The invariance principle for ϕ-mixing sequences

Summary In this paper we investigate the invariance principle for -mixing sequences, satisfying restrictions on the variances which are a weak form of stationarity. No mixing rate is assumed. For -mixing strictly stationary sequences we give a necessary and sufficient condition for the invariance principle.     

A self-normalized invariance principle for a ϕ-mixing sequence

In this paper, we investigate a self-normalized invariance principle for a ?-mixing stationary sequence {X _j , j ≥ 1} of random variables such that L(x):= E(X ₁ ² I{|X ₁| ≤ x}) is a slowly varying function at ∞.     

The convergence of moments in the central limit theorem for stationary ϕ-mixing processes

Пусть {X_j} - строго стац ионарная последоват ельностьс ?перемешиванием, EXj-Q,E¦-X _j¦^r<∞ для некоторогоr>2. Положим \(S_n = \mathop \sum \limits_{j = 1}^n X_j \) . Ибрагимов (1962) доказал, что если приn →∞, то 1 $$\mathop {\lim }\limits_{n \to \infty } P\{ S_n /\sigma _n< x\} = (2\pi )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \mathop \smallint \limits_{ - \infty }^x e^{{{ - u^2 } \mathord{\left/ {\vphantom {{ - u^2 } 2}} \right. \kern-\nulldelimiterspace} 2}} du.$$ В работе установлено, что при указанных выш е условиях в этой центральной пр едельной теореме имеет место т акже и сходимостьr-ых абсолютных моментов, т.е. если σ _n ² →∞ приn→ ∞, то $$\mathop {\lim }\limits_{n \to \infty } E|S_n /\sigma _n |^r = (2\pi )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \mathop \smallint \limits_{ - \infty }^{ + \infty } |u|^r e^{ - u^2 /2} du.$$ Этот результат обобщ ает один более ранний результат автора (1980 г.).     

On the Picard principle for Δ?+?μ

Given a (local) Kato measure?μ on ${{\mathbb{R}^d} \setminus \{0\},\,d \ge 2}$ , let ${{\mathcal H}_0^{\Delta+\mu}(U)}$ be the convex cone of all continuous real solutions u?≥ 0 to the equation Δu?+?u μ?=?0 on the punctured unit ball U satisfying ${\lim_{|x|\to 1} u(x)=0}$ . It is shown that ${{\mathcal H}_0^{\Delta+\mu}(U)\ne \{0\}}$ if and only if the operator ${f\mapsto \int_U G(\cdot,y)f(y)\,d\mu(y)}$ , where G denotes the Green function on U, is bounded on ${\mathcal L^2(U,\mu)}$ and has a norm which is at most one. Moreover, extremal rays in ${{\mathcal H}_0^{\Delta+\mu}(U)}$ are characterized and it is proven that Δ?+?μ satisfies the Picard principle on U, that is, that ${{\mathcal H}_0^{\Delta+\mu}(U)}$ consists of one ray, provided there exists a suitable sequence of shells in U such that, on these shells,?μ is either small or not too far from being radial. Further, it is shown that the verification of the Picard principle can be localized. Several results on L ²-(sub)eigenfunctions and 3G-inequalities which are used in the paper, but may be of independent interest, are proved at the end of the paper.     

Moderate deviations and functional limits for random processes with stationary and independent increments

We first give a functional moderate deviation principle for random processes with stationary and independent increments under the Ledoux's condition. Then we apply the result to the functional limits for increments of the processes and obtain some Csorgo-Revesz type functional laws of the iterated logarithm.     

On the hyperdeterminant for 2×2×3 arrays

We use the representation theory of Lie algebras and computational linear algebra to determine the simplest non-constant invariant polynomial in the entries of a general 2?×?2?×?3 array. This polynomial is homogeneous of degree 6 and has 66 terms with coefficients ±1, ±2 in the 12 indeterminates x _ijk where i, j?=?1,?2 and k?=?1,?2,?3. This invariant can be regarded as a natural generalization of Cayley's hyperdeterminant for 2?×?2?×?2 arrays.     

On the “bang-bang” principle for nonlinear evolution inclusions

Summary In this paper we establish the existence of extremal solutions for a class of nonlinear evolution inclusions defined on an evolution triple of Hilbert spaces. Then we show that these extremal solutions are in fact dense in the solutions of the original system. Subsequently we use this density result to derive nonlinear and infinite dimensional versions of the bang-bang principle for control systems. An example of a nonlinear parabolic distributed parameter system is also worked out in detail.     

Pontryagin's maximum principle for optimal control of the stationary Navier–Stokes equations

This paper deals with Pontryagin's maximum principle of the optimal control governed by stationary Navier–Stokes equation. Some kind of state constraint is involved.     

