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).     

Embeddings of weighted sobolev spaces and generalized Caffarelli-Kohn-Nirenberg inequalities

We characterize all the real numbers a, b, c and 1 ?? p, q, r < ?? such that the weighted Sobolev space $$W_{\{ a,b\} }^{\{ q,q\} }({R^N}\backslash \{ 0\} ): = \{ u \in L_{loc}^1({R^N}\backslash \{ 0\} ):{\left| x \right|^{a/q}} \in {L^q}({R^{N),}}{\left| x \right|^{b/p}}\nabla u \in {({L^p}({R^N}))^N}\} $$ is continuously embedded into $${L^r}({R^N};{\left| x \right|^c}dx): = \{ u \in L_{loc}^1({R^N}\backslash \{ 0\} ):{\left| x \right|^{c/r}}u \in {L^r}({R^N})\} $$ with norm ??·?? _c,r . It turns out that, except when N ?? 2 and a = c = b ? p = ?N, such an embedding is equivalent to the multiplicative inequality $${\left\| u \right\|_{c,r}} \le C\left\| {\nabla u} \right\|_{b,p}^\theta \left\| u \right\|_{a,q}^{1 - \theta }$$ for some suitable ?? ?? [0, 1], which is often but not always unique. If a, b, c > ?N, then C ₀ ^?? (? ^N ) ? W _{a,b} ^(q,p) (? ^N {0}) ?? L ^r (? ^N ; |x| ^c dx) and such inequalities for u ?? C ₀ ^?? (? ^N ) are the well-known Caffarelli-Kohn-Nirenberg inequalities; but their generalization to W _{a,b} ^(q,p) (? ^N {0}) cannot be proved by a denseness argument. Without the assumption a, b, c > ?N, the inequalities are essentially new, even when u ?? C ₀ ^?? (? ^N {0}), although a few special cases are known, most notably the Hardy-type inequalities when p = q. In a different direction, the embedding theorem easily yields a generalization when the weights |x| ^a , |x| ^b and |x| ^c are replaced with more general weights w _a ,w _b and w _c , respectively, having multiple power-like singularities at finite distance and at infinity.     

Euler–Maclaurin Expansions for Integrals with Arbitrary Algebraic-Logarithmic Endpoint Singularities

In this paper, we provide the Euler?CMaclaurin expansions for (offset) trapezoidal rule approximations of the finite-range integrals $I[f]=\int^{b}_{a}f(x)\,dx$ , where f??C ^??(a,b) but can have general algebraic-logarithmic singularities at one or both endpoints. These integrals may exist either as ordinary integrals or as Hadamard finite part integrals. We assume that f(x) has asymptotic expansions of the general forms where $\widehat{P}(y),P_{s}(y)$ and $\widehat{Q}(y),Q_{s}(y)$ are polynomials in y. The ?? _s and ?? _s are distinct, complex in general, and different from ?1. They also satisfy The results we obtain in this work extend the results of a recent paper [A.?Sidi, Numer. Math. 98:371?C387, 2004], which pertain to the cases in which $\widehat{P}(y)\equiv0$ and $\widehat{Q}(y)\equiv0$ . They are expressed in very simple terms based only on the asymptotic expansions of f(x) as x??a+ and x??b?. The results we obtain in this work generalize, and include as special cases, all those that exist in the literature. Let $D_{\omega}=\frac{d}{d\omega}$ , h=(b?a)/n, where n is a positive integer, and define $\check{T}_{n}[f]=h\sum^{n-1}_{i=1}f(a+ih)$ . Then with $\widehat{P}(y)=\sum^{\hat{p}}_{i=0}{\hat{c}}_{i}y^{i}$ and $\widehat{Q}(y)=\sum^{\hat{q}}_{i=0}{\hat{d}}_{i}y^{i}$ , one of these results reads where ??(z) is the Riemann Zeta function and ?? _i are Stieltjes constants defined via $\sigma_{i}= \lim_{n\to\infty}[\sum^{n}_{k=1}\frac{(\log k)^{i}}{k}-\frac{(\log n)^{i+1}}{i+1}]$ , i=0,1,???.     

Базисы из экспонент в пространствахL p

Let the rootsλ _n of an entire functionL(z) be separated and lie in some horizontal strip ¦Im z¦ ≦h, and suppose that $$0< c \leqq |L(z)|(1 + |z|)^{ - b} \exp ( - a|\operatorname{Im} z|) \leqq C< \infty$$ for ¦Imz¦≧H>h. If 1<p<2 and - 1/pq (1/q+1/p=1), then the system {exp (iλ _n x)} _n=0 ^∞ constitutes a basis нn the spaceL ^p (-a,a). In the caseb=1/q orb=?1/p the theorem fails, Equivalence of the following two statements is also proved:

1. {exp (iλ _n x)} _n=0 ^∞ is an extendable convergence system inL ^p from the interval (-a, a).
2. {exp (iλ _n x)} _n=0 ^∞ is a continuable basis inL ^p (-a,a).

     

A Congruence Connecting Latin Rectangles and Partial Orthomorphisms

A partial orthomorphism of ${\mathbb{Z}_{n}}$ is an injective map ${\sigma : S \rightarrow \mathbb{Z}_{n}}$ such that ${S \subseteq \mathbb{Z}_{n}}$ and ??(i)?Ci ? ??(j)? j (mod n) for distinct ${i, j \in S}$ . We say ?? has deficit d if ${|S| = n - d}$ . Let ??(n, d) be the number of partial orthomorphisms of ${\mathbb{Z}_{n}}$ of deficit d. Let ??(n, d) be the number of partial orthomorphisms ?? of ${\mathbb{Z}_n}$ of deficit d such that ??(i) ? {0, i} for all ${i \in S}$ . Then ??(n, d) =???(n, d)n ²/d ² when ${1\,\leqslant\,d < n}$ . Let R _{k, n} be the number of reduced k ×?n Latin rectangles. We show that $$R_{k, n} \equiv \chi (p, n - p)\frac{(n - p)!(n - p - 1)!^{2}}{(n - k)!}R_{k-p,\,n-p}\,\,\,\,(\rm {mod}\,p)$$ when p is a prime and ${n\,\geqslant\,k\,\geqslant\,p + 1}$ . In particular, this enables us to calculate some previously unknown congruences for R _{n, n} . We also develop techniques for computing ??(n, d) exactly. We show that for each a there exists??? _a such that, on each congruence class modulo??? _a , ??(n, n-a) is determined by a polynomial of degree 2a in n. We give these polynomials for ${1\,\leqslant\,a\,\leqslant 6}$ , and find an asymptotic formula for ??(n, n-a) as n ?? ??, for arbitrary fixed a.     

Isothermic Submanifolds

We propose an answer to a question raised by F. Burstall: Is there any interesting theory of isothermic submanifolds of ? ⁿ of dimension greater than two? We call an n-immersion f(x) in ? ^m isothermic _k if the normal bundle of f is flat and x is a line of curvature coordinate system such that its induced metric is of the form $\sum_{i=1}^{n} g_{ii}\,\mathrm{d} x_{i}^{2}$ with $\sum_{i=1}^{n} \epsilon_{i} g_{ii}=0$ , where ?? _i =1 for 1??i??n?k and ?? _i =?1 for n?k<i??n. A smooth map (f ₁,??,f _n ) from an open subset ${\mathcal{O}}$ of ? ⁿ to the space of m×n matrices is called an n-tuple of isothermic _k n-submanifolds in ? ^m if each f _i is an isothermic _k immersion, $(f_{i})_{x_{j}}$ is parallel to $(f_{1})_{x_{j}}$ for all 1??i,j??n, and there exists an orthonormal frame (e ₁,??,e _n ) and a GL(n)-valued map (a _ij ) such that $\mathrm{d}f_{i}= \sum_{j=1}^{n} a_{ij} e_{j}\,\mathrm {d} x_{j}$ for 1??i??n. Isothermic₁ surfaces in ?³ are the classical isothermic surfaces in ?³. Isothermic _k submanifolds in ? ^m are invariant under conformal transformations. We show that the equation for n-tuples of isothermic _k n-submanifolds in ? ^m is the $\frac{O(m+n-k,k)}{O(m)\times O(n-k,k)}$ -system, which is an integrable system. Methods from soliton theory can therefore be used to construct Christoffel, Ribaucour, and Lie transforms, and to describe the moduli spaces of these geometric objects and their loop group symmetries.     

On ≤k-Edges, Crossings, and Halving Lines of Geometric Drawings of K n

Let P be a set of points in general position in the plane. Join all pairs of points in P with straight line segments. The number of segment-crossings in such a drawing, denoted by $\operatorname {cr}(P)$ , is the rectilinear crossing number of P. A halving line of P is a line passing through two points of P that divides the rest of the points of P in (almost) half. The number of halving lines of P is denoted by h(P). Similarly, a k-edge, 0??k??n/2?1, is a line passing through two points of P and leaving exactly k points of P on one side. The number of ??k-edges of P is denoted by E _??k (P). Let $\overline {\mathrm {cr}}(n)$ , h(n), and E _??k (n) denote the minimum of $\operatorname {cr}(P)$ , the maximum of h(P), and the minimum of E _??k (P), respectively, over all sets P of n points in general position in the plane. We show that the previously best known lower bound on E _??k (n) is tight for k<?(4n?2)/9? and improve it for all k???(4n?2)/9?. This in turn improves the lower bound on $\overline {\mathrm {cr}}(n)$ from $0.37968\binom{n}{4}+\varTheta (n^{3})$ to $\frac{277}{729}\binom{n}{4}+\varTheta (n^{3})\geq 0.37997\binom{n}{4}+\varTheta (n^{3})$ . We also give the exact values of $\overline {\mathrm {cr}}(n)$ and h(n) for all n??27. Exact values were known only for n??18 and odd n??21 for the crossing number, and for n??14 and odd n??21 for halving lines.     

Asymptotics of eigenvalues of A regular boundary-value problem

We study a boundary-value problem x ⁽ⁿ⁾ + Fx = λx, U _h(x) = 0, h = 1,..., n, where functions x are given on the interval [0, 1], a linear continuous operator F acts from a Hölder space H ^y into a Sobolev space W ₁ ^n+s , U _h are linear continuous functional defined in the space $H^{k_h } $ , and k _h ≤ n + s - 1 are nonnegative integers. We introduce a concept of k-regular-boundary conditions U _h(x)=0, h = 1, ..., n and deduce the following asymptotic formula for eigenvalues of the boundary-value problem with boundary conditions of the indicated type: $\lambda _v = \left( {i2\pi v + c_ \pm + O(|v|^\kappa )} \right)^n $ , v = ± N, ± N ± 1,..., which is true for upper and lower sets of signs and the constants κ≥0 and c _± depend on boundary conditions.     

Generalized monotone approximation inL p Space

Letf(x) ∈L _p[0,1], 1?p? ∞. We shall say that functionf(x)∈Δ^k (integerk?1) if for anyh ∈ [0, 1/k] andx ∈ [0,1?kh], we have Δ _h ^k f(x)?0. Denote by ∏ _n the space of algebraic polynomials of degree not exceedingn and define $$E_{n,k} (f)_p : = \mathop {\inf }\limits_{\mathop {P_n \in \prod _n }\limits_{P_n^{(\lambda )} \geqslant 0} } \parallel f(x) - P_n (x)\parallel _{L_p [0,1]} .$$ We prove that for any positive integerk, iff(x) ∈ Δ ^k ∩ L _p[0, 1], 1?p?∞, then we have $$E_{n,k} (f)_p \leqslant C\omega _2 \left( {f,\frac{1}{n}} \right)_p ,$$ whereC is a constant only depending onk.     

On coefficients of null-series and on sets of uniqueness of trigonometric and Walsh systems

В работе доказываютс я следующие утвержде ния. Теорема I.Пусть ? _n ↓0u \(\sum\limits_{n = 0}^\infty {\varepsilon _n^2 = + \infty } \) .Тогд а существует множест во Е?[0, 1]с μЕ=0 такое что:1. Существует ряд \(\sum\limits_{n = 0}^\infty {a_n W_n } (t)\) с к оеффициентами ¦а _n ¦≦{in¦n¦, который сх одится к нулю всюду вне E и ε∥a_n∥>0.2. Если b _n ¦=о(ε _n )и ряд \(\sum\limits_{n = 0}^\infty {b_n W_n (t)} \) сх одится к нулю всюду вн е E за исключением быть может некоторого сче тного множества точе к, то b _n =0для всех п. Теорема 3.Пусть ? _n ↓0u \(\mathop {\lim \sup }\limits_{n \to \infty } \frac{{\varepsilon _n }}{{\varepsilon _{2n} }}< \sqrt 2 \) Тогд а существует множест во E?[0, 1] с υ E=0 такое, что:

1. Существует ряд \(\sum\limits_{n = - \infty }^{ + \infty } {a_n e^{inx} ,} \sum\limits_{n = - \infty }^{ + \infty } {\left| {a_n } \right|} > 0,\) кот орый сходится к нулю в сюду вне E и ¦a_n≦¦n¦ для n=±1, ±2, ...
2. Если ряд \(\sum\limits_{n = - \infty }^{ + \infty } {b_n e^{inx} } \) сходится к нулю всюду вне E и ¦b_v¦=о(ε ¦n¦), то b_n=0 для всех я. Теорема 5. Пусть послед овательности S⁽¹⁾={ε ₀ ⁽¹⁾ , ε ₁ ⁽¹⁾ , ε ₂ ⁽¹⁾ , ...} u S²=ε ₀ ⁽²⁾ , ε ₁ ⁽²⁾ . ε ₂ ⁽²⁾ монотонно стремятся к нулю, \(\mathop {\lim \sup }\limits_{n \to \infty } \varepsilon ^{(i)} /\varepsilon _{2n}^{(i)}< 2,i = 1,2\) , причем \(\mathop {\lim }\limits_{n \to \infty } \varepsilon _n^{(2)} /\varepsilon _n^{(i)} = + \infty \) . Тогда для каждого ε>O н айдется множество Е? [-π,π], μE >2π — ε, которое является U(S¹), но не U(S¹) — множеством для тригонометричес кой системы. Аналог теоремы 5 для си стемы Уолша был устан овлен в [7].

     

Asymptotic expansions of solutions of equations with a deviating argument in Banach spaces

For the equation $$Lu = \frac{1}{i}\frac{{du}}{{dt}}\sum\nolimits_{j = 0}^m {A_j u} (l - h_j^0 - h_j^1 (t)) = f(t),$$ whereh ₀ ^o =0,h ₀ ¹ =0 (t) ≡ 0,h _j ^o = const > 0,h ₁ ^j (t),j= 1, ...,m are nonnegