A nonclassical LIL for sums of B-valued random variables when extreme terms are excluded

Let {X,X _n ; n≧1} be a sequence of B-valued i.i.d. random variables. Denote $X_{{n}}^{(r)}=X_{{m}}$ if ∥X _m ∥ is the r-th maximum of {∥X _k ∥; k≦n}, and let ${}^{(r)}S_{{n}}=S_{{n}}-(X_{{n}}^{(1)}+\cdots+X_{{n}}^{(r)})$ be the trimmed sums, where $S_{{n}}=\sum_{ k=1}^{n}X_{{k}}$ . Given a sequence of positive constants {h(n), n≧1}, which is monotonically approaching infinity and not asymptotically equivalent to loglogn, a limit result for $^{(r)}S_{{n}}/\sqrt{2nh(n)}$ is derived.     

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.     

Complex Interpolation of Operators and Optimal Domains

Let X ₀ and X ₁ be two order continuous Banach function spaces on a finite measure space, (E ₀, E ₁) a Banach space interpolation pair, and \({T: X_0 + X_1 \to E_0 + E_1}\) an admissible operator between the pairs (X ₀,X ₁) and (E ₀,E ₁). If \({T_{\theta} : [X_0, X_1]_{[\theta ]} \to [E_0, E_1]_{[\theta]}}\) is the interpolated operator by the first complex method of Calderón and m ₀, m ₁ and m _θ are the vector measures coming from \({{T\vert}_{X_0}}\) and \({{T\vert}_{X_1}}\) and T _θ, respectively, then we study the relationship between the optimal domain \({L^1(m_{\theta})}\) of T _θ and the complex interpolation space \({[L^1(m_0),L^1(m_1)]_{[\theta]}}\) of the optimal domains of \({{T\vert}_{X_0}}\) and \({{T\vert}_{X_1}}\) . Then, we apply the obtained result to study interpolation of p-th power factorable and bidual (p,q)-power-concave operators.     

Quasikonvexe Multiplikatoren starker Konvergenz

пУсть {f _k; f _k ^* ?X×X^* — пОлНАь БИОРтОгОНАльНАь сИс тЕМА В БАНАхОВОМ пРОстРАН стВЕ X (X^* — сОпРьжЕННОЕ пРОст РАНстВО). пУсть (?→+0) $$\begin{gathered} S_n f = \sum\limits_{k = 0}^n {f_k^* (f)f_k ,} K(f,t) = \mathop {\inf }\limits_{g \in Z} (\left\| {f - g} \right\|_x + t\left| g \right|_z ), \hfill \\ X_0 = \{ f \in X:\mathop {\lim }\limits_{n \to \infty } \left\| {S_n f - f} \right\|_x = 0\} ,X_\omega = \{ f \in X:K(f,t) = 0(\omega (t))\} , \hfill \\ \end{gathered} $$ гДЕZ?X — НЕкОтОРОЕ пОД пРОстРАНстВО с пОлУН ОРМОИ ¦·¦ И Ω — МОДУль НЕпРЕРыВНО стИ УДОВлЕтВОРьУЩИИ Усл ОВИУ sup Ω(t)/t=∞. пОслЕДОВАтЕ льНОстьΤ={Τ _k} кОМплЕксНых ЧИ сЕл НАжыВАЕтсь МНОжИтЕл ЕМ сИльНОИ схОДИМОст И ДльX _Τ, жАпИсьΤ?М[X _Τ,X _Τ], ЕслИ Д ль кАжДОгО ЁлЕМЕНтАf?X _Τ сУЩЕстВ УЕт тАкОИ ЁлЕМЕНтf ^τ?х₀, ЧтОf _k ^* (f ^τ)=Τ_kf _k ^* (f) Дль ВсЕхk. ДОкА жАНО сРЕДИ ДРУгИх слЕДУУЩ ЕЕ УтВЕРжДЕНИЕ. тЕОРЕМА. пУсmь {f_k; f _k ^* } —Н ЕкОтОРыИ (с, 1)-БАжИс тАк ОИ, ЧтО ВыпОлНьУтсь НЕРАВЕН стВА тИпА НЕРАВЕНстВА ДжЕ ксОНА с пОРьДкОМ O(?_n) u тИ пА НЕРАВЕНстВА БЕРНшmЕИ НА с пОРьДкОМ O(1/?_n). ЕслИ пОслЕДОВАтЕл ьНОсть Τ кВАжИВыпУкл А И ОгРАНИЧЕНА, тО $$\tau \in M[X_{\omega ,} X_0 ] \Leftrightarrow \omega (\varphi _n )\tau _n \left\| {S_n } \right\|_{[X,X]} = o(1).$$ ЁтОт ОБЩИИ пОДхОД НЕМ ЕДлЕННО ДАЕт клАссИЧ ЕскИЕ РЕжУльтАты, ОтНОсьЩИ Есь к ОДНОМЕРНыМ тРИгОНОМЕтРИЧЕскИМ РьДАМ. НО тЕпЕРь ВОжМО жНы ДАльНЕИшИЕ пРИлОжЕН Иь, НАпРИМЕР, к РАжлОжЕНИьМ пО пОлИ НОМАМ лЕжАНДРА, лАгЕР РА ИлИ ЁРМИтА.     

