On the Problem of the Optimal Choice of Record Values

Let the independent random variables X1, X2, … have the same continuous distribution function. The upper record values X(1) = X₁ < X(2) < … generated by this sequence of variables, as well as the lower record values x(1) = X₁ > x(2) > …, are considered. It is known that in this situation, the mean value c(n) of the total number of the both types of records among the first n variables X is given by the equality c(n)=2(1+1/2+…+1/n), n = 1, 2, …. The problem considered here is following: how, sequentially obtaining the observed values x₁, x₂, … of variables X and selecting one of them as the initial point, to obtain the maximal mean value e(n) of the considered numbers of records among the rest random variables. It is not possible to come back to rejected elements of the sequence. Some procedures of the optimal choice of the initial element X _r are discussed. The corresponding tables for the values e(n) and differences δ(n)= e(n)–c(n) are presented for different values of n. The value of δ= lim_n→∞δ(n)is also given. In some sense, the considered problem and optimization procedure presented in this paper are quite similar to the classical “secretary problem,” in which the probability of selecting the last record value in the set of independent identically distributed X is maximized.     

Expansive homoclinic classes of generic <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-vector fields

Let γ be a hyperbolic closed orbit of a C ¹ vector field X on a compact C ^∞ manifold M of dimension n ≥ 3, and let H _X(γ) be the homoclinic class of X containing γ. In this paper, we prove that C ¹-generically, if H _X(γ) is expansive and isolated, then it is hyperbolic.     

On Linear Preservers of Two-Sided Gut-Majorization On <Emphasis Type="Bold">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">m</Emphasis></Subscript>

For X, Y ∈ M_n,m it is said that X is gut-majorized by Y, and we write X ?_gutY, if there exists an n-by-n upper triangular g-row stochastic matrix R such that X = RY. Define the relation ~_gut as follows. X ~_gutY if X is gut-majorized by Y and Y is gut-majorized by X. The (strong) linear preservers of ?_gut on ?ⁿ and strong linear preservers of this relation on M_n,m have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of ~_gut on ?ⁿ and M_n,m.     

W-Gröbner basis and monomial ideals under polynomial composition

The notion of weakly relatively prime and W-Gröbner basis in K[x ₁, x ₂, …, x _n ] are given. The following results are obtained: for polynomials f ₁, f ₂, …, f _m , \(\{ f_1^{\lambda _1 } ,f_2^{\lambda _2 } ,...,f_m^{\lambda _m } \} \) is a Gröbner basis if and only if f ₁, f ₂, …, f _m are pairwise weakly relatively prime with λ ₁, λ ₂, …, λ _m arbitrary non-negative integers; polynomial composition by Θ = (θ ₁, θ ₂, …, θ _n ) commutes with monomial-Gröbner bases computation if and only if θ ₁, θ ₂, …, θ _m are pairwise weakly relatively prime.     

A Generalization of Strassen’s Functional LIL

Let X ₁,X ₂,… be a sequence of i.i.d. mean zero random variables and let S _n denote the sum of the first n random variables. We show that whenever we have with probability one, lim?sup? _n→∞|S _n |/c _n =α ₀<∞ for a regular normalizing sequence {c _n }, the corresponding normalized partial sum process sequence is relatively compact in C[0,1] with canonical cluster set. Combining this result with some LIL type results in the infinite variance case, we obtain Strassen type results in this setting.     

Expectation of the truncated randomly weighted sums with dominatedly varying summands

We consider the asymptotic behavior of the values P(S > x), E(S 1_{S>x}), and E(S | S > x). Here S = θ₁X₁ + θ₂X₂ + · · · + θ_nX_n is a randomly weighted sum of the basic random variables X₁,X₂, . . . , X_n, which follow some special dependence structure, and {θ₁, θ₂, . . . , θ_n} is a collection of nonnegative and arbitrarily dependent random weights; the collections {X₁,X₂, . . .,X_n} and {θ₁, θ₂, . . . , θ_n} are supposed to be independent. We derive asymptotic formulas in the case where the number of summands n is fixed and the distributions of the basic random variables are dominatedly varying.We apply them to some values related to the risk measures of certain weighted sums.     

Finite Groups Whose <Emphasis Type="Italic">n</Emphasis>-Maximal Subgroups Are Modular

Let G be a finite group. If M_n< M_n?1< · · · < M₁< M₀ = G with Mi a maximal subgroup of M_i?1 for all i = 1,..., n, then M_n (n > 0) is an n-maximal subgroup of G. A subgroup M of G is called modular provided that (i) 〈X,M ∩ Z〉 = 〈X,M〉 ∩ Z for all X ≤ G and Z ≤ G such that X ≤ Z, and (ii) 〈M,Y ∩ Z〉 = 〈M,Y 〉 ∩ Z for all Y ≤ G and Z ≤ G such that M ≤ Z. In this paper, we study finite groups whose n-maximal subgroups are modular.     

