A Monte Carlo Estimation of the Entropy for Markov Chains

We introduce an estimate of the entropy of the marginal density p ^t of a (eventually inhomogeneous) Markov chain at time t≥1. This estimate is based on a double Monte Carlo integration over simulated i.i.d. copies of the Markov chain, whose transition density kernel is supposed to be known. The technique is extended to compute the external entropy , where the p ₁ ^t s are the successive marginal densities of another Markov process at time t. We prove, under mild conditions, weak consistency and asymptotic normality of both estimators. The strong consistency is also obtained under stronger assumptions. These estimators can be used to study by simulation the convergence of p ^t to its stationary distribution. Potential applications for this work are presented: (1) a diagnostic by simulation of the stability property of a Markovian dynamical system with respect to various initial conditions; (2) a study of the rate in the Central Limit Theorem for i.i.d. random variables. Simulated examples are provided as illustration.      

Parameter estimation for a misspecified arma model with infinite variance innovations

We consider a general linear model , where the innovations Z_t belong to the domain of attraction of an α-stable law for α<2, so that neither Z_t nor X_t have a finite variance. We do not assume that (X_t) is a standardARMA process of the form φ(B)X_t=ϕ(B)Z_t, but we fit anARMA process of a given order to the data X₁,...,X_n by estimating the coefficients of φ and ϕ. Given that (X_t) is anARMA process, it has been proved that the Whittle estimator is a consistent estimator of the true coefficients of ϕ and φ. Moreover, it then has a heavytailed limit distribution and the rate of convergence is (n/logn)^1/α, which compares favorably with the L² situation with rate . In this note we study the limit properties of the Whittle estimator when the underlying model is not necessarily anARMA process. Under general conditions we show that the Whittle estimate converges in probability. It converges weakly to a distribution which does not have a finite moment of order a and the rate of convergence is again (n/logn)^1/α. We also give an analytic expression for the limit distribution. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II, Eger, Hungary, 1994.     

The Increment Ratio statistic under deterministic trends

The Increment Ratio (IR) statistic (see (1.1) below) was introduced in Surgailis et al. [16]. The IR statistic can be used for testing nonparametric hypotheses for d-integrated (−1/2 < d < 5/4) behavior of time series, including short memory (d = 0), (stationary) long-memory (0 < d < 1/2), and unit roots (d = 1). For stationary/stationary increment Gaussian observations, in [16], a rate of decay of the bias of the IR statistic and a central limit theorem are obtained. In this paper, we study the asymptotic distribution of the IR statistic under the model X _t = X _t⁰ + g _N(t) (t = 1, …, N), where X _t⁰ is a stationary/stationary increment Gaussian process as in [16], and g _N(t) is a slowly varying deterministic trend. In particular, we obtain sufficient conditions on X _t⁰ and g _N(t) under which the IR test has the same asymptotic confidence intervals as in the absence of the trend. We also discuss the asymptotic distribution of the IR statistic under change-points in mean and scale parameters. Partially supported by the bilateral France-Lithuania scientific project Gilibert and Lithuanian State Science and Studies Foundation, grant No. T-25/08.     

Equilibrium fluctuations of asymmetric simple exclusion processes in dimension d≥3

We consider an asymmetric exclusion process in dimension d≥ 3 under diffusive rescaling starting from the Bernoulli product measure with density 0 < α < 1. We prove that the density fluctuation field Y ^N _t converges to a generalized Ornstein–Uhlenbeck process, which is formally the solution of the stochastic differential equatin dY _t = ?Y _t dt + dB ^∇ _t , where ? is a second order differential operator and B ^∇ _t is a mean zero Gaussian field with known covariances. Received: 31 May 1999 / Revised version: 15 June 2000 / Published online: 24 January 2001     