Pfaffian point process for the Gaussian real generalised eigenvalue problem

The generalised eigenvalues for a pair of N?× N matrices (X ₁, X ₂) are defined as the solutions of the equation det (X ₁ ? λX ₂)?=?0, or equivalently, for X ₂ invertible, as the eigenvalues of ${X_{2}^{-1}X_{1}}$ . We consider Gaussian real matrices X ₁, X ₂, for which the generalised eigenvalues have the rotational invariance of the half-sphere, or after a fractional linear transformation, the rotational invariance of the unit disk. In these latter variables we calculate the joint eigenvalue probability density function, the probability p _N,k of finding k real eigenvalues, the densities of real and complex eigenvalues (the latter being related to an average over characteristic polynomials), and give an explicit Pfaffian formula for the higher correlation functions ${\rho_{(k_1,k_2)}}$ . A limit theorem for p _N,k is proved, and the scaled form of ${\rho_{(k_1,k_2)}}$ is shown to be identical to the analogous limit for the correlations of the eigenvalues of real Gaussian matrices. We show that these correlations satisfy sum rules characteristic of the underlying two-component Coulomb gas.     

Tail homogeneity of invariant measures of multidimensional stochastic recursions in a critical case

We consider the stochastic recursion ${X_{n+1} = M_{n+1}X_{n} + Q_{n+1}, (n \in \mathbb{N})}$ , where ${Q_n, X_n \in \mathbb{R}^d }$ , M _n are similarities of the Euclidean space ${ \mathbb{R}^d }$ and (Q _n , M _n ) are i.i.d. We study asymptotic properties at infinity of the invariant measure for the Markov chain X _n under assumption ${\mathbb{E}{[\log|M|]}=0}$ i.e. in the so called critical case.     

Circular law theorem for random Markov matrices

Let (X _jk ) _jk≥1 be i.i.d. nonnegative random variables with bounded density, mean m, and finite positive variance σ ². Let M be the n?× n random Markov matrix with i.i.d. rows defined by ${M_{jk}=X_{jk}/(X_{j1}+\cdots+X_{jn})}$ . In particular, when X ₁₁ follows an exponential law, the random matrix M belongs to the Dirichlet Markov Ensemble of random stochastic matrices. Let λ₁, . . . , λ _n be the eigenvalues of ${\sqrt{n}M}$ i.e. the roots in ${\mathbb{C}}$ of its characteristic polynomial. Our main result states that with probability one, the counting probability measure ${\frac{1}{n}\delta_{\lambda_1}+\cdots+\frac{1}{n}\delta_{\lambda_n}}$ converges weakly as n→∞ to the uniform law on the disk ${\{z\in\mathbb{C}:|z|\leq m^{-1}\sigma\}}$ . The bounded density assumption is purely technical and comes from the way we control the operator norm of the resolvent.     

