Randomly weighted sums of dependent subexponential random variables*

We consider the randomly weighted sums $ \sum\nolimits_{k = 1}^n {{\theta_k}{X_k},n \geqslant 1} $ , where $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ are n real-valued random variables with subexponential distributions, and $ \left\{ {{\theta_k},1 \leqslant k \leqslant n} \right\} $ are other n random variables independent of $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ and satisfying $ a \leqslant \theta \leqslant b $ for some $ 0 < a \leqslant b < \infty $ and all $ 1 \leqslant k \leqslant n $ . For $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ satisfying some dependent structures, we prove that $$ {\text{P}}\left( {\mathop {{\max }}\limits_{1 \leqslant m \leqslant n} \sum\limits_{k = 1}^m {{\theta_k}{X_k} > x} } \right)\sim {\text{P}}\left( {\sum\limits_{k = 1}^m {{\theta_k}{X_k} > x} } \right)\sim {\text{P}}\left( {\mathop {{\max }}\limits_{1 \leqslant k \leqslant n} {\theta_k}{X_k} > x} \right)\sim \sum\limits_{k = 1}^m {{\text{P}}\left( {{\theta_k}{X_k} > x} \right)} $$ as x??????.     

The Continuity of M and N in Greedy Lattice Animals

Let {X _v: v ∈ Z ^d}, d≥2, be i.i.d. positive random variables with the common distribution F which satisfy, for some a>0, ∫ x ^d (log⁺ x) ^d+a dF(x)<∞ Define $$M_n = \max \left\{ {\sum\limits_{\upsilon \in \pi } {X_\upsilon } {\kern 1pt} :\pi {\text{ a selfavoiding path of length }}n{\text{ starting at the origin}}} \right\}$$ $$N_n = \max \left\{ {\sum\limits_{\upsilon \in \xi } {X_\upsilon } {\kern 1pt} :\xi {\text{ a lattice animal of size }}n{\text{ containing the origin}}} \right\}$$ Then it has been shown that there exist positive finite constants M = M[F] and N = N[F] such that $${\mathop {\lim }\limits_{n \to \infty }} \frac{{M_n }}{n} = M{\text{ and }}{\mathop {\lim }\limits_{n \to \infty }} \frac{{N_n }}{n} = N{\text{ a}}{\text{.s}}{\text{. and in }}L^1 $$      

Asymptotic properties for the loglog laws under positive association

Let {X _n : n ?? 1} be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set $S_n = \sum\limits_{k = 1}^n {X_k }$ , $Mn = \mathop {\max }\limits_{k \leqslant n} \left| {S_k } \right|$ , n ?? 1. Suppose that $0 < \sigma ^2 = EX_1^2 + 2\sum\limits_{k = 2}^\infty {EX_1 X_k < \infty }$ . In this paper, we prove that if E|X ₁|^2+?? < for some ?? ?? (0, 1], and $\sum\limits_{j = n + 1}^\infty {Cov\left( {X_1 ,X_j } \right) = O\left( {n^{ - \alpha } } \right)}$ for some ?? > 1, then for any b > ?1/2 $$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2b + 1} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^{b - 1/2} }} {{n^{3/2} \log n}}} E\left\{ {M_n - \sigma \varepsilon \sqrt {2n\log \log n} } \right\}_ + = \frac{{2^{ - 1/2 - b} E\left| N \right|^{2(b + 1)} }} {{(b + 1)(2b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2(b + 1)} }}}$$ and $$\mathop {\lim }\limits_{\varepsilon \nearrow \infty } \varepsilon ^{ - 2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^b }} {{n^{3/2} \log n}}E\left\{ {\sigma \varepsilon \sqrt {\frac{{\pi ^2 n}} {{8\log \log n}}} - M_n } \right\}} _ + = \frac{{\Gamma (b + 1/2)}} {{\sqrt 2 (b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2b + 2} }}} ,$$ where x ₊ = max{x, 0}, N is a standard normal random variable, and ??(·) is a Gamma function.     