Iterated Brownian Motion in Parabola-Shaped Domains

Iterated Brownian motion Z_t serves as a physical model for diffusions in a crack. If τ_D(Z) is the first exit time of this processes from a domain D⊂ℝⁿ, started at z∈D, then P_z[τ_D(Z)>t] is the distribution of the lifetime of the process in D. In this paper we determine the large time asymptotics of which gives exponential integrability of for parabola-shaped domains of the form P_α={(x,Y)∈ℝ×ℝⁿ⁻¹:x>0, |Y|<Ax^α}, for 0<α<1, A>0. We also obtain similar results for twisted domains in ℝ² as defined in DeBlassie and Smits: Brownian motion in twisted domains, Preprint, 2004. In particular, for a planar iterated Brownian motion in a parabola we find that for z∈℘

I.i.d. representations for the bivariate product limit estimators and the bootstrap versions

Let F(s, t) = P(X > s, Y > t) be the bivariate survival function which is subject to random censoring. Let be the bivariate product limit estimator (PL-estimator) by Campbell and Földes (1982, Proceedings International Colloquium on Non-parametric Statistical Inference, Budapest 1980, North-Holland, Amsterdam). In this paper, it was shown that

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, where {ζ_i(s, t)} is i.i.d. mean zero process and R_n(s, t) is of the order O((n⁻¹log n)^3/4) a.s. uniformly on compact sets. Weak convergence of the process {n⁻¹ Σ_{i = 1}ⁿ ζ_i(s, t)} to a two-dimensional-time Gaussian process is shown. The covariance structure of the limiting Gaussian process is also given. Corresponding results are also derived for the bootstrap estimators. The result can be extended to the multivariate cases and are extensions of the univariate case of Lo and Singh (1986, Probab. Theory Relat. Fields, 71, 455–465). The estimator is also modified so that the modified estimator is closer to the true survival function than in supnorm.     

Regularity of weak solutions to the Navier–Stokes equations in exterior domains

Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u ₀, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u ₀ and f or on the solution u itself such as Serrin’s condition || u ||_{L^s(0,T; L^q(W))} < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||_{L^s￠(t-d,t; L^q(W))}{\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||₂²{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.     

Measure-valued Markov branching processes conditioned on non-extinction

We consider a particular class of measure-valued Markov branching processes that are constructed as “superprocesses” over some underlying Markov process. Such a processX dies out almost surely, so we introduce various conditioning schemes which keepX alive at large times. Under suitable hypotheses, which include the convergence of the semigroup for the underlying process to some limiting probability measureν, we show that the conditional distribution oft ⁻¹ X _t converges to that ofZν ast → ∞, whereZ is some strictly positive, real random variable. Research supported in part by NSF grant DMS 8701212. Research supported in part by an NSERC operating grant.     

The multiplier conjecture for elementary abelian groups

Applying the method that we presented in [19], in this article we prove: “Let G be an elementary abelian p-group. Let n = dn₁. If d(≠ p) is a prime not dividing n₁, and the order w of d mod p satisfies $ w > \frac{{d^2}}{3} $, then the Second Multiplier Theorem holds without the assumption n₁ > λ, except that only one case is yet undecided: w ≤ d², and $ \frac{{p - 1}}{{2w}} \ge 3 $, and t is a quadratic residue mod p, and t is not congruent to $ x^{\frac{{p - 1}}{{2w}}j} $ (mod p) (1 ≤ j < 2w), where t is an integer meeting the conditions of Second Multiplier Theorem, and x is a primitive root of p.”. © 1994 John Wiley & Sons, Inc.     

Stationary distributions of the simplest dynamical systems in R perturbed by generalized Poisson process

Let z(t) Rⁿ be a generalized Poisson process with parameter λ and let A: Rⁿ → Rⁿ be a linear operator. The conditions of existence and limiting properties as λ → ∞ or as λ → 0 of the stationary distribution of the process x(t) Rⁿ which satisfies the equation dx(t) = Ax(t)dt + dz(t) are investigated.     

