A duality relation for entrance and exit laws for Markov processes

Markov processes X_t on (X, F_X) and Y_t on (Y, F_Y) are said to be dual with respect to the function f(x, y) if E_xf(X_t, y) = E_yf(x, Y_t for all x ? X, y ? Y, t ? 0. It is shown that this duality reverses the role of entrance and exit laws for the processes, and that two previously published results of the authors are dual in precisely this sense. The duality relation for the function f(x, y) = 1_{x<y} is established for one-dimensional diffusions, and several new results on entrance and exit laws for diffusions, birth-death processes, and discrete time birth-death chains are obtained.     

A note on random walk in random scenery

We consider a random walk in random scenery {Xn=η(S₀)+?+η(Sn),n∈N}, where a centered walk {Sn,n∈N} is independent of the scenery {η(x),x∈Zd}, consisting of symmetric i.i.d. with tail distribution P(η(x)>t)∼exp(−cαtα), with 1?α<d/2. We study the probability, when averaged over both randomness, that {Xn>ny} for y>0, and n large. In this note, we show that the large deviation estimate is of order exp(−ca(ny)), with a=α/(α+1).     

An inverse first-passage problem for one-dimensional diffusions with random starting point

We consider an inverse first-passage time (FPT) problem for a homogeneous one-dimensional diffusion X(t), starting from a random position η. Let S(t) be an assigned boundary, such that P(η≥S(0))=1, and F an assigned distribution function. The problem consists of finding the distribution of η such that the FPT of X(t) below S(t) has distribution F. We obtain some generalizations of the results of Jackson et al., 2009, which refer to the case when X(t) is Brownian motion and S(t) is a straight line across the origin.     

Crossing velocities for an annealed random walk in a random potential

We consider a random walk in an i.i.d. non-negative potential on the d-dimensional integer lattice. The walk starts at the origin and is conditioned to hit a remote location y on the lattice. We prove that the expected time under the annealed path measure needed by the random walk to reach y grows only linearly in the distance from y to the origin. In dimension 1 we show the existence of the asymptotic positive speed.     

On a continuous analogue of the stochastic difference equation Xn=ρXn-1+Bn

Let B₁, B₂, ... be a sequence of independent, identically distributed random variables, letX₀ be a random variable that is independent ofB_n forn?1, let ρ be a constant such that 0<ρ<1 and letX₁,X₂, ... be another sequence of random variables that are defined recursively by the relationshipsX_n=ρX_n-1+B_n. It can be shown that the sequence of random variablesX₁,X₂, ... converges in law to a random variableX if and only ifE[log⁺¦B₁¦]<∞. In this paper we let {B(t):0≦t<∞} be a stochastic process with independent, homogeneous increments and define another stochastic process {X(t):0?t<∞} that stands in the same relationship to the stochastic process {B(t):0?t<∞} as the sequence of random variablesX₁,X₂,...stands toB₁,B₂,.... It is shown thatX(t) converges in law to a random variableX ast →+∞ if and only ifE[log⁺¦B(1)¦]<∞ in which caseX has a distribution function of class L. Several other related results are obtained. The main analytical tool used to obtain these results is a theorem of Lukacs concerning characteristic functions of certain stochastic integrals.     

Catalytic branching random walks in ℤ d with branching at the origin

A time-continuous branching random walk on the lattice ? ^d , d ≥ 1, is considered when the particles may produce offspring at the origin only. We assume that the underlying Markov random walk is homogeneous and symmetric, the process is initiated at moment t = 0 by a single particle located at the origin, and the average number of offspring produced at the origin is such that the corresponding branching random walk is critical. The asymptotic behavior of the survival probability of such a process at moment t → ∞ and the presence of at least one particle at the origin is studied. In addition, we obtain the asymptotic expansions for the expectation of the number of particles at the origin and prove Yaglom-type conditional limit theorems for the number of particles located at the origin and beyond at moment t.     

Local times for stochastic processes which are subordinate to Gaussian processes

Let X and Y be random vectors of the same dimension such that Y has a normal distribution with mean vector O and covariance matrix R. Let g(x), x≥0, be a bounded nonincreasing function. X is said to be g-subordinate to Y if |Ee^iu′X| ≤ g(u′Ru) for all real vectors u of the same dimension as X. This is used to define the g-subordination of a real stochastic process X(t), 0 ≤ t ≤ 1, to a Gaussian process Y(t), 0 ≤ t ≤ 1. It is shown that the basic local time properties of a given Gaussian process are shared by all the processes that age g-subordinate to it. It is shown in particular that certain random series, including some random Fourier series, are g-subordinate to Gaussian processes, and so have their local time properties.     

Exact asymptotics of supremum of a stationary Gaussian process over a random interval

Let {X(t):t∈[0,∞)} be a centered stationary Gaussian process. We study the exact asymptotics of P(sup_s∈[0,T]X(s)>u), as u→∞, where T is an independent of {X(t)} nonnegative random variable. It appears that the heaviness of T impacts the form of the asymptotics, leading to three scenarios: the case of integrable T, the case of T having regularly varying tail distribution with parameter λ∈(0,1) and the case of T having slowly varying tail distribution.     