Lévy processes with values in a space of continuous functions arising as limits of independent triangular arrays

In this paper we investigate the limit distribution of the functions of independent triangular arrays X_nj, 1≤j≤k(n), n≥1. According to LeCam's theorem, if f belongs to the class of functionsPD[0,2] (which is slightly weaker than the assumptions that f(0)=0, and f has the second derivative at zero), then the distribution of is shift convergent. He also gives the explicit form of the characteristic function of the limit infinitely divisible distribution. We consider the class of functionsPD[0,1] and prove a similar statement. Since in the definition of the sequence of centering constants the truncation points depend only on the value of X_nj and not on the function f, this makes the analysis of the joint distribution of random variables in the above form considerably easier. Also we analyze the process of partial sums , 0≤u≤1. where f(x,t) is a parametric family of functions depending continuously on the parameter t. In the case of power functions we give an explicit representation of the limit process in term of Poissonian integrals. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II. Eger. Hungary. 1994.     

On scaling invariance and type-Ⅰ singularities for the compressible Navier-Stokes equations

We find a new scaling invariance of the barotropic compressible Navier-Stokes equations. Then it is shown that type-Ⅰ singularities of solutions with■ can never happen at time T for all adiabatic number γ 1. Here κ 0 does not depend on the initial data.This is achieved by proving the regularity of solutions under■ This new scaling invariance also motivates us to construct an explicit type-Ⅱ blowup solution for γ 1.     

On the stability of the Kalman–Bucy filter with stationary time varying parameters

Linear filtering systems with stationary parameters are considered in continuous time. We generalize the conditionally Gaussian result of Lipster – Shiryaev and Haussmann – Pardoux. Moreover, we show that the Kalman – Bucy filter is exponentially stable under weak conditions of stabile ability and detect ability. This filter will have a well defined asymptotic behaviour     

On the homogenization principle in a time-periodic problem for the Navier–Stokes equations with rapidly oscillating mass force

We study the behavior of the set of time-periodic solutions of the three-dimensional system of Navier–Stokes equations in a bounded domain as the frequency of the oscillations of the right-hand side tends to infinity. It is established that the set of periodic solutions tends to the solution set of the homogenized stationary equation.     

Complete convergence of weighted sums for arrays of rowwise φ-mixing random variables

In this paper, we establish the complete convergence and complete moment convergence of weighted sums for arrays of rowwise φ-mixing random variables, and the Baum-Katz-type result for arrays of rowwise φ-mixing random variables. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of φ-mixing random variables is obtained. We extend and complement the corresponding results of X. J. Wang, S. H. Hu (2012).     

On the Agmon-Miranda maximum principle for solutions of strongly elliptic equations in domains of ℝn with conical points

In this paper the Agmon-Miranda maximum principle for solutions of strongly elliptic differential equations Lu = 0 in a bounded domain G with a conical point is considered. Necessary and sufficient conditions for the validity of this principle are given both for smooth solutions of the equation Lu = 0 in G and for the generalized solution of the problem Lu = 0 in G, D _k ^v u = gk on G (k = 0,...,m-1). It will be shown that for every elliptic operator L of order 2m > 2 there exists such a cone in ⁿ (n4) that the Agmon-Miranda maximum principle fails in this cone.     

On some invariance of the quotient mean with respect to Makó–Páles means

Given a continuous strictly monotone function \(\varphi \) defined on an open real interval I and a probability measure \(\mu \) on the Borel subsets of [0, 1], the Makó–Páles mean is defined by

$$\begin{aligned} {\mathcal {M}}_{\varphi ,\mu }(x,y):=\varphi ^{-1}\left( \int ^1_0\varphi (tx+(1-t)y)\, d\mu (t)\right) ,\quad x,y\in I. \end{aligned}$$

Under some conditions on the functions \(\varphi \) and \(\psi \) defined on I, the quotient mean is given by

$$\begin{aligned} Q_{\varphi ,\psi }(x,y):=\left( \frac{\varphi }{\psi }\right) ^{-1}\left( \frac{\varphi (x)}{\psi (y)}\right) , \quad x,y\in I. \end{aligned}$$

In this paper, we study some invariance of the quotient mean with respect to Makó–Páles means.

     

On a Lower Bound of the Uniform Distance in the Central Limit Theorem for ϕ-Mixing Random Variables

We derive a lower bound of the uniform distance in the central limit theorem for real -mixing random variables under the finiteness of the eighth moments of summands. The main result of the present paper generalizes the corresponding author"s result obtained in 1997 for m-dependent random variables to the case of -mixing random variables.