ is the inductive limit of the spaces Vn=M(n,C), and the group K∞ is the inductive limit of the product groups Kn=U(n)×U(n), where U(n) is the unitary group.

ASPECTS OF RECURRENCE AND TRANSIENCE FOR LÉVY PROCESSES IN TRANSFORMATION GROUPS AND NONCOMPACT RIEMANNIAN SYMMETRIC PAIRS

AbstractWe study recurrence and transience for Lévy processes induced by topological transformation groups acting on complete Riemannian manifolds. In particular the transience–recurrence dichotomy in terms of potential measures is established and transience is shown to be equivalent to the potential measure having finite mass on compact sets when the group acts transitively. It is known that all bi-invariant Lévy processes acting in irreducible Riemannian symmetric pairs of noncompact type are transient. We show that we also have ‘harmonic transience’, that is, local integrability of the inverse of the real part of the characteristic exponent which is associated to the process by means of Gangolli’s Lévy–Khinchine formula.

Nonparametric estimate of the ruin probability in a pure-jump Lévy risk model

In this paper, we propose a nonparametric estimator of ruin probability in a Lévy risk model. The aggregate claims process X={Xt,≥0} is modeled by a pure-jump Lévy process. Assume that high-frequency observed data on X are available. The estimator is constructed based on the Pollaczek–Khinchin formula and Fourier transform. Risk bounds as well as a data-driven cut-off selection methodology are presented. Simulation studies are also given to show the finite sample performance of our estimator.

Prelimit and limit generalizations of the Pollaczek–Khinchin formula

Prelimit and limit generalizations of the Pollaczek–Khinchin formula

Lévy Khinchin Formula on Commutative Hypercomplex System

Abstract. A commutative hypercomplex system L 1 (Q,m) is, roughly speaking, a spacewhich is defined by a structure measure (c(A,B,r),(A,B ∈ β(Q)). Such space has beenstudied by Berezanskii and Krein. Our main purpose is to establish a generalization ofconvolution semigroups and to discuss the role of the L´evy measure in the L´evy-Khinchinrepresentation in terms of continuous negative definite functions on the dual hypercomplexsystem. 1. IntroductionThe integral representation of negative definite functions is known in the liter-ature as the L´evy-Khinchin formula. This was established for G= Rin the late1930’s by L´evy and Khinchin. It had been extended to Lie groups by Hunt [9] andby Parthasarathy et al [13] to locally compact abelian groups with a countable case.In 1969 Harzallah [7] gave a representation formula for an arbitrary locally compactabelian group. Hazod [8] obtained a L´evy-Khinchin formula for an arbitrary locallycompact group. The general L´evy-Khinchin formula and the special case, where theinvolution is identical are due to Berg [4]. Lasser [12] deduced the L´evy-Khinchinformula for commutative hypergroups. Now these contribution may be viewed asa L´evy-Khinchin formula for negative definite functions defined on commutativehypercomplex systems.Let Q be a complete separable locally compact metric space of pointsp,q,r··· ,β(Q) be the σ-algebra of Borel subsets, and β

An M/ G/1 queue with two phases of service subject to the server breakdown and delayed repair

This paper deals with the steady-state behaviour of an M/ G/1 queue with an additional second phase of optional service subject to breakdowns occurring randomly at any instant while serving the customers and delayed repair. This model generalizes both the classical M/ G/1 queue subject to random breakdown and delayed repair as well as M/ G/1 queue with second optional service and server breakdowns. For this model, we first derive the joint distributions of state of the server and queue size, which is one of chief objectives of the paper. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch as a classical generalization of Pollaczek–Khinchin formula. Next, we derive Laplace Stieltjes transform of busy period distribution and waiting time distribution. Finally, we obtain some important performance measures and reliability indices of this model.

Low-frequency noise spectrum of cyclo-stationary random telegraph signals

The noise spectrum of random telegraph signals (RTS) under cyclo-stationary excitation is evaluated through rigorous analytical calculations. First the autocorrelation of the RTS signal is calculated, and then the Wiener–Khinchin formula applied, leading to an analytical formulation for the RTS spectrum. The model is valid for any periodic excitation signal. Numerical simulations that corroborate the analytical results and explore the noise behavior under cyclo-stationary excitation are provided.

Lévy–Khinchin Formula and Existence of Densities for Convolution Semigroups on Symmetric Spaces

We prove the existence of a smooth density for a convolution semigroup on a symmetric space and obtain its spherical representation.

Sensitivity analysis on ruin probabilities with heavy-tailed claims

In this note, we consider the classical insurance risk model with heavy-tailed claim distributions. By using the Pollaczek–Khinchin Formula, we provide some sensitivity analysis on the ruin probability.

Ruin probabilities for competing claim processes

Let C1, C2,…,Cm be independent subordinators with finite expectations and denote their sum by C. Consider the classical risk process X(t) = x + ct - C(t). The ruin probability is given by the well-known Pollaczek–Khinchin formula. If ruin occurs, however, it will be caused by a jump of one of the subordinators Ci. Formulae for the probability that ruin is caused by Ci are derived. These formulae can be extended to perturbed risk processes of the type X(t) = x + ct - C(t) + Z(t), where Z is a Lévy process with mean 0 and no positive jumps.

Ruin probabilities for competing claim processes

Let C 1, C 2,…,C m be independent subordinators with finite expectations and denote their sum by C. Consider the classical risk process X(t) = x + ct - C(t). The ruin probability is given by the well-known Pollaczek–Khinchin formula. If ruin occurs, however, it will be caused by a jump of one of the subordinators C i . Formulae for the probability that ruin is caused by C i are derived. These formulae can be extended to perturbed risk processes of the type X(t) = x + ct - C(t) + Z(t), where Z is a Lévy process with mean 0 and no positive jumps.

Beurling–Deny formula of semi-Dirichlet forms

In this Note we announce a structure result for non-symmetric Dirichlet forms and semi-Dirichlet forms. Our result is regarded as an extension of the celebrated Beurling–Deny formula which is up to now available only for symmetric Dirichlet forms. The result can also be regarded as an extension of Lévy–Khinchine formula or more generally, an extension of Courrège's Theorem in the semi-Dirichlet forms setting. To cite this article: Z.-C. Hu, Z.-M. Ma, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

A single-server retrial queue with general retrial times and Bernoulli schedule

We consider an M/ G/1 retrial queue with general retrial times where the blocked customers either with probability q join the infinite waiting room (called priority queue) or with complementary probability p leave the service area and enter the retrial group (called orbit) in accordance with an FCFS discipline. We assume that only the customer at the head of the orbit is allowed to retry for service. We study the ergodicity of the embedded Markov chain, its stationary distribution function and the joint generating function of the number of customers in both groups in the steady-state regime. Moreover, we obtain the generating function of the system size distribution, which generalizes the well-known Pollaczek–Khinchin formula. Also we obtain the stochastic decomposition law and as an application we study the asymptotic behaviour under high rate of retrials. Besides the optimal control of the priority policy is investigated. The results agree with known special cases. Finally, numerical calculations are used to observe system performance.