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1.
Let {X n ,?n≧1} be a sequence of nondegenerate, symmetric, i.i.d. random variables which are in the domain of attraction of the normal?law?with zero means and possibly infinite variances. Denote ${S_{n}=\sum_{i=1}^{n} X_{i}}$ , ${V_{n}^{2}=\sum_{i=1}^{n} X_{i}^{2}}$ . Then we prove that there is a sequence of positive constants {b(n),?n≧1} which is defined by Klesov and Rosalsky [11], is monotonically approaching infinity and is not asymptotically equivalent to loglogn but is such that $\displaystyle \limsup_{n\to\infty} \frac{|S_n|}{\sqrt{2V_n^2b(n)}}= 1$ almost surely if some additional technical assumptions are imposed.  相似文献   

2.
Let (X, Λ) be a pair of random variables, where Λ is an Ω (a compact subset of the real line) valued random variable with the density functiong(Θ: α) andX is a real-valued random variable whose conditional probability function given Λ=Θ is P {X=x|Θ} withx=x 0, x1, …. Based onn independent observations ofX, x (n), we are to estimate the true (unknown) parameter vectorα=(α 1, α2, ...,αm) of the probability function ofX, Pα(X=∫ΩP{X=x|Θ}g(Θ:α)dΘ. A least squares estimator of α is any vector \(\hat \alpha \left( {X^{\left( n \right)} } \right)\) which minimizes $$n^{ - 1} \sum\limits_{i = 1}^n {\left( {P_\alpha \left( {x_i } \right) - fn\left( {x_i } \right)} \right)^2 } $$ wherex (n)=(x1, x2,…,x n) is a random sample ofX andf n(xi)=[number ofx i inx (n)]/n. It is shown that the least squares estimators exist as a unique solution of the normal equations for all sufficiently large sample size (n) and the Gauss-Newton iteration method of obtaining the estimator is numerically stable. The least squares estimators converge to the true values at the rate of \(O\left( {\sqrt {2\log \left( {{{\log n} \mathord{\left/ {\vphantom {{\log n} n}} \right. \kern-0em} n}} \right)} } \right)\) with probability one, and has the asymptotically normal distribution.  相似文献   

3.
Let T : J → J be an expanding rational map of the Riemann sphere acting on its Julia set J andf : J →R denote a Hölder continuous function satisfyingf(x)?log | T′(x vb for allx in J. Then for any pointz 0 in J define the set Dz 0(f) of “well-approximable” points to be the set of points in J which lie in the Euclidean ball $B(\gamma ,{\text{ exp(}} - \sum {_{i - 0}^{\mathfrak{n} - 1} } f(T^\ell x)))$ for infinitely many pairs (y, n) satisfying T n (y)=z0. We prove that the Hausdorff dimension of Dz 0(f) is the unique positive numbers(f) satisfying the equation P(T,?s(f).f)=0, where P is the pressure on the Julia set. This result is then shown to have consequences for the limsups of ergodic averages of Hölder continuous functions. We also obtain local counting results which are analogous to the orbital counting results in the theory of Kleinian groups.  相似文献   

4.
Iff∈C[?1, 1] is real-valued, letE R mn (f) andE C mn (f) be the errors in best approximation tof in the supremum norm by rational functions of type (m, n) with real and complex coefficients, respectively. We show that formn?1≥0 $$\gamma _{mn} = \inf \{ {{E_{mn}^C (f)} \mathord{\left/ {\vphantom {{E_{mn}^C (f)} {E_{mn}^R (f)}}} \right. \kern-\nulldelimiterspace} {E_{mn}^R (f)}}:f \in C[ - 1,1]\} = \tfrac{1}{2}.$$   相似文献   

5.
We construct a sequence (n k ) such that n k + 1n k → ∞ and for any ergodic dynamical system (X, Σ, μ, T) and f ε L 1(μ) the averages converge to X f dμ for μ almost every x. Since the above sequence is of zero Banach density this disproves a conjecture of J. Rosenblatt and M. Wierdl about the nonexistence of such sequences. Research supported by the Hungarian National Foundation for Scientific research T049727.  相似文献   

6.
This paper generalizes the penalty function method of Zang-will for scalar problems to vector problems. The vector penalty function takes the form $$g(x,\lambda ) = f(x) + \lambda ^{ - 1} P(x)e,$$ wheree ?R m, with each component equal to unity;f:R nR m, represents them objective functions {f i} defined onX \( \subseteq \) R n; λ ∈R 1, λ>0;P:R nR 1 X \( \subseteq \) Z \( \subseteq \) R n,P(x)≦0, ∨xR n,P(x) = 0 ?xX. The paper studies properties of {E (Z, λ r )} for a sequence of positive {λ r } converging to 0 in relationship toE(X), whereE(Z, λ r ) is the efficient set ofZ with respect tog(·, λr) andE(X) is the efficient set ofX with respect tof. It is seen that some of Zangwill's results do not hold for the vector problem. In addition, some new results are given.  相似文献   