Steckin-marchaud-type inequalities in connection with bernstein polynomials

The purpose of this paper is to derive the estimate (0≤α≤2,n∈N,?(x)=[x(1?x)]^1/2) $$\omega _\alpha (n^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ,f) \leqslant M_\alpha n^{ - 1} \sum\limits_{k = 1}^n {\left\| {\varphi ^{ - \alpha } (B_k f - f)} \right\|} c$$ in terms of the modulus of continuity (of second order) $$\omega _\alpha (t,f): = \sup \{ \varphi ^{ - \alpha } (x)|\Delta _{h\varphi (x)}^ * f(x)|:x,x \pm h\varphi (x) \in [0,1],0< h \leqslant t\} $$ and the Bernstein polynomial B_nf for ?^?αf∈C[0,1].     

On the best constants in the Khintchine inequality for Steinhaus variables

Let S _j : (Ω, P) → S ¹ ? ? be an i.i.d. sequence of Steinhaus random variables, i.e. variables which are uniformly distributed on the circle S ¹. We determine the best constants a _p in the Khintchine-type inequality $${a_p}{\left\| x \right\|_2} \leqslant {\left( {{\text{E}}{{\left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|}^p}} \right)^{1/p}} \leqslant {\left\| x \right\|_2};{\text{ }}x = ({x_j})_{j = 1}^n \in {{\Bbb C}^n}$$ for 0 < p < 1, verifying a conjecture of U. Haagerup that $${a_p} = \min \left( {\Gamma {{\left( {\frac{p}{2} + 1} \right)}^{1/p}},\sqrt 2 {{\left( {{{\Gamma \left( {\frac{{p + 1}}{2}} \right)} \mathord{\left/ {\vphantom {{\Gamma \left( {\frac{{p + 1}}{2}} \right)} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right)}^{1/p}}} \right)$$ . Both expressions are equal for p = p ₀ }~ 0.4756. For p ≥ 1 the best constants a _p have been known for some time. The result implies for a norm 1 sequence x ∈ ? ⁿ , ‖x‖₂ = 1, that $${\text{E}}\ln \left| {\frac{{{S_1} + {S_2}}}{{\sqrt 2 }}} \right| \leqslant {\text{E}}\ln \left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|$$ , answering a question of A. Baernstein and R. Culverhouse.     

The algebraic independence of certain transcendental continued fractions

In the present note the algebraic independence of certain continued fractions is proved. Especially, we prove that the Böhmer-Mahler's series \(\sum\limits_{K = 1}^\infty {\left[ {\omega _v k} \right]} {\text{ }}g_\mu ^{ - k} \left( {1 \leqslant \mu \leqslant s,1 \leqslant v \leqslant t} \right)\) are algebraically independent, where \(\mathop \omega \nolimits_1 {\text{ , }}...{\text{ , }}\mathop \omega \nolimits_{\text{t}} \) , ..., \(\mathop g\nolimits_1 {\text{ , }}...{\text{ , }}\mathop g\nolimits_s \) are some irrational numbers andg ₁, ...,g _s are distinct positive integers.     

Subspace Arrangements of Type B n and D n

Let ${\mathcal{D}}_{n,k} $ be the family of linear subspaces of ?ⁿ given by all equations of the form $\varepsilon _1 x_{i_1 } = \varepsilon _2 x_{i_2 } = \cdot \cdot \cdot \varepsilon _k x_{i_k } ,$ for 1 ≤ < ? ? ? < i _k ≤ i and $\left( {\varepsilon _1 ,...,\varepsilon _k } \right)\varepsilon \left\{ { + 1, - 1} \right\}^k $ Also let ${\mathcal{B}}_{n,k,h} $ be ${\mathcal{D}}_{n,k} $ enlarged by the subspaces $x_{j_1 } = x_{j_2 } = \cdot \cdot \cdot x_{j_h } = 0,$ for 1 ≤. The special cases ${\mathcal{B}}_{n,2,1} $ and ${\mathcal{D}}_{n,2} $ are well known as the reflection hyperplane arrangements corresponding to the Coxeter groups of type B _nand D _n respectively. In this paper we study combinatorial and topological properties of the intersection lattices of these subspace arrangements. Expressions for their Möbius functions and characteristic polynomials are derived. Lexicographic shellability is established in the case of ${\mathcal{B}}_{n,k,h,} 1 \leqslant h < k$ , which allows computation of the homology of its intersection lattice and the cohomology groups of the manifold $\begin{gathered} {\mathcal{D}}_{n,2} \\ M_{n,k,h,} = {\mathbb{R}}^n \backslash \bigcup {{\mathcal{B}}_{n,k,h,} } \\ \end{gathered} $ . For instance, it is shown that $H^d \left( {M_{n,k,k - 1} } \right)$ is torsion-free and is nonzero if and only if d = t(k ? 2) for some $t,0 \leqslant t \leqslant \left[ {{n \mathord{\left/ {\vphantom {n k}} \right. \kern-0em} k}} \right]$ . Torsion-free cohomology follows also for the complement in ?ⁿof the complexification ${\mathcal{B}}_{n,k,h}^C ,1 \leqslant h < k$ .     

A remark on the extended Hermite—Fejér type interpolation of higher order

This paper is devoted to study the interpolation of higher order by the nodes $$\left\{ { \pm 1} \right\} \cup \left\{ {\cos \frac{{(2k - 1)\pi }}{{2n}}} \right\}_{k = 1}^n ,n = 1,2,...\,.$$      

Some consequences of the Lindelöf conjecture

Suppose that the Lindelöf conjecture is valid in the following quantitative form: $$|\zeta (\frac{1}{2} + it)| \leqslant c_0 |t|^{\varepsilon (|t|)} $$ , where ε(t) is a monotone decreasing function, $\varepsilon (2t) \geqslant \tfrac{1}{2}\varepsilon (t),\varepsilon (t) \geqslant \tfrac{1}{{\sqrt {log t} }}$ . Then it is proved that for |t|≥T₀ the disk $\{ s:|s - \tfrac{1}{2} - it| \leqslant v\} $ contains at most 20v log |t| zeros of ζ(s) if $\tfrac{1}{2} \geqslant v \geqslant \sqrt {\varepsilon (t)} $ . There exists an absolute constant A such that for |t|≥T₁ the disk $\{ s:|s - \tfrac{1}{2} - it| \leqslant A\varepsilon ^{\tfrac{1}{3}} (t)\} $ contains at least one zero of ζ(s). Bibliography: 2 titles.     

Exact Non-Self-Similar Solutions of the Equation $$u_t = \Delta \ln u$$

In this paper, we obtain new exact non-self-similar solutions of the nonlinear diffusion equation $$\begin{gathered} {\text{ }}u_t = \Delta \ln u, \hfill \\ u \triangleq u\left( {x,t} \right):\Omega \times \mathbb{R}^ + \to \mathbb{R},{\text{ }} x \in \mathbb{R}^n , \hfill \\ \end{gathered} $$ where $\Omega \subset \mathbb{R}^n $ is the domain and $\mathbb{R}^ + = \left\{ {t:0 \leqslant t < + \infty } \right\},{\text{ }}u\left( {x,t} \right) \geqslant 0$ is the temperature of the medium.     

Volume Formulas in Lp-Spaces

We extend classical volume formulas for ellipsoids and zonoids to p-sums of segments $${vol}\left( {\sum\limits_{i=1}^m { \oplus_p } [ -x_i ,x_i ]} \right)^{1/n} \sim_{c_p} n^{ - \frac{1}{{p'}}} \left( {\sum\limits_{card(I) = n} {|\det (x_i)_i |^p}} \right)^{\frac{1}{{pn}}}$$ where x1,...,xm are m vectors in $\mathbb{R}^n ,\frac{1}{p} + \frac{1}{{p\prime }} = 1$ . According to the definition of Firey, the Minkowski p-sum of segments is given by $$\sum\limits_{i = 1}^m { \oplus _p [ - x_{i,} x_i ]} = \left\{ {\sum\limits_{i = 1}^m {\alpha _i } x_i \left| {\left( {\sum\limits_{i = 1}^m {|\alpha _i |^{p^\prime } } } \right)} \right.^{\frac{1}{{p^\prime }}} \leqslant 1} \right\}.$$ We describe related geometric properties of the Lewis maps associated to classical operator norms.     