Exceptional times and invariance for dynamical random walks

Consider a sequence of i.i.d. random variables. Associate to each X _i (0) an independent mean-one Poisson clock. Every time a clock rings replace that X-variable by an independent copy and restart the clock. In this way, we obtain i.i.d. stationary processes {X _i (t)} _{t ≥0} (i=1,2,···) whose invariant distribution is the law ν of X ₁(0). Benjamini et al. (2003) introduced the dynamical walk S _n (t)=X ₁(t)+···+X _n (t), and proved among other things that the LIL holds for n↦S _n (t) for all t. In other words, the LIL is dynamically stable. Subsequently (2004b), we showed that in the case that the X _i (0)'s are standard normal, the classical integral test is not dynamically stable. Presently, we study the set of times t when n↦S _n (t) exceeds a given envelope infinitely often. Our analysis is made possible thanks to a connection to the Kolmogorov ɛ-entropy. When used in conjunction with the invariance principle of this paper, this connection has other interesting by-products some of which we relate. We prove also that the infinite-dimensional process converges weakly in to the Ornstein–Uhlenbeck process in For this we assume only that the increments have mean zero and variance one. In addition, we extend a result of Benjamini et al. (2003) by proving that if the X _i (0)'s are lattice, mean-zero variance-one, and possess (2+ɛ) finite absolute moments for some ɛ>0, then the recurrence of the origin is dynamically stable. To prove this we derive a gambler's ruin estimate that is valid for all lattice random walks that have mean zero and finite variance. We believe the latter may be of independent interest. The research of D. Kh. is partially supported by a grant from the NSF.     

About a construction and some analysis of time inhomogeneous diffusions on monotonely moving domains

We construct and analyze in a very general way time inhomogeneous (possibly also degenerate or reflected) diffusions in monotonely moving domains E⊂R×R^d, i.e. if E_t?{x∈R^d|(t,x)∈E}, t∈R, then either E_s⊂E_t, ∀s?t, or E_s⊃E_t, ∀s?t, s,t∈R. Our major tool is a further developed L²(E,m)-analysis with well chosen reference measure m. Among few examples of completely different kinds, such as e.g. singular diffusions with reflection on moving Lipschitz domains in R^d, non-conservative and exponential time scale diffusions, degenerate time inhomogeneous diffusions, we present an application to what we name skew Bessel process on γ. Here γ is either a monotonic function or a continuous Sobolev function. These diffusions form a natural generalization of the classical Bessel processes and skew Brownian motions, where the local time refers to the constant function γ≡0.     

On function spaces related to fractional diffusions on d-sets

We prove that all the Dirichlet forms associated with certain diffusions on a d-set are equivalent and that their common domain is an integral Lipschitz space. We also provide an analytic characterisation of the walk dimension d_w of a d-set F and show that all fractional diffusions on F share d_w as their walk dimension.     

On the limiting behavior of the characteristic function of the ergodic distribution of the semi-Markov walk with two boundaries

The semi-Markov walk (X(t)) with two boundaries at the levels 0 and β > 0 is considered. The characteristic function of the ergodic distribution of the processX(t) is expressed in terms of the characteristics of the boundary functionals N(z) and S _N(z), where N(z) is the firstmoment of exit of the random walk {Sn}, n ≥ 1, from the interval (?z, β ? z), z ∈ [0, β]. The limiting behavior of the characteristic function of the ergodic distribution of the process W _β (t) = 2X(t)/β ? 1 as β → ∞ is studied for the case in which the components of the walk (η _i) have a two-sided exponential distribution.     

Clustering of linearly interacting diffusions and universality of their long-time limit distribution

Let K⊂ℝ ^d (d≥ 1) be a compact convex set and Λ a countable Abelian group. We study a stochastic process X in K ^Λ, equipped with the product topology, where each coordinate solves a SDE of the form dX _i (t) = ∑ _j a(j−i) (X _j (t) −X _i (t))dt + σ (X _i (t))dB _i (t). Here a(·) is the kernel of a continuous-time random walk on Λ and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel a _S (i) = a(i) + a(−i) is recurrent, then each component X _i (∞) is concentrated on {x∈K : σ(x) = 0 and the coordinates agree, i.e., the system clusters. Both these statements fail if a _S is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components. This distribution turns out to be universal in all recurrent kernels a _S on Abelian groups Λ. Received: 10 May 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000     

Simulation de lois conjuguées et application à l'estimation de la transformée de Cramer dans ℝd

Let X, X₁, X₂,… be a sequence of i.i.d. ℝ^d-valued random variables with distribution F. An algorithm for the simulation of random vectors with distribution dF_t (x):= e〈t,x〉dF(x)/(t), where (t):= Ee^ť,X〉 (cf. [2]) is used for the estimation of the Cramer transform H (x):= sup_t (〈t, X〉 − log(t)). This method, which belongs to the class of “acceptance-rejection” techniques, is fast and uses a random sieve on the sequence (X_i)_{i ≥ 1}; it does not assume any prior knowledge on F or . We state the asymptotic properties of this estimator calculated on a n-sample of simulated r.v. 's with distribution F_t. We also present some numerical simulations.     