On the absolute continuity of multidimensional Ornstein-Uhlenbeck processes

Let X be a n-dimensional Ornstein-Uhlenbeck process, solution of the S.D.E. $${\rm d}X_{t}\; =\; AX_{t} {\rm d}t \; +\; {\rm d}B_t$$ where A is a real n?× n matrix and B a Lévy process without Gaussian part. We show that when A is non-singular, the law of X ₁ is absolutely continuous in ${\mathbb{R}^n}$ if and only if the jumping measure of B fulfils a certain geometric condition with respect to A, which we call the exhaustion property. This optimal criterion is much weaker than for the background driving Lévy process B, which might be singular and sometimes even have a one-dimensional discrete jumping measure. This improves on a result by Priola and Zabczyk.     

A Note on Drazin Invertibility for Upper Triangular Block Operators

A bounded linear operator A acting on a Banach space X is said to be an upper triangular block operators of order n, and we write ${A \in \mathcal{UT}_{n}(X)}$ , if there exists a decomposition of ${X = X_{1} \oplus . . . \oplus X_{n}}$ and an n × n matrix operator ${(A_{i,j})_{\rm 1 \leq i, j \leq n}}$ such that ${A = (A_{i, j})_{1 \leq i, j \leq n}, A_{i, j} = 0}$ for i > j. In this note we characterize a large set of entries A _{i, j} with j > i such that ${\sigma_{\rm D} (A) = {\bigcup\limits_{i = 1}^{n}} \sigma_{\rm D} (A_{i, i})}$ ; where σ_D(.) is the Drazin spectrum. Some applications concerning the Fredholm theory and meromorphic operators are given.     

Tail Behavior of Sums and Maxima of Sums of Dependent Subexponential Random Variables

In this paper, we consider dependent random variables X _k , k=1,2,?? with supports on [?b _k ,??), respectively, where the b _k ??0 are some finite constants. We derive asymptotic results on the tail probabilities of the quantities $S_{n}=\sum_{k=1}^{n} X_{k}$ , X _(n)=max?_1??k??n X _k and S _(n)=max?_1??k??n S _k , n??1 in the case where the random variables are dependent with heavy-tailed (subexponential) distributions, which substantially generalize the results of Ko and Tang (J. Appl. Probab. 45, 85?C94, 2008).     

Approximation of the Tail Probability of Dependent Random Sums Under Consistent Variation and Applications

In this paper, we consider the random sums of one type of asymptotically quadrant sub-independent and identically distributed random variables {X, X _i , i?=?1, 2,???} with consistently varying tails. We obtain the asymptotic behavior of the tail $\textsf{P}(X_1+\cdots+X_\eta>x)$ under different cases of the interrelationships between the tails of X and η, where η is an integer-valued random variable independent of {X, X _i , i?=?1, 2,???}. We find out that the asymptotic behavior of $\textsf{P}(X_1+\cdots+X_\eta>x)$ is insensitive to the dependence assumed in the present paper. We state some applications of the asymptotic results to ruin probabilities in the compound renewal risk model under dependent risks. We also state some applications to a compound collective risk model under the Markov environment.     

Randomly weighted sums of subexponential random variables with application to capital allocation

We are interested in the tail behavior of the randomly weighted sum \( \sum _{i=1}^{n}\theta _{i}X_{i}\) , in which the primary random variables X ₁, …, X _n are real valued, independent and subexponentially distributed, while the random weights ?? ₁, …, ?? _n are nonnegative and arbitrarily dependent, but independent of X ₁, …, X _n . For various important cases, we prove that the tail probability of \(\sum _{i=1}^{n}\theta _{i}X_{i}\) is asymptotically equivalent to the sum of the tail probabilities of ?? ₁ X ₁, …, ?? _n X _n , which complies with the principle of a single big jump. An application to capital allocation is proposed.     