Asymptotics for the finite-time ruin probability in a discrete-time risk model with dependent insurance and financial risks<Superscript>*</Superscript>

We consider a discrete-time risk model with insurance and financial risks. Within period i ≥ 1, the real-valued net insurance loss caused by claims is the insurance risk, denoted by X _i , and the positive stochastic discount factor over the same time period is the financial risk, denoted by Y _i . Assume that {(X, Y), (X _i , Y _i ), i ≥ 1} form a sequence of independent identically distributed random vectors. In this paper, we investigate a discrete-time risk model allowing a dependence structure between the two risks. When (X, Y ) follows a bivariate Sarmanov distribution and the distribution of the insurance risk belongs to the class ?(γ) for some γ > 0, we derive the asymptotics for the finite-time ruin probability of this discrete-time risk model.     

Moderate deviation principle for <Emphasis Type="Italic">m</Emphasis>-dependent random variables<Superscript>*</Superscript>

Let {X _n }_n?≥?1 be a sequence of strictly stationary m-dependent random variableswith EX₁ = 0 and \( \mathrm{E}{X}_1^2<\infty \), and let (b _n ) be an increasing sequence of positive numbers such that b _n ?↑?∞ and \( {b}_n/\sqrt{n}\downarrow 0\kern0.5em \mathrm{as}\kern0.5em n\to \infty \). We establish a moderate deviation principle of \( {\left({b}_n\sqrt{n}\right)}^{-1}{\sum}_{i=1}^n{X}_i \) under the condition

$$ \underset{n\to \infty }{\lim \sup}\frac{1}{b_n^2}\log \left[n\mathbf{P}\left(\left|{X}_1\right|>{b}_n\sqrt{n}\right)\right]=-\infty, $$

which is weaker than the classical exponential integrability condition. The results in the present paper weaken the assumptions of Chen [5] and extend partially the results of Eichelsbacher and Löwe [10].     

On Tverberg partitions

A theorem of Tverberg from 1966 asserts that every set X ? ? ^d of n = T(d, r) = (d + 1)(r ? 1) + 1 points can be partitioned into r pairwise disjoint subsets, whose convex hulls have a point in common. Thus every such partition induces an integer partition of n into r parts (that is, r integers a ₁,..., a _r satisfying n = a ₁ + ··· + a _r ), in which the parts a _i correspond to the number of points in every subset. In this paper, we prove that for any partition of n where the parts satisfy a _i ≤ d + 1 for all i = 1,..., r, there exists a set X ? ? of n points, such that every Tverberg partition of X induces the same partition on n, given by the parts a ₁,..., a _r .     

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.

     

Some <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Rigidity Results for Complete Manifolds with Harmonic Curvature

Let (M ⁿ , g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R m? the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R m? goes to zero uniformly at infinity if for \(p\geq \frac n2\), the L ^p -norm of R m? is finite. Moreover, If R is positive, then (M ⁿ , g) is compact. As applications, we prove that (M ⁿ , g) is isometric to a spherical space form if for \(p\geq \frac n2\), R is positive and the L ^p -norm of R m? is pinched in [0, C ₁), where C ₁ is an explicit positive constant depending only on n, p, R and the Yamabe constant. We give an isolation theorem of the trace-free Ricci curvature tensor of compact locally conformally flat Riemannian n-manifolds with constant positive scalar curvature, which extends Theorem 1 of Hebey and M. Vaugon (J. Geom. Anal. 6, 531–553, 1996). This result is sharp, and we can precisely characterize the case of equality. In particular, when n = 4, we recover results by Gursky (Indiana Univ. Math. J. 43, 747–774, 1994; Ann. Math. 148, 315–337, 1998).     

Thom-Sebastiani properties of Kohn-Rossi cohomology of compact connected strongly pseudoconvex CR manifolds

Let X_1 and X_2 be two compact connected strongly pseudoconvex embeddable Cauchy-Riemann(CR) manifolds of dimensions 2m-1 and 2n-1 in C~(m+1)and C~(n+1), respectively. We introduce the ThomSebastiani sum X = X_1 ⊕X_2which is a new compact connected strongly pseudoconvex embeddable CR manifold of dimension 2m+2n+1 in C~(m+n+2). Thus the set of all codimension 3 strongly pseudoconvex compact connected CR manifolds in Cn+1for all n 2 forms a semigroup. X is said to be an irreducible element in this semigroup if X cannot be written in the form X_1 ⊕ X_2. It is a natural question to determine when X is an irreducible CR manifold. We use Kohn-Rossi cohomology groups to give a necessary condition of the above question. Explicitly,we show that if X = X_1 ⊕ X_2, then the Kohn-Rossi cohomology of the X is the product of those Kohn-Rossi cohomology coming from X_1 and X_2 provided that X_2 admits a transversal holomorphic S~1-action.     