A law of the iterated logarithm for random walk in random scenery with deterministic normalizers

LetX, X _i ,i≥1, be a sequence of independent and identically distributed ? ^d -valued random vectors. LetS _o=0 and \(S_n = \sum\nolimits_{i = 1}^n {X_i } \) forn≤1. Furthermore letY, Y(α), α∈? ^d , be independent and identically distributed ?-valued random variables, which are independent of theX _i . Let \(Z_n = \sum\nolimits_{i = 0}^n {Y(S_i )} \) . We will call (Z _n ) arandom walk in random scenery. In this paper, we consider the law of the iterated logarithm for random walk in random scenery where deterministic normalizers are utilized. For example, we show that if (S _n ) is simple, symmetric random walk in the plane,E[Y]=0 andE[Y ²]=1, then $$\mathop {\overline {\lim } }\limits_{n \to \infty } \frac{{Z_n }}{{\sqrt {2n\log (n)\log (\log (n))} }} = \sqrt {\frac{2}{\pi }} a.s.$$      

On diffusivity of a tagged particle in asymmetric zero-range dynamics

Consider a distinguished, or tagged particle in zero-range dynamics on Zd with rate g whose finite-range jump probabilities p possess a drift ∑jp(j)≠0. We show, in equilibrium, that the variance of the tagged particle position at time t is at least order t in all d?1, and at most order t in d=1 and d?3 for a wide class of rates g. Also, in d=1, when the jump distribution p is totally asymmetric and nearest-neighbor, and the rate g(k) increases, and g(k)/k either decreases or increases with k, we show the diffusively scaled centered tagged particle position converges to a Brownian motion with a homogenized diffusion coefficient in the sense of finite-dimensional distributions. Some characterizations of the tagged particle variance are also given.     

The topology of continua that are approximated by disjoint subcontinua

Suppose that is a collection of disjoint subcontinua of continuum X such that limi→∞dH(Yi,X)=0 where dH is the Hausdorff metric. Then the following are true:

(1)

X is non-Suslinean.

(2)

If each Yi is chainable and X is finitely cyclic, then X is indecomposable or the union of 2 indecomposable subcontinua.

(3)

If X is G-like, then X is indecomposable.

(4)

If all lie in the same ray and X is finitely cyclic, then X is indecomposable.

     

Limit theorems for supremum of Gaussian processes over a random interval

Let {X(t), t ≥ 0} be a centered stationary Gaussian process with correlation r(t)such that 1-r(t) is asymptotic to a regularly varying function. With T being a nonnegative random variable and independent of X(t), the exact asymptotics of P(sup_(t∈[0,T])X(t) x) is considered, as x →∞.     

Asymptotics of random contractions

In this paper we discuss the asymptotic behaviour of random contractions X=RS, where R, with distribution function F, is a positive random variable independent of S∈(0,1). Random contractions appear naturally in insurance and finance. Our principal contribution is the derivation of the tail asymptotics of X assuming that F is in the max-domain of attraction of an extreme value distribution and the distribution function of S satisfies a regular variation property. We apply our result to derive the asymptotics of the probability of ruin for a particular discrete-time risk model. Further we quantify in our asymptotic setting the effect of the random scaling on the Conditional Tail Expectations, risk aggregation, and derive the joint asymptotic distribution of linear combinations of random contractions.     

Splitting trees with neutral Poissonian mutations I: Small families

We consider a neutral dynamical model of biological diversity, where individuals live and reproduce independently. They have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate b. Such a genealogical tree is usually called a splitting tree [9], and the population counting process (N_t;t≥0) is a homogeneous, binary Crump-Mode-Jagers process.We assume that individuals independently experience mutations at constant rate θ during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called an allele, to its carrier. We are interested in the allele frequency spectrum at time t, i.e., the number A(t) of distinct alleles represented in the population at time t, and more specifically, the numbers A(k,t) of alleles represented by k individuals at time t, k=1,2,…,N_t.We mainly use two classes of tools: coalescent point processes, as defined in [15], and branching processes counted by random characteristics, as defined in [11] and [13]. We provide explicit formulae for the expectation of A(k,t) conditional on population size in a coalescent point process, which apply to the special case of splitting trees. We separately derive the a.s. limits of A(k,t)/N_t and of A(t)/N_t thanks to random characteristics, in the same vein as in [19].Last, we separately compute the expected homozygosity by applying a method introduced in [14], characterizing the dynamics of the tree distribution as the origination time of the tree moves back in time, in the spirit of backward Kolmogorov equations.