7.
В работе в качестве ха рактеристики функци иf рассматриваются сле дующие ее модули: ωk(f; x; δ)=sup {¦Δ h k f(t¦: t, t+kh∈[x-kδ/2, x+kδ/2][a, b], ωk(f; δ)={sup ωk(f; x; δ): х∈[а, Ь]. Получены оценки погр ешности квадратурны х формул с помощью модулят. Нап ример, справедливо следующ ее утверждение. Пусть квадратурная формул а точна на отрезке [а, Ь] д ля всех алгебраическ их многочленов степени не вышеk- 1 иR n (f) — погрешностьn-соста вной квадратуры, поро жденнойL(f). Тогда $$L(f) = \sum\limits_{i = 0}^m {\sum\limits_{j = 0}^{\alpha _i } {A_i^j f^{(j)} (x_i )} } $$ . гдеs=max αi, аС не зависит отп, f и [а, Ь]. Для погрешности инте рполяционных формул получены подобные оценки, выра женные с помощью модуля непр ерывности, а для погре шности численного дифферен цирования – с помощью локального м одуля непрерывностиω k (f; x; δ). Для полученных оцено к характерно, что усло вия, накладываемые на при ближаемую функцию, не более огра ничительны, чем этого требует сама постановка зада чи (например, для квадратурных фор мул — условие интегри руемости по Риману). Для тех случае в, когда функция достаточное число раз дифференци руема, из этих оценок как следс твия вытекают все известные оценки.  相似文献   

8.
Suppose that X={X t :t≥0} is a supercritical super Ornstein-Uhlenbeck process, that is, a superprocess with an Ornstein-Uhlenbeck process on $\mathbb{R}^{d}$ corresponding to $L=\frac{1}{2}\sigma^{2}\Delta-b x\cdot\nabla$ as its underlying spatial motion and with branching mechanism ψ(λ)=?αλ+βλ 2+∫(0,+∞)(e ?λx ?1+λx)n(dx), where α=?ψ′(0+)>0, β≥0, and n is a measure on (0,∞) such that ∫(0,+∞) x 2 n(dx)<+∞. Let $\mathbb{P} _{\mu}$ be the law of X with initial measure μ. Then the process W t =e ?αt X t ∥ is a positive $\mathbb{P} _{\mu}$ -martingale. Therefore there is W such that W t W , $\mathbb{P} _{\mu}$ -a.s. as t→∞. In this paper we establish some spatial central limit theorems for X. Let $\mathcal{P}$ denote the function class $$ \mathcal{P}:=\bigl\{f\in C\bigl(\mathbb{R}^d\bigr): \mbox{there exists } k\in\mathbb{N} \mbox{ such that }|f(x)|/\|x\|^k\to 0 \mbox{ as }\|x\|\to\infty \bigr\}. $$ For each $f\in\mathcal{P}$ we define an integer γ(f) in term of the spectral decomposition of f. In the small branching rate case α<2γ(f)b, we prove that there is constant $\sigma_{f}^{2}\in (0,\infty)$ such that, conditioned on no-extinction, $$\begin{aligned} \biggl(e^{-\alpha t}\|X_t\|, ~\frac{\langle f , X_t\rangle}{\sqrt{\|X_t\|}} \biggr) \stackrel{d}{\rightarrow}\bigl(W^*,~G_1(f)\bigr), \quad t\to\infty, \end{aligned}$$ where W ? has the same distribution as W conditioned on no-extinction and $G_{1}(f)\sim \mathcal{N}(0,\sigma_{f}^{2})$ . Moreover, W ? and G 1(f) are independent. In the critical rate case α=2γ(f)b, we prove that there is constant $\rho_{f}^{2}\in (0,\infty)$ such that, conditioned on no-extinction, $$\begin{aligned} \biggl(e^{-\alpha t}\|X_t\|, ~\frac{\langle f , X_t\rangle}{t^{1/2}\sqrt{\|X_t\|}} \biggr) \stackrel{d}{\rightarrow}\bigl(W^*,~G_2(f)\bigr), \quad t\to\infty, \end{aligned}$$ where W ? has the same distribution as W conditioned on no-extinction and $G_{2}(f)\sim \mathcal{N}(0, \rho_{f}^{2})$ . Moreover W ? and G 2(f) are independent. We also establish two central limit theorems in the large branching rate case α>2γ(f)b. Our central limit theorems in the small and critical branching rate cases sharpen the corresponding results in the recent preprint of Mi?o? in that our limit normal random variables are non-degenerate. Our central limit theorems in the large branching rate case have no counterparts in the recent preprint of Mi?o?. The main ideas for proving the central limit theorems are inspired by the arguments in K. Athreya’s 3 papers on central limit theorems for continuous time multi-type branching processes published in the late 1960’s and early 1970’s.  相似文献   

9.
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 τ0, ЧтОf k * (f τ)=Τkf k * (f) Дль ВсЕхk. ДОкА жАНО сРЕДИ ДРУгИх слЕДУУЩ ЕЕ УтВЕРжДЕНИЕ. тЕОРЕМА. пУсmь {fk; 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.  相似文献   

10.
Let \(f(z) = \sum\limits_{h = 0}^\infty {f_h z^h } \) be a power series with positive radius of convergenceR f ≤1,f h algebraic and lacunary in the following sense: Let {r n }, {s n } be two infinite sequences of integers, satisfying $$0 = s_0 \leqslant r_1< s_1 \leqslant r_2< s_2 \leqslant r_3< s_3 \leqslant ..., \mathop {lim}\limits_{n \to \infty } (s_n /F(n)) = \infty $$ such that $$f_h = 0 if r_n< h< s_n ,f_{r_n } \ne 0,f_{s_n } \ne 0 for n = 1,2,3,...;$$ F(n) denotes a certain function ofn, dependent onr n and \(f_0 ,f_1 ,f_2 , \ldots f_{r_n } \) . Using ideas from a note ofK. Mahler, among other results the following main theorem is proved: The function valuef(α) (with α algebraic, 0<|α|<R f ) is algebraic if and only if there exists a positive integerN=N(α) such that $$P_n (\alpha ): = \sum\limits_{h = s_n }^{r_{n + 1} } {f_h \alpha ^h = 0 for all n \geqslant N.} $$   相似文献   

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