Deterministic equivalents of additive functionals of recurrent diffusions and drift estimation

Let X _t be a one-dimensional Harris recurrent diffusion, with a drift depending on an unknown parameter θ belonging to some metric compact Θ. We firstly show that all integrable additive functionals of X _t are asymptotically equivalent in probability to some deterministic process v _t . Then we use this result to study the behavior of the maximum likelihood estimator for the parameter θ. Under mild regularity assumptions, we find an upper rate of its convergence as a function of v _t , extending some recent results for ergodic diffusions.      

A generalization of the rogosinski-rogosinski theorem

We establish necessary and sufficient conditions for numerical functions αj(x), j ∈ N, x ∈ X, under which the conditions K(f _j ⊂ K(f ₁) ∀j≥2 and yield The functions fj(x) are uniformly bounded on the set X and take values in a boundedly compact space L, and K(fj) is the kernel of the function fj. The well-known Rogosinski-Rogosinski theorem follows from the proved statements in the case where X = N, α _j (x) ≡ α_j, and the space L is the m-dimensional Euclidean space.     

Approximations for a Three Dimensional Scan Statistic

Let X _ijk ,1 ≤ i ≤ N ₁,1 ≤ j ≤ N ₂, 1 ≤ k ≤ N ₃ be a sequence of independent and identically distributed 0 − 1 Bernoulli trials. X _ijk  = 1 if an event has occurred at the i,j,k ^th location in a three dimensional rectangular region and X _ijk  = 0, otherwise. For 2 ≤ m _j  ≤ N _j  − 1,1 ≤ j ≤ 3, a three dimensional discrete scan statistic is defined as the maximum number of events in any m ₁×m ₂×m ₃ rectangular sub-region in the entire N ₁×N ₂×N ₃ rectangular region. In this article, a product-type approximation and three Poisson approximations are derived for the distribution of this three dimensional scan statistic. Numerical results are presented to evaluate the accuracy of these approximations and their use in testing for randomness.     

Small deviations of the maximal element of a sequence of weighted independent random variables

Consider a sequence {X _i } of independent copies of a nonnegative random variable X and let M = sup _{j ≥ 1}λ _j X _j , where {λ _j } is a nonincreasing sequence of positive numbers for which P(M < ∞) = 1. The asymptotic behavior of -logP(M < r) as r → 0 is studied.     

Limiting distribution of sums of nonnegative stationary random variables

Summary Let {X _n,j,−∞<j<∞∼,n≧1, be a sequence of stationary sequences on some probability space, with nonnegative random variables. Under appropriate mixing conditions, it is shown thatS _n=X_n,1+…+X _n,n has a limiting distribution of a general infinitely divisible form. The result is applied to sequences of functions {f _n(x)∼ defined on a stationary sequence {X _j∼, whereX _n.f=f_n(X_j). The results are illustrated by applications to Gaussian processes, Markov processes and some autoregressive processes of a general type. This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Sciences Foundation, Grant MCS 82-01119.     

Record values and the exponential distribution

A sequence {X _n,n≧1} of independent and identically distributed random variables with continuous cumulative distribution functionF(x) is considered.X _j is a record value of this sequence ifX _j>max (X ₁, …,X _j−1). Let {X _L(n) n≧0} be the sequence of such record values. Some properties ofX _L(n) andX _L(n)−X_L(n−1) are studied when {X _n,n≧1} has the exponential distribution. Characterizations of the exponential distribution are given in terms of the sequence {X _L(n),n≧0} The work was partly completed when the author was at the Department of Statistics, University of Brasilia, Brazil.     

On the Potential Theory of One-dimensional Subordinate Brownian Motions with Continuous Components

Suppose that S is a subordinator with a nonzero drift and W is an independent 1-dimensional Brownian motion. We study the subordinate Brownian motion X defined by X _t  = W(S _t ). We give sharp bounds for the Green function of the process X killed upon exiting a bounded open interval and prove a boundary Harnack principle. In the case when S is a stable subordinator with a positive drift, we prove sharp bounds for the Green function of X in (0, ∞ ), and sharp bounds for the Poisson kernel of X in a bounded open interval.     

