Probabilistic Extensions of the Erdős–Ko–Rado Property

The classical Erdős–Ko–Rado (EKR) Theorem states that if we choose a family of subsets, each of size k, from a fixed set of size , then the largest possible pairwise intersecting family has size . We consider the probability that a randomly selected family of size t=t _n has the EKR property (pairwise nonempty intersection) as n and k=k _n tend to infinity, the latter at a specific rate. As t gets large, the EKR property is less likely to occur, while as t gets smaller, the EKR property is satisfied with high probability. We derive the threshold value for t using Janson’s inequality. Using the Stein–Chen method we show that the distribution of X ₀, defined as the number of disjoint pairs of subsets in our family, can be approximated by a Poisson distribution. We extend our results to yield similar conclusions for X _i , the number of pairs of subsets that overlap in exactly i elements. Finally, we show that the joint distribution (X ₀, X ₁, ..., X _b ) can be approximated by a multidimensional Poisson vector with independent components.      

Conditioned Diffusions which are Brownian Bridges

Let X _t be a one-dimensional diffusion of the form dX _t=dB _t+(X _t)dt. Let Tbe a fixed positive number and let be the diffusion process which is X _t conditioned so that X ₀=X _T=x. If the drift is constant, i.e., , then the conditioned diffusion process is a Brownian bridge. In this paper, we show the converse is false. There is a two parameter family of nonlinear drifts with this property.     

On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems

We study a quasilinear elliptic problem

Asymptotic properties of the norm of extremum values of normal random elements in the space C[0, 1]

We prove that

On Uniform Laws of Large Numbers for Ergodic Diffusions and Consistency of Estimators

Consider a regular diffusion process X with finite speed measure m. Denote the normalized speed measure by μ. We prove that the uniform law of large numbers holds if the class has an envelope function that is μ-integrable, or if is bounded in L _p(μ) for some p>1. In contrast with uniform laws of large numbers for i.i.d. random variables, we do not need conditions on the ‘size’ of the class in terms of bracketing or covering numbers. The result is a consequence of a number of asymptotic properties of diffusion local time that we derive. We apply our abstract results to improve consistency results for the local time estimator (LTE) and to prove consistency for a class of simple M-estimators. This revised version was published online in June 2006 with corrections to the Cover Date.     

Convergence rate to traveling waves for generalized Benjamin–Bona–Mahony–Burgers equations

This paper is concerned with the large time behavior of traveling wave solutions to the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers equations

Logarithmic Estimates for Submartingales and Their Differential Subordinates

In the paper we determine, for any K>0 and α∈[0,1], the optimal constant L(K,α)∈(0,∞] for which the following holds: If X is a nonnegative submartingale and Y is α-strongly differentially subordinate to X, then

Precise asymptotics in Chung’s law of the iterated logarithm

Let X, X1, X2,... be i.i.d, random variables with mean zero and positive, finite variance σ^2, and set Sn = X1 ＋... ＋ Xn, n≥1. The author proves that, if EX^2I{｜X｜≥t} = 0（（log log t）^-1） as t→∞, then for any a〉-1 and b〉 -1,lim ε↑1/√1＋a（1/√1＋a-ε）b＋1 ∑n=1^∞（logn）^a（loglogn）^b/nP{max κ≤n｜Sκ｜≤√σ^2π^2n/8loglogn（ε＋an）}=4/π（1/2（1＋a）^3/2）^b＋1 Г（b＋1）,whenever an = o（1/log log n）. The author obtains the sufficient and necessary conditions for this kind of results to hold.     

Properties of kagi and renko moments for homogeneous diffusion processes

For a homogeneous diffusion process (X _t ) _t?0, we consider problems related to the distribution of the stopping times $\begin{gathered} \gamma _{\max } = \inf \{ t \geqslant 0:\mathop {\sup }\limits_{s \leqslant t} X_s - X_t \geqslant H\} ,\gamma _{\min } = \inf \{ t \geqslant 0:X_t - \mathop {\inf }\limits_{s \leqslant t} X_s \geqslant H\} , \hfill \\ \kappa _0 = \inf \{ t \geqslant 0:\mathop {\sup }\limits_{s \leqslant t} X_s - \mathop {\inf }\limits_{s \leqslant t} X_s \geqslant H\} . \hfill \\ \end{gathered} $ . The results obtained are used to construct an inductive procedure allowing us to find the distribution of the increments of the process X between two adjacent kagi and renko instants of time.     

The precise asymptotics of the complete convergence for moving average processes of <Emphasis Type="Italic">m</Emphasis>-dependent <Emphasis Type="Italic">B</Emphasis>-valued elements

Let {εt;t ∈ Z} be a sequence of m-dependent B-valued random elements with mean zeros and finite second moment. {a3;j ∈ Z} is a sequence of real numbers satisfying ∑j=-∞^∞｜aj｜ 〈 ∞. Define a moving average process Xt = ∑j=-∞^∞aj＋tEj,t ≥ 1, and Sn = ∑t=1^n Xt,n ≥ 1. In this article, by using the weak convergence theorem of { Sn/√ n _〉 1}, we study the precise asymptotics of the complete convergence for the sequence {Xt; t ∈ N}.     