Asymptotics and Arithmetical Properties of Complexity for Circulant Graphs

Abstract—We study analytical and arithmetical properties of the complexity function for infinite families of circulant C _n (s₁, s_2,…, s _k ) C_2n(s₁, s_2,…, s _k , n). Exact analytical formulas for the complexity functions of these families are derived, and their asymptotics are found. As a consequence, we show that the thermodynamic limit of these families of graphs coincides with the small Mahler measure of the accompanying Laurent polynomials.     

Refined Self-normalized Large Deviations for Independent Random Variables

Let X ₁,X ₂,…?, be independent random variables with EX _i =0 and write \(S_{n}=\sum_{i=1}^{n}X_{i}\) and \(V_{n}^{2}=\sum_{i=1}^{n}X_{i}^{2}\). This paper provides new refined results on the Cramér-type large deviation for the so-called self-normalized sum S _n /V _n . The major techniques used to derive these new findings are different from those used previously.     

On necessary and sufficient conditions for the self-normalized central limit theorem

Let X₁, X₂, … be a sequence of independent random variables and S_n = Σ _i=1 ⁿ X_i and V _n ² = Σ _i=1 ⁿ X _i ² . When the elements of the sequence are i.i.d., it is known that the self-normalized sum S_n=V_n converges to a standard normal distribution if and only if max_1?i?n|X_i|/V_n→0 in probability and the mean of X₁ is zero. In this paper, sufficient conditions for the self-normalized central limit theorem are obtained for general independent random variables. It is also shown that if max_1?i?n|X_i|/V_n→0 in probability, then these sufficient conditions are necessary.     

Representation and Approximation of Positivity Preservers

We consider a closed set S?? ⁿ and a linear operator

$\Phi \colon \mathbb{R}[X_1,\ldots,X_n]\rightarrow \mathbb{R}[X_1,\ldots,X_n]$

that preserves nonnegative polynomials, in the following sense: if f≥0 on S, then Φ(f)≥0 on S as well. We show that each such operator is given by integration with respect to a measure taking nonnegative functions as its values. This can be seen as a generalization of Haviland’s Theorem, which concerns linear functionals on ?[X ₁,…,X _n ]. For compact sets S we use the result to show that any nonnegativity preserving operator is a pointwise limit of very simple nonnegativity preservers with finite dimensional range.

     

A renewal approach to Markovian <Emphasis Type="Italic">U</Emphasis>-statistics

In this paper we describe a novel approach to the study of U-statistics in the Markovian setup based on the (pseudo-) regenerative properties of Harris Markov chains. Exploiting the fact that any sample path X ₁, ..., X _n of a general Harris chain X may be divided into asymptotically i.i.d. data blocks \(\mathcal{B}_1 , \ldots \mathcal{B}_N\) of random length corresponding to successive (pseudo-) regeneration times, we introduce the notion of regenerative U-statistic Ω _N = Σ _k≠l ω _h \(\left( {\mathcal{B}_k ,\mathcal{B}_l } \right)\)/(N(N ? 1)) related to a U-statistic U _n = Σ _i≠j h(X _i , X _j )/(n(n ? 1)). We show that, under mild conditions, these two statistics are asymptotically equivalent up to the order O _?(n?1). This result serves as a basis for establishing limit theorems related to statistics of the same form as U _n . Beyond its use as a technical tool for proving results of a theoretical nature, the regenerative method is also employed here in a constructive fashion for estimating the limiting variance or the sampling distribution of certain U-statistics through resampling. The proof of the asymptotic validity of this statistical methodology is provided, together with an illustrative simulation result.     

On (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)-absorbing ideals of commutative rings

Let R be a commutative ring with 1 ≠ 0 and U(R) be the set of all unit elements of R. Let m, n be positive integers such that m > n. In this article, we study a generalization of n-absorbing ideals. A proper ideal I of R is called an (m, n)-absorbing ideal if whenever a ₁?a _m ∈I for a ₁,…, a _m ∈R?U(R), then there are n of the a _i ’s whose product is in I. We investigate the stability of (m, n)-absorbing ideals with respect to various ring theoretic constructions and study (m, n)-absorbing ideals in several commutative rings. For example, in a Bézout ring or a Boolean ring, an ideal is an (m, n)-absorbing ideal if and only if it is an n-absorbing ideal, and in an almost Dedekind domain every (m, n)-absorbing ideal is a product of at most m ? 1 maximal ideals.