Convergence of Point Processes with Weakly Dependent Points

For each n≥1, let {X _j,n }_1≤j≤n be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process $N_{n}=\sum_{j=1}^{n}\delta_{X_{j,n}}For each n≥1, let {X _j,n }_1≤j≤n be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process N_n=?_j=1ⁿd_Xj,nN_{n}=\sum_{j=1}^{n}\delta_{X_{j,n}} to an infinitely divisible point process. From the point process convergence we obtain the convergence in distribution of the partial sum sequence S _n =∑ _j=1 ⁿ X _j,n to an infinitely divisible random variable whose Lévy measure is related to the canonical measure of the limiting point process. As examples, we discuss the case of triangular arrays which possess known (row-wise) dependence structures, like the strong mixing property, the association, or the dependence structure of a stochastic volatility model.     

On a characterization of the exponential distribution by spacings

Summary LetX be a non-negative random variable with probability distribution functionF. SupposeX _i,n (i=1,…,n) is theith smallest order statistics in a random sample of sizen fromF. A necessary and sufficient condition forF to be exponential is given which involves the identical distribution of the random variables (n−i)(X _i+1,n−X_i,n) and (n−j)(X _j+1,n−X_j,n) for somei, j andn, (1≦i<j<n). The work was partly completed when the author was at the Dept. of Statistics, University of Brasilia, Brazil.     

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     

Modeling teletraffic arrivals by a Poisson cluster process

In this paper we consider a Poisson cluster process N as a generating process for the arrivals of packets to a server. This process generalizes in a more realistic way the infinite source Poisson model which has been used for modeling teletraffic for a long time. At each Poisson point Γ _j , a flow of packets is initiated which is modeled as a partial iid sum process , with a random limit K _j which is independent of (X _ji ) and the underlying Poisson points (Γ _j ). We study the covariance structure of the increment process of N. In particular, the covariance function of the increment process is not summable if the right tail P(K _j > x) is regularly varying with index α∊ (1, 2), the distribution of the X _ji ’s being irrelevant. This means that the increment process exhibits long-range dependence. If var(K _j ) < ∞ long-range dependence is excluded. We study the asymptotic behavior of the process (N(t)) _{t≥ 0} and give conditions on the distribution of K _j and X _ji under which the random sums have a regularly varying tail. Using the form of the distribution of the interarrival times of the process N under the Palm distribution, we also conduct an exploratory statistical analysis of simulated data and of Internet packet arrivals to a server. We illustrate how the theoretical results can be used to detect distribution al characteristics of K _j , X _ji , and of the Poisson process. AMS Subject Classifications Primary—60K30; Secondary—60K25 A large part of this research was done with support of Institut Mittag-Leffler of the Royal Swedish Academy of Sciences when the authors participated in the Fall 2004 program on Queuing Theory and Teletraffic Theory. Mikosch’s research is also partially supported by MaPhySto, the Danish research network for mathematical physics and stochastics and the Danish Research Council (SNF) Grant No 21-04-0400. Samorodnitsky’s research is also partially supported by NSF grant DMS-0303493 and NSA grant MSPF-02G-183 at Cornell University. González-Arévalo’s research is partially supported by BoRSF grant LEQSF(2004-2007)-RD-A-31 at the University of Louisiana at Lafayette.     

Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

The dynamical behavior of multi-spot solutions in a two-dimensional domain Ω is analyzed for the two-component Schnakenburg reaction–diffusion model in the singularly perturbed limit of small diffusivity ε for one of the two components. In the limit ε→0, a quasi-equilibrium spot pattern in the region away from the spots is constructed by representing each localized spot as a logarithmic singularity of unknown strength S _j for j=1,…,K at unknown spot locations x _j ∈Ω for j=1,…,K. A formal asymptotic analysis, which has the effect of summing infinite logarithmic series in powers of −1/log ε, is then used to derive an ODE differential algebraic system (DAE) for the collective coordinates S _j and x _j for j=1,…,K, which characterizes the slow dynamics of a spot pattern. This DAE system involves the Neumann Green’s function for the Laplacian. By numerically examining the stability thresholds for a single spot solution, a specific criterion in terms of the source strengths S _j , for j=1,…,K, is then formulated to theoretically predict the initiation of a spot-splitting event. The analytical theory is illustrated for spot patterns in the unit disk and the unit square, and is compared with full numerical results computed directly from the Schnakenburg model.      

