On characterization of multivariate stable distributions via random linear statistics

LetX ₁,X ₂, ...,X _n be independent and identically distributed random vectors inR ^d , and letY=(Y ₁,Y ₂, ...,Y _n )′ be a random coefficient vector inR ⁿ , independent ofX _j ^/′ . We characterize the multivariate stable distributions by considering the independence of the random linear statistic $$U = Y_1 X_1 + Y_2 X_2 + \cdot \cdot \cdot + Y_n X_n $$ and the random coefficient vectorY.     

On infinite-rank singular perturbations of the Schrödinger operator

Schrödinger operators with infinite-rank singular potentials V=Σ _i,j=1 ^∞ b _ij〈φ_j,·〉φ_i are studied under the condition that the singular elements ψ _j are ξ _j(t)-invariant with respect to scaling transformationsin ?³.     

Padé tables of entire functions of very slow and smooth growth

Letf(z)=σ _j?o ^∞ a _j z ^j be entire with $$|a_{j - 1} a_{j + 1} /a_j^2 | \leqslant \rho _0^2 ,j = 1,2,3, \ldots ,$$ whereρ ₀=0.4559... is the positive root of the equation $$2\sum\limits_{j = 1}^\infty {\rho ^{j^2 } = 1.}$$ . It is shown that the Padé table off is normal, and asL→∞, [L/M _L ](z) converges uniformly in compact subsets ofC tof, for any sequence of nonnegative integers {M _L } _L=1 ^∞. In particular, the diagonal sequence {[L/L]} converges uniformly in compact subsets ofC tof. Furthermore, the constantρ ₀ is shown to be best possible in a strong sense.     

The Law of the Iterated Logarithm for Finitely Inhomogeneous Random Walks

The Hartman–Wintner–Strassen law of the iterated logarithm states that if X ₁, X ₂,… are independent identically distributed random variables and S _n =X ₁+???+X _n , then

$\limsup_{n}S_{n}/\sqrt{2n\log \log n}=1\quad \text{a.s.},\qquad \liminf_{n}S_{n}/\sqrt{2n\log \log n}=-1\quad \text{a.s.}$

if and only if EX ₁ ² =1 and EX ₁=0. We extend this to the case where the X _n are no longer identically distributed, but rather their distributions come from a finite set of distributions.

     

Time-Varying Fractionally Integrated Processes with Finite or Infinite Variance and Nonstationary Long Memory

Recently, Philippe et al. (C.R. Acad. Sci. Paris. Ser. I 342, 269–274, 2006; Theory Probab. Appl., 2007, to appear) introduced a new class of time-varying fractionally integrated filters A(d)x _t =∑ _j=0 ^∞ a _j (t)x _t?j , B(d)x _t =∑ _j=0 ^∞ b _j (t)x _t?j depending on arbitrary given sequence d=(d _t ,t∈?) of real numbers, such that A(d)^?1=B(?d), B(d)^?1=A(?d) and such that when d _t ≡d is a constant, A(d)=B(d)=(1?L) ^d is the usual fractional differencing operator. Philippe et al. studied partial sums limits of (nonstationary) filtered white noise processes X _t =B(d)ε _t and Y _t =A(d)ε _t in the case when (1) d is almost periodic having a mean value $\bar{d}\in (0,1/2)$ , or (2) d admits limits d _±=lim? _t→±∞ d _t ∈(0,1/2) at t=±∞. The present paper extends the above mentioned results of Philippe et al. into two directions. Firstly, we consider the class of time-varying processes with infinite variance, by assuming that ε _t ,t∈? are iid rv’s in the domain of attraction of α-stable law (1<α≤2). Secondly, we combine the classes (1) and (2) of sequences d=(d _t ,t∈?) into a single class of sequences d=(d _t ,t∈?) admitting possibly different Cesaro limits $\bar{d}_{\pm}\in(0,1-(1/\alpha))$ at ±∞. We show that partial sums of X _t and Y _t converge to some α-stable self-similar processes depending on the asymptotic parameters $\bar{d}_{\pm}$ and having asymptotically stationary or asymptotically vanishing increments.     

The Heyde theorem for locally compact Abelian groups

We prove a group analogue of the well-known Heyde theorem where a Gaussian measure is characterized by the symmetry of the conditional distribution of one linear form given another. Let X be a locally compact second countable Abelian group containing no subgroup topologically isomorphic to the circle group T, G be the subgroup of X generated by all elements of order 2, and Aut(X) be the set of all topological automorphisms of X. Let αj,βj∈Aut(X), j=1,2,…,n, n?2, such that for all i≠j. Let ξj be independent random variables with values in X and distributions μj with non-vanishing characteristic functions. If the conditional distribution of L₂=β₁ξ₁+?+βnξn given L₁=α₁ξ₁+?+αnξn is symmetric, then each μj=γj∗ρj, where γj are Gaussian measures, and ρj are distributions supported in G.     

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 ∞.     

On estimation of analytic density functions in L p

Let X ₁,X ₂, … be a sequence of independent identically distributed random variables with an unknown density function f on R. The function f is assumed to belong to a certain class of analytic functions. The problem of estimation of f using L _p -risk, 1 ≤ p < ∞, is considered. A kernel-type estimator f _n based on X ₁, …, X _n is proposed and the upper bound on its asymptotic local maximum risk is established. Our result is consistent with a conjecture of Guerre and Tsybakov [7] and augments previous work in this area.     

Precise large deviations for sums of random vectors with dependent components of consistently varying tails

Let {X _i = (X _1,i ,...,X _m,i )^?, i ≥ 1} be a sequence of independent and identically distributed nonnegative m-dimensional random vectors. The univariate marginal distributions of these vectors have consistently varying tails and finite means. Here, the components of X ₁ are allowed to be generally dependent. Moreover, let N(·) be a nonnegative integer-valued process, independent of the sequence {X _i , i ≥ 1}. Under several mild assumptions, precise large deviations for S _n = Σ _i=1 ⁿ X _i and S _N(t) = Σ _i=1 ^N(t) X _i are investigated. Meanwhile, some simulation examples are also given to illustrate the results.     

Expansion over the signum system on a sigma-finite space

A generalization of the notion of expansion over the signum system to the case of a sigma-finite space is considered in the paper. Let {E _k } _k=1 ^∞ be an exhaustion of X, then under the condition \(\sum\limits_{k = 1}^\infty {\tfrac{1}{{\mu E_k }}} = \infty \) the expansion converges to the expanded function in the metric of L ₂, and under the conditions \(\sum\limits_{k = 1}^\infty {\tfrac{1}{{\mu E_k }}} = \infty \) and \(\mathop {\lim }\limits_{k \to \infty } \tfrac{{\mu E_{k - 1} }}{{\mu E_k }} = 1\) the convergence holds almost everywhere for functions from L _∞(X).     

On weighted approximations in D[0, 1] with applications to self-normalized partial sum processes

Let X,X ₁,X ₂,… be a sequence of non-degenerate i.i.d. random variables with mean zero. The best possible weighted approximations are investigated in D[0, 1] for the partial sum processes {S _[nt], 0 ≦ t ≦ 1} where S _n = Σ _j=1 ⁿ X _j , under the assumption that X belongs to the domain of attraction of the normal law. The conclusions then are used to establish similar results for the sequence of self-normalized partial sum processes {S _[nt]=V _n , 0 ≦ t ≦ 1}, where V _n ² = Σ _j=1 ⁿ X _j ² . L _p approximations of self-normalized partial sum processes are also discussed.     

A class of Fibonacci-type sequences

Let {L_n:n ? 1} be a sequence of the form

where {a_j} and {b_j} are positive integers, and e=max_i,j{a_i, b_j}. A necessary and sufficient condition on the integers {a_j} and {b_j} is given so that, for all choices of positive initial values L₁, L₂,…,L_e,L_n=Σ^p_j=1L_n?aj for all large enough n.     

On the normal approximation of a binomial random sum

We present upper bounds of the L _s norms of the normal approximation for random sums of independent identically distributed random variables X ₁ , X ₂ , . . . with finite absolute moments of order 2 + δ, 0 < δ ≤ 1, where the number of summands N is a binomial random variable independent of the summands X ₁ , X ₂ , . . . . The upper bounds obtained are of order (E N) ^?δ/2 for all 1 ≤ s ≤ ∞.     

Erdős-Rényi-Shepp laws and weighted sums of independent identically distributed random variables

LetX ₀,X ₁,X ₂,... be i.i.d. random variables withE(X ₀)=0,E(X ₀ ² )=1,E(exp{tX _o}<∞ (|t|<t ₀) and partial sumsS _n . Starting from Shepp's version of the well-known Erd?s-Rényi-Shepp law $$\mathop {\lim }\limits_{n \to \infty } \sup ([c\log n])^{ - 1} )(S_{n + [c\log n]} - S_n ) = \alpha {\text{a}}{\text{.s}}{\text{.}}$$ where α is a number depending uponc and the distribution ofX ₀, we show that other weighted sumsV(n)=Σa _j (n)X _j exhibit a similar lim sup behavior, if the weights satisfy certain regularity conditions. We also prove for such weighted sums certain versions of the classical Erd?s-Rényi law.     

Characterization of similarities between two n-frames of projectors

An n-frame on a Banach space $\text{X}$ is E=(E₁,?, E_n) where the E_j's are bounded linear operators on $\text{X}$ such that E_j≠0,

$\text{∑}\text{j=1}\text{n}\text{E}{}_{\mathrm{j}}$

, and E_jE_k=δ_jkE_k (j, k=1,?, n). It is known that if two n-frames E and F are sufficiently close to each other, then they are similar, that is, F_j=TE_jT^-1 with T a bounded linear operator. Among the operators which realize the similarity of the two frames, there is the balanced transformation U(F, E)=(Σⁿ_j=1F_jE_j)(Σⁿ_j=1E_jF_jE_j)^{$\text{-1}\text{2}$}. One of our main results is a local characterization of the balanced transformation. Another operator which implements the similarity between E and F is the direct rotation R(F, E). It comes up in connection with the study of the set of all n-frames as a Banach manifold with an affine connection. Finally, it is shown that for quite a large set of pairs of 2-frames, the direct rotation has a global characterization.     

Absolutely continuous operators on function spaces and vector measures

Let (Ω, Σ, μ) be a finite atomless measure space, and let E be an ideal of L ⁰(μ) such that ${L^\infty(\mu) \subset E \subset L^1(\mu)}$ . We study absolutely continuous linear operators from E to a locally convex Hausdorff space ${(X, \xi)}$ . Moreover, we examine the relationships between μ-absolutely continuous vector measures m : Σ → X and the corresponding integration operators T _m : L ^∞(μ) → X. In particular, we characterize relatively compact sets ${\mathcal{M}}$ in ca _μ (Σ, X) (= the space of all μ-absolutely continuous measures m : Σ → X) for the topology ${\mathcal{T}_s}$ of simple convergence in terms of the topological properties of the corresponding set ${\{T_m : m \in \mathcal{M}\}}$ of absolutely continuous operators. We derive a generalized Vitali–Hahn–Saks type theorem for absolutely continuous operators T : L ^∞(μ) → X.     

Theorems on Large Deviations for Randomly Indexed Sum of Weighted Random Variables

In this paper, we consider a random variable \(Z_{t}=\sum_{i=1}^{N_{t}}a_{i}X_{i}\), where \(X, X_{1}, X_{2}, \ldots\) are independent identically distributed random variables with mean E X=μ and variance D X=σ ²>0. It is assumed that Z ₀=0, 0≤a _i <∞, and N _t , t≥0 is a non-negative integer-valued random variable independent of X _i , i=1,2,…?. The paper is devoted to the analysis of accuracy of the standard normal approximation to the sum \(\tilde{Z}_{t}=(\mathbf{D}Z_{t})^{-1/2}(Z_{t}-\mathbf{E}Z_{t})\), large deviation theorems in the Cramer and power Linnik zones, and exponential inequalities for \(\mathbf{P}(\tilde{Z}_{t}\geq x)\).     

Compactification of the space of branched coverings of the two-dimensional sphere

For a closed oriented surface Σ we define its degenerations into singular surfaces that are locally homeomorphic to wedges of disks. Let X_Σ,n be the set of isomorphism classes of orientation-preserving n-fold branched coverings Σ → S ² of the two-dimensional sphere. We complete X_Σ,n with the isomorphism classes of mappings that cover the sphere by the degenerations of Σ. In the case Σ = S ², the topology that we define on the obtained completion \({\overline X _{\Sigma ,n}}\) coincides on \({X_{{s^2},n}}\) with the topology induced by the space of coefficients of rational functions P/Q, where P and Q are homogeneous polynomials of degree n on ?P¹ ≌ S ². We prove that \({\overline X _{\Sigma ,n}}\) coincides with the Diaz–Edidin–Natanzon–Turaev compactification of the Hurwitz space H(Σ, n) ? X _Σ,n consisting of isomorphism classes of branched coverings with all critical values being simple.     

Separate asymptotics of two series of eigenvalues for a single elliptic boundary-value problem

The spectral problem in a bounded domain Ω?Rⁿ is considered for the equation Δu= λu in Ω, ?u=λ?υ/?ν on the boundary of Ω (ν the interior normal to the boundary, Δ, the Laplace operator). It is proved that for the operator generated by this problem, the spectrum is discrete and consists of two series of eigenvalues {λ _j ⁰ } _j=1 ^∞ and {λ _j ^∞ } _j=1 ^∞ , converging respectively to 0 and +∞. It is also established that $$N^0 (\lambda ) = \sum\nolimits_{\operatorname{Re} \lambda _j^0 \geqslant 1/\lambda } {1 \approx const} \lambda ^{n - 1} , N^\infty (\lambda ) \equiv \sum\nolimits_{\operatorname{Re} \lambda _j^\infty \leqslant \lambda } {1 \approx const} \lambda ^{n/1} .$$ The constants are explicitly calculated.