Weak Convergence of a Nonlinear Functional of a Stationary Gaussian Process. Application to the Local Time

Let {X _t : 0 ≦ t ≦ 1} be a centered stationary Gaussian process, with correlation function satisfying the condition ρ(t) = 1 − t ^β L(t), 0 < β < 2, and let L be a slowly varying function at zero. Observing the process at points i/N, i = 0,1,..., N and considering |X _i/N − X _(i-1)/N | ^p with p > 0, we study the properties of the Donsker line associated with p-th order variations . We also study the relationship between the number of crossings of a regularization of the initial process and the local time of the initial process. The results depend on the values of β. This revised version was published online in June 2006 with corrections to the Cover Date.     

Confinement of Brownian motion among Poissonian obstacles in ℝ d , d≥ 3

Consider Brownian motion among random obstacles obtained by translating a fixed compact nonpolar subset of ℝ ^d , d≥ 1, at the points of a Poisson cloud of constant intensity v <: 0. Assume that Brownian motion is absorbed instantaneously upon entering the obstacle set. In SZN-conf Sznitman has shown that in d = 2, conditionally on the event that the process does not enter the obstacle set up to time t, the probability that Brownian motion remains within distance ∼t ^1/4 from its starting point is going to 1 as t goes to infinity. We show that the same result holds true for d≥ 3, with t ^1/4 replaced by t ^{1/( d +2)}. The proof is based on Sznitmans refined method of enlargement of obstacles [10] as well as on a quantitative isoperimetric inequality due to Hall [4]. Received: 6 July 1998     

Asymptotic closeness to limiting shapes for expanding embedded plane curves

We show that for embedded or convex plane curves expansion, the difference u(x,t)-r(t) in support functions between the expanding curves γ_t and some expanding circles C_t (with radius r(t)) has its asymptotic shape as t→∞. Moreover the isoperimetric difference L²-4πA is decreasing and it converges to a constant if the expansion speed is asymptotically a constant and the initial curve is not a circle. For convex initial curves, if the expansion speed is asymptotically infinite, then L²-4πA decreases to and there exists an asymptotic center of expansion for γ_t. Mathematics Subject Classification (2000) 35K15, 35K55     

A local limit theorem for the number of nodes,the height,and the number of final leaves in a critical branching process tree

Let Z_i be the number of particles in the ith generation of a non-degenerate critical Bienaymé-Galton-Watson process with offspring distribution $ p_r = P \{\hbox{a given individual has {\it r} children}\},\kern2em r\geq 0. $ Let ν = Σ^infinity₀ Z_j be the total progeny and let ζ = inf{r: Z_r = 0} be the extinction time. Equivalently, ν and ζ are the total number of nodes and (1 + the height), respectively, of the family tree of the branching process. Assume that E{Z₁} = Σ p_rr = 1 and E{Z₁^{3 + δ}} = Σ p_rr^{3 + δ} < infinity for some δ ϵ (0, 1). We find an asymptotic formula with remainder term for k⁴P{ζ = k + 1, Z_k = ℓ ν = n} when k→ infinity, which is uniform over n and ℓ. This is used to confirm a conjecture by Wilf that the number of leaves in the last generation of a randomly chosen rooted tree converges in distribution. More precisely, in the terminology introduced above, there exists a probability distribution {q₁} such that for n → infinity $ P\{Z_{\zeta-1} = l | \nu=n\} = q_l + O \left({{\log^3 n } \over {n^{1/2}}}\right), $ uniformly over ℓ ≥1. The limiting distribution is identified by means of a functional equation for the generating function Σ^infinity₁ q s^ℓ. Numerically, q₁ ≅ 0.0602, q₂ ≅ 0.248, q₃ ≅ 0.094, and q₄ ≅ 0.035. Our method can also be used to find lim _{k→ infinity} k⁴P{ζ = k + 1, Z_k = ℓ ν = n} when only E{Z₁^{2 + δ}} < infinity for some 0 ≤δ≤1, but we do not treat this case here; it goes without saying that the fewer moment assumptions one makes, the poorer the estimates become. © 1996 John Wiley & Sons, Inc.     