Solvability of nonlinear systems including (γ, δ)-comparison pairs

Let $\gamma ,\delta \in \mathbb{R}^n $ with $\gamma _j ,\delta _j \in \{ 0,1\} $ . A comparison pair for a system of equations f_i(u₁,…,u_n)=0 (i=1,…,n) is a pair of vectors $v,w \in \mathbb{R}^n ,v \leqslant w$ , such that $$\begin{array}{*{20}c} {\gamma _i f_i (u_1 , \ldots ,u_{i - 1} ,v_i ,u_i + 1, \ldots ,u_n ) \leqslant 0,} \\ {\delta _i f_i (u_1 , \ldots ,u_{i - 1} ,w_i ,u_i + 1, \ldots ,u_n ) \geqslant 0} \\ \end{array} $$ for $\gamma _j u_j \geqslant v_j ,\delta _j u_j \leqslant w_j (j = 1, \ldots ,n)$ . The presence of comparison pairs enables one to essentially weaken the assumptions of the existence theorem. Bibliography: 1 title.     

Approximation by M. Riesz Kernels in L p for p<1

Let α > 0. We consider the linear span $\mathfrak{X}_\alpha \left( {\mathbb{R}^n } \right)$ of scalar Riesz's kernels $\left\{ {\tfrac{1}{{\left| {x - a} \right|^\alpha }}} \right\}_{a \in \mathbb{R}^n }$ and the linear span $\mathfrak{Y}_\alpha \left( {\mathbb{R}^n } \right)$ of vector Riesz's kernels $\left\{ {\tfrac{1}{{\left| {x - a} \right|^{\alpha + 1} }}\left( {x - a} \right)} \right\}_{a \in \mathbb{R}^n }$ . We study the following problems. (1) When is the intersection $\mathfrak{X}_\alpha \left( {\mathbb{R}^n } \right) \cap L^p \left( {\mathbb{R}^n } \right)$ dense in L^p(?ⁿ)? (2) When is the intersection $\mathfrak{Y}_\alpha \left( {\mathbb{R}^n } \right) \cap L^p \left( {\mathbb{R}^n ,\mathbb{R}^n } \right)$ dense in L^p(?ⁿ, ?ⁿ)? Bibliography: 15 titles.     

Small complete arcs in Galois planes

In this paper we construct a class of k-arcs in PG(2, q), q=p ^h, h>1, p≠3 and prove its completeness for h large enough. The main result states that this class contains complete k-arcs with $$k \leqslant 2 \cdot q^{{9 \mathord{\left/ {\vphantom {9 {10}}} \right. \kern-\nulldelimiterspace} {10}}} {\text{ }}\left( {10{\text{ divides }}h{\text{ and }}q{\text{ }} \geqslant {\text{ }}q_{\text{0}} } \right).$$ Such complete k-arcs are the unique known complete k-arcs with $$k \leqslant {q \mathord{\left/ {\vphantom {q 4}} \right. \kern-\nulldelimiterspace} 4}.$$      

Inequalities for the beta function

We prove the following inequalities involving Euler’s beta function. (i) Let α and β be real numbers. The inequalities $\left( {\frac{{y^{z - x} }} {{x^{z - y} z^{y - x} }}} \right)^\alpha \leqslant \frac{{B(x,x)^{z - y} B(z,z)^{y - x} }} {{B(y,y)^{z - x} }} \leqslant \left( {\frac{{y^{z - x} }} {{x^{z - y} z^{y - x} }}} \right)^\beta $ hold for all x, y, z with 0 < x ≤ y ≤ z if and only if α ≤ 1/2 and β ≥ 1. (ii) Let a and b be non-negative real numbers. For all positive real numbers x and y we have $\delta (a,b) \leqslant \frac{{x^a B(x + b,y) + y^a B(x,y + b)}} {{(x + y)^a B(x,y)}} \leqslant \Delta (a,b) $ with the best possible bounds $\delta (a,b) = \min \{ 2^{ - a} ,2^{1 - a - b} \} and\Delta (a,b) = \max \{ 1,2^{1 - a - b} \} . $ .     

On the strong summation of Fourier series with variable exponents

Пустьf 2π-периодическ ая суммируемая функц ия, as _k (x) еë сумма Фурье порядк аk. В связи с известным ре зультатом Зигмунда о сильной суммируемости мы уст анавливаем, что если λn→∞, то сущес твует такая функцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _{2n} } } \right\}^{1/\lambda _{2n} } = \infty .$$ Отсюда, в частности, вы текает, что если λn?∞, т о существует такая фун кцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } } \right\}^{1/\lambda _n } = \infty .$$ Пусть, далее, ω-модуль н епрерывности и $$H^\omega = \{ f:\parallel f(x + h) - f(x)\parallel _c \leqq K_f \omega (h)\} .$$ . Мы доказываем, что есл и λ _n ?∞, то необходимым и достаточным условие м для того, чтобы для всехf∈H ^ω выполнялос ь соотношение $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _n } } \right\}^{1/\lambda _n } = 0(x \in [0;2\pi ])$$ является условие $$\omega \left( {\frac{1}{n}} \right) = o\left( {\frac{1}{{\log n}} + \frac{1}{{\lambda _n }}} \right).$$ Это же условие необхо димо и достаточно для того, чтобы выполнялось соотнош ение $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } = 0(f \in H^\omega ,x \in [0;2\pi ]).$$      

On the strong approximation by the (C, α)-means of Fourier series. II

стАтьь ьВльЕтсь пРОД ОлжЕНИЕМ пРЕДыДУЩЕИ ОДНОИМЕННОИ РАБОты АВтОРА, гДЕ ИжУ ЧАлсь пОРьДОк ВЕлИЧИН пРИ УслОВИьх, ЧтО α>-1/2, Рα >- 1 И ЧтО МАтРИцАt _nk УДОВлЕтВОРьЕт НЕкОт ОРОМУ УслОВИУ РЕгУльРНОстИ. жДЕсь ДОкАжыВАЕтсь, Ч тО ЕслИf∈H ^Ω, тО ВыпОлНь Етсь ОцЕНкА $$\left\{ {\frac{1}{{\lambda _n }}\mathop \Sigma \limits_{k = n - \lambda _n + 1}^n \left| {\sigma _k^\alpha \left( x \right) - f\left( x \right)} \right|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} = O\left( {\left\{ {\frac{1}{{\lambda _n }}\mathop \Sigma \limits_{k = n - \lambda _n + 1}^n \left( {\frac{1}{k}\mathop \smallint \limits_{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}}^{2\pi } \frac{{\omega \left( t \right)}}{{t^2 }}dt} \right)^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} + \left( {\frac{{\lambda _n }}{n}} \right)^\alpha \omega \left( {\frac{1}{n}} \right)} \right)$$ (λ₁=1, λ_n+1-λ_n≦1), А тАкжЕ ЧтО Ёт А ОцЕНкА ОкОНЧАтЕльН А В сВОИх тЕРМИНАх; пОДОБ НыИ РЕжУль-тАт спРАВЕДлИВ тАкжЕ И Дль сОпРьжЕННОИ ФУНкцИИ . ДОкАжыВАЕтсь, ЧтО Усл ОВИьα>?1/2 Иpα>?1, кОтОРыЕ Б ылИ НАлОжЕНы В УпОМьНУтО И ВышЕ ЧАстИ I, сУЩЕстВЕН Ны.     

Symmetry breaking bifurcation from solutions concentrating on the equator of \mathbb{S}^N

We are concerned with the elliptic problem $${\varepsilon ^2}{\Delta _{{S^n}}}u - u + {u^p} = 0{\text{ in }}{S^n},u > 0{\text{ in }}{S^n}$$ , where ${\Delta _{{S^n}}}$ is the Laplace-Beltrami operator on $\mathbb{S}^n : = \left\{ {x \in \mathbb{R}^{n + 1} ;\left\| x \right\| = 1} \right\}\left( {n \geqslant 3} \right)$ , and p ? 2. We construct a smooth branch C of solutions concentrating on the equator S ⁿ ∩ {x _{n+1 = 0}}. Using the Crandall-Rabinowitz bifurcation theorem, we show that C has infinitely many bifurcation points from which continua of nonradial solutions emanate. In applying the bifurcation theorem, we verify the transversality condition directly.