An infinite-dimensional law of large numbers in Cesaro's sense

Let (X _k) _k≥0 be a sequence of independent copies of a random variableX taking its values in a real separable Banach space (B, ¦ ¦). For every real number β>?1 one defines the following coefficients: $$A_0^\beta = 1, A_1^\beta = \beta + 1,..., A_k^\beta = (\beta + 1) \cdots (\beta + k)/k!,...$$ It is shown that for all α∈]0, 1[ the sequenceV _n =(1/A _n ^α )∑_0?k?n A _n?k ^α?1 X _k converges almost surely toE(X) if and only if ‖X‖^1/α is integrable. This extends results obtained earlier by several authors for scalar-valued random variables: Lorentz (case 1/2<α<1), Chow and Lai (case 0<α<1/2), Déniel and Derriennic (case α=1/2).     

On Solvability of Hermitian Solutions to a System of Five Matrix Equations

In this paper, we establish the formulas of the maximal and minimal ranks of the Hermitian matrix expression ${C_{5}-A_{3}X_{1}A_{3}*-A_{4}X_{2}A_{4}*}$ where X ₁ and X ₂ are Hermitian solutions to two systems of matrix equations A ₁ X ₁ = C ₁,X ₁ B ₁ = C ₂ and A ₂ X ₂ = C ₃,X ₂ B ₂ = C ₄, respectively. Using this result and matrix rank method, we give necessary and sufficient conditions for the existence of Hermitian solutions to a system of five matrix equations ${A_{1}X_{1}=C_{1},X_{1}B_{1}=C_{2},A_{2}X_{2}=C_{3},X_{2}B_{2}=C_{4} ,A_{3}X_{1}A_{3}*+A_{4}X_{2}A_{4}*=C_{5}}$ by rank equalities. The general expressions and extreme ranks of the Hermitian solutions X ₁ and X ₂ to the system mentioned above are also presented.     

Large Deviations for Proportions of Observations Which Fall in Random Sets Determined by Order Statistics

Let {X _n :n?≥?1} be independent random variables with common distribution function F and consider $K_{h:n}(D)=\sum_{j=1}^n1_{\{X_j-X_{h:n}\in D\}}$ , where h?∈?{1,...,n}, X _1:k ?≤???≤?X _k:k are the order statistics of the sample X ₁,...,X _k and D is some suitable Borel set of the real line. In this paper we prove that, if F is continuous and strictly increasing in the essential support of the distribution and if $\lim_{n\to\infty}\frac{h_n}{n}=\lambda$ for some λ?∈?[0,1], then $\{K_{h_n:n}(D)/n:n\geq 1\}$ satisfies the large deviation principle. As a by product we derive the large deviation principle for order statistics $\{X_{h_n:n}:n\geq 1\}$ . We also present results for the special case of Bernoulli distributed random variables with mean p?∈?(0,1), and we see that the large deviation principle holds only for p?≥?1/2. We discuss further almost sure convergence of $\{K_{h_n:n}(D)/n:n\geq 1\}$ and some related quantities.     

Schauder estimates for sub-elliptic equations

We establish Hölder estimates of second derivatives for a class of sub-elliptic partial differential operators in ${\mathbb{R}^{N}}$ of the kind $$\mathcal L=\sum_{i,j=1}^{m}a_{ij}(x)X_{i}X_{j}+X_{0},$$ where the X _j ’s are smooth vector fields in ${\mathbb{R}^{N}}$ , and a _ij is a uniformly elliptic matrix. It is assumed that the X _j ’s satisfy homogeneity conditions with respect to a group of dilations δ _r which yield the existence of a composition law ${\circ}$ in ${\mathbb{R}^{N}}$ making the triplet ${\mathbb G=(\mathbb{R}^{N},\circ,\delta_{r})}$ an homogeneous Lie group on which the X _j ’s are left translation invariant. The Hölder norms are defined in terms of this composition law. The main tools used are the Taylor formula for smooth functions on ${\mathbb{G}}$ , some properties of the corresponding Taylor polynomials, and an orthogonality theorem that extends to homogeneous Lie groups a classical theorem of Calderón and Zygmund in the Euclidean setting.     

Local solvability on {\mathbb{H}_1}: non-homogeneous operators

Local solvability and non-solvability are classified for left-invariant differential operators on the Heisenberg group ${\mathbb{H}_1}$ of the form L?=?P _n (X, Y)?+?Q(X, Y) where the P _n are certain homogeneous polynomials of order n?≥ 2 and Q is of lower order with ${X=\partial_x,\,Y=\partial_y+x\partial_w}$ on ${\mathbb{R}^3}$ . We extend previous studies of operators of the form P _n (X, Y) via representations involving ordinary differential operators with a parameter.     

2-Reducible Two Paths and an Edge Constructing a Path in (2k + 1)-Edge-Connected Graphs

Let k ≥ 5 be an odd integer and G = (V(G), E(G)) be a k-edge-connected graph. For ${X\subseteq V(G),e(X)}$ denotes the number of edges between X and V(G) ? X. We here prove that if ${\{s_i,t_i\}\subseteq X_i\subseteq V(G)(i=1,2),f}$ is an edge between s ₁ and ${s_2,X_1\cap X_2=\emptyset,e(X_1)\le 2k-3,e(X_2)\le 2k-2}$ , and e(Y) ≥ k + 1 for each ${Y\subseteq V(G)}$ with ${Y\cap\{s_1,t_1,s_2,t_2\}=\{s_1,t_2\}}$ , then there exist paths P ₁ and P ₂ such that P _i joins s _i and ${t_i,V(P_i)\subseteq X_i}$ (i = 1, 2) and ${G-f-E(P_1\cup P_2)}$ is (k ? 2)-edge-connected, and in fact we give a generalization of this result.     

A generalization of the katona theorem for crosst-intersecting families

LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:

1. There are two crosst-intersecting Hamming spheresA ₀,? ₀ with centerX such that |A| ≤ |A ₀| and|?| ≤ |? ₀| hold.
2. Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.

Using these claims, the following conjecture of Frankl is proven:

1. Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK _l ⁿ is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
2. Ifn + t = 2k then |A| |? ≤ (K _k ⁿ )² holds.

The extremal configurations are also determined.     

On Berry-Esseen type bounds and asymptotic expansions for slightly trimmed means

The paper is concerned with the second-order asymptotics of distributions of trimmed means $T_n = (\sum\nolimits_{i = k_n + 1}^{n - m_n } {X_{i:n} } )/n$ , where k _n , m _n are sequences of integers, 0 ≤ k _n < n ? m _n ≤ n, and r _n := min(k _n , m _n ) → ∞ as n → ∞, X _i:n are order statistics that correspond to the sample X ₁, …, X _n of independent identically distributed random variables with distribution function F. We focus on the case of slightly trimmed means, when max(k _n , m _n )/n → 0 as n → ∞. In [11], Berry-Esseen-type bounds were obtained for the normal approximation of T _n ; under certain regularity conditions, these bounds are of the order O(r _n ^?1/2 ). In [11], it is also shown that this order cannot be improved if E X ₁ ² = ∞. Moreover, asymptotic Edgeworth-type expansions were found in [11] for slightly trimmed means and their Studentized versions. In the present paper, we supplement the results of [11] by Berry-Esseen-type bounds and asymptotic approximations for the case E X ₁ ² < ∞.