On deviations between empirical and quantile processes for mixing random variables

Let {X_n} be a strictly stationary φ-mixing process with Σ_j=1^∞ φ^1/2(j) < ∞. It is shown in the paper that if X₁ is uniformly distributed on the unit interval, then, for any t [0, 1], |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log log n)^3/4) a.s. and sup_0≤t≤1 |F_n⁻¹(t) − t + F_n(t) − t| = (O(n^−3/4(log n)^1/2(log log n)^1/4) a.s., where F_n and F_n⁻¹(t) denote the sample distribution function and tth sample quantile, respectively. In case {X_n} is strong mixing with exponentially decaying mixing coefficients, it is shown that, for any t [0, 1], |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log n)^1/2(log log n)^3/4) a.s. and sup_0≤t≤1 |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log n)(log log n)^1/4) a.s. The results are further extended to general distributions, including some nonregular cases, when the underlying distribution function is not differentiable. The results for φ-mixing processes give the sharpest possible orders in view of the corresponding results of Kiefer for independent random variables.     

The Blow-up Locus of Heat Flows for Harmonic Maps

Abstract Let M and N be two compact Riemannian manifolds. Let u _k (x, t) be a sequence of strong stationary weak heat flows from M×R ⁺ to N with bounded energies. Assume that u _k→u weakly in H ^{1, 2}(M×R ⁺, N) and that Σ^t is the blow-up set for a fixed t > 0. In this paper we first prove Σ^t is an H ^m−2-rectifiable set for almost all t∈R ⁺. And then we prove two blow-up formulas for the blow-up set and the limiting map. From the formulas, we can see that if the limiting map u is also a strong stationary weak heat flow, Σ^t is a distance solution of the (m− 2)-dimensional mean curvature flow [1]. If a smooth heat flow blows-up at a finite time, we derive a tangent map or a weakly quasi-harmonic sphere and a blow-up set ∪_t<0Σ^t× {t}. We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion. This work is supported by NSF grant     

Positive solutions for mixed problems of singular fractional differential equations

We investigate the existence of positive solutions to the singular fractional boundary value problem: $^c\hspace{-1.0pt}D^{\alpha }u +f(t,u,u^{\prime },^c\hspace{-2.0pt}D^{\mu }u)=0$, u′(0) = 0, u(1) = 0, where 1 < α < 2, 0 < μ < 1, f is a L^q‐Carathéodory function, $q > \frac{1}{\alpha -1}$, and f(t, x, y, z) may be singular at the value 0 of its space variables x, y, z. Here $^c \hspace{-1.0pt}D$ stands for the Caputo fractional derivative. The results are based on combining regularization and sequential techniques with a fixed point theorem on cones.     

Limiting size index distributions for ball-bin models with Zipf-type frequencies

We consider a random ball-bin model where balls are thrown randomly and sequentially into a set of bins. The frequency of choices of bins follows the Zipf-type (power-law) distribution; that is, the probability with which a ball enters the ith most popular bin is asymptotically proportional to 1/i ^α , α > 0. In this model, we derive the limiting size index distributions to which the empirical distributions of size indices converge almost surely, where the size index of degree k at time t represents the number of bins containing exactly k balls at t. While earlier studies have only treated the case where the power α of the Zipf-type distribution is greater than unity, we here consider the case of α ≤ 1 as well as α > 1. We first investigate the limiting size index distributions for the independent throw models and then extend the derived results to a case where bins are chosen dependently. Simulation experiments demonstrate not only that our analysis is valid but also that the derived limiting distributions well approximate the empirical size index distributions in a relatively short period.     

Renormalization and blow up for charge one equivariant critical wave maps

We prove the existence of equivariant finite time blow-up solutions for the wave map problem from ℝ²⁺¹→S ² of the form where u is the polar angle on the sphere, is the ground state harmonic map, λ(t)=t ^-1-ν, and is a radiative error with local energy going to zero as t→0. The number can be prescribed arbitrarily. This is accomplished by first “renormalizing” the blow-up profile, followed by a perturbative analysis. Mathematics Subject Classification (1991) 35L05, 35Q75, 35P25