Drift estimation for a periodic mean reversion process

In this paper we propose a periodic, mean-reverting Ornstein–Uhlenbeck process of the form

An extension of the Hoeffding inequality to unbounded random variables

Let S = X ₁ + ⋯ + X _n be a sum of independent random variables such that 0 ⩽ X _k ⩽ 1 for all k. Write p = E S/n and q = 1 − p. Let 0 < t < q. In this paper, we extend the Hoeffding inequality [16, Theorem 1]

Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales

By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria and study the asymptotic behavior of the nonoscillatory solutions of the second-order nonlinear neutral delay dynamic equation

Symmetric Tensors And Geometry of {\mathbb{P}} ^N Subvarieties

This paper using a geometric approach produces vanishing and nonvanishing results concerning the spaces of twisted symmetric differentials on subvarieties , with k ≤ m. Emphasis is given to the case of k = m which is special and whose nonvanishing results on the dimensional range dim X > 2/3(N − 1) are related to the space of quadrics containing X and the variety of all tangent trisecant lines of X. The paper ends with an application showing that the twisted symmetric plurigenera, along smooth families of projective varieties X_t are not invariant even for α arbitrarily large. Received: September 2006, Revision: May 2007, Accepted: June 2007     

Testing for Tail Independence in Extreme Value models

Let (X,Y) be a random vector which follows in its upper tail a bivariate extreme value distribution with reverse exponentialmargins. We show that the conditional distribution function (df) of X + Y, given that X + Y>c, converges to the df F (t) = t ², , as if and only if X,Y are tail independent. Otherwise, the limit is F (t) = t. This is utilized to test for the tail independence of X, Y via various tests, including the one suggested by the Neyman–Pearson lemma. Simulations show that the Neyman–Pearson test performs best if the threshold c is close to 0, whereas otherwise it is the Kolmogorov–Smirnov test that performs best. The mathematical conditions are studied under which the Neyman–Pearson approach actually controls the type I error. Our considerations are extended to extreme value distributions in arbitrary dimensions as well as to distributions which are in a differentiable spectral neighborhood of an extreme value distribution.     

Nonparametric estimation of trend for stochastic differential equations driven by fractional Brownian motion

Consider a stochastic process {X _t , 0 ≤ t ≤ T} governed by a stochastic differential equation given by

Existence of limits of maximal means

For the class II(ℝ ^m ) of continuous almost periodic functionsf: ℝ ^m → ℝ, we consider the problem of the existence of the limit

Asymptotic Independence of Distant Partial Sums of Linear Processes

We investigate the joint weak convergence (f.d.d. and functional) of the vector-valued process (U _n ⁽¹⁾ (τ), U _n ⁽²⁾ (τ)) for τ ∈ [0, 1], where and are normalized partial-sum processes separated by a large lag m, m/n → ∞, and (X _t , t ∈ ℤ) is a stationary moving-average process with i.i.d. (or martingale-difference) innovations having finite variance. We consider the cases where (X _t ) is a process with long memory, short memory, or negative memory. We show that, in all these cases, as n → ∞ and m/n → ∞, the bivariate partial-sum process (U _n ⁽¹⁾ (τ), U _n ⁽²⁾ (τ)) tends to a bivariate fractional Brownian motion with independent components. The result is applied to prove the consistency of certain increment-type statistics in moving-average observations. This work supported by the joint Lithuania-French research program Gilibert. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 479–500, October–December, 2005.     

Exponentiating a Bundle of Linear Operators

This paper deals with the concept of exponentiability for a special class of multivalued maps. To be more precise, we discuss the exponentiability of a multivalued map F: X⇉X expressible in the form F(x) = {Ax:A ∈ Ξ}, with Ξ denoting a collection of linear continuous operators defined on a Banach space X. Among other results, we prove that, under suitable assumptions on Ξ, the Painlevé–Kuratowski limit

The Factor Decomposition Theorem of Bounded Generalized Inverse Modules and Their Topological Continuity

In this paper we obtain a Douglas type factor decomposition theorem about certain important bounded module maps. Thus, we come to the discussion of the topological continuity of bounded generalized inverse module maps. Let X be a topological space, x →Tx ： X→L（E） be a continuous map, and each R（Tx） be a closed submodule in E, for every fixed x C X. Then the map x→ Tx^＋： X→L（E） is continuous if and only if ｜｜Tx^＋｜｜ is locally bounded, where Tx^＋ is the bounded generalized inverse module map of Tx. Furthermore, this is equivalent to the following statement： For each x0 in X, there exists a neighborhood ∪0 at x0 and a positive number λ such that （0, λ^2）lohtatn in ∩x∈∪0C／σ（Tx^＋Tx）, where a（T） denotes the spectrum of operator T.