Generation of Hauptmoduln of Γ<Subscript>1</Subscript>(<Emphasis Type="Italic">N</Emphasis>) by Weierstrass units and application to class fields

We show that the modular functions j _1,N generate function fields of the modular curve X ₁(N), N ∈ {7; 8; 9; 10; 12}, and apply them to construct ray class fields over imaginary quadratic fields.     

Small deviation probabilities for sums of independent positive random variables

In this note, we give estimates of small deviation probabilities of the sum ∑_j≥1 λ_j X_j, where {λ_j} are nonnegative numbers and {X_j} are i.i.d. positive random variables that satisfy mild assumptions at zero and infinity. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 151–167.     

Exact laws for sums of ratios of order statistics from the Pareto distribution

Consider independent and identically distributed random variables {X _nk, 1 ≤ k ≤ m, n ≤ 1} from the Pareto distribution. We select two order statistics from each row, X _n(i) ≤ X _n(j), for 1 ≤ i < j ≤ = m. Then we test to see whether or not Laws of Large Numbers with nonzero limits exist for weighted sums of the random variables R _ij = X _n(j)/X _n(i).     

Uniform bound for asymptotic normality of discounted <Emphasis Type="Italic">m</Emphasis>-dependent random variables using Heinrich's method

We estimate the difference | F_Zv(x) - F(x) | \left| {{F_{{Z_v}}}(x) - \Phi (x)} \right| , where F_Zv(x) {F_{{Z_v}}}(x) is the distribution function of normalized series Z _v = B ⁻¹ _v Σ^∞ _j=0 v ^j X _j with B ² _v = \mathbb E \mathbb {E} (Σ^∞ _j=0 v ^j X _j ) > 0 and the discount factor v, 0 < v < 1; X ₀,X ₁,X ₂,… is a sequence of m-dependent random variables, and Φ(x) is the standard normal distribution function. In a particular case, the obtained upper bound is of order O((1−v)^1/2).     

A property of (v,k, λ)-designs

It is shown that if the subsetsX ₁,...,X _v of a setX form a (v, k, λ)-design, then there does not exist another subsetX _v+1 ofX havingany cardinalityk ₁ and intersecting each of theX _j, 1≦j≦v, inany number λ₁ of elements, where 0<k ₁<v and 0<λ₁<k (in order to avoid uninteresting cases).     

Weierstrass points on X 0(p ℓ) and arithmetic properties of Fourier coefficients of cusp forms

Generally it is unknown, whether or not ∞ is a Weierstrass point on the modular curve X ₀(N) if N is squarefree. A classical result of Atkin and Ogg states that ∞ is not a Weierstrass point on X ₀(N), if N=pM with p prime, p ∤ M and the genus of X ₀(M) zero. We use results of Kohnen and Weissauer to show that there is a connection between this question and the p-adic valuation of cusp forms under the Atkin–Lehner involution. This gives, in a sense, a generalization of Ogg’s Theorem in some cases.      

On the Kaplan–Meier Estimator of Long-Range Dependent Sequences

The limiting distributions are obtained for the Kaplan–Meier estimator of unknown distribution function of stationary time series of the form G(X _j), where X _j is stationary Gaussian process with long-range dependence and G(·) is non-random function.     

Asymptotic Behavior of the Maximum of Sums of I.I.D. Random Variables along Monotone Blocks

Let {X_i, Y_i}_i=1,2,... be an i.i.d. sequence of bivariate random vectors with P(Y₁ = y) = 0 for all y. Put M_n(j) = max_0≤k≤n-j (X_k+1 + ... X_k+j)I_k,j, where I_k,k+j = I{Y_k+1 < ⋯ < Y_k+j} denotes the indicator function for the event in brackets, 1 ≤ j ≤ n. Let L_n be the largest index l ≤ n for which I_k,k+l = 1 for some k = 0, 1, ..., n - l. The strong law of large numbers for “the maximal gain over the longest increasing runs,” i.e., for M_n(L_n) has been recently derived for the case where X₁ has a finite moment of order 3 + ε, ε > 0. Assuming that X₁ has a finite mean, we prove for any a = 0, 1, ..., that the s.l.l.n. for M(L_n - a) is equivalent to EX ₁ ^3+a I{X₁ > 0} < ∞. We derive also some new results for the a.s. asymptotics of L_n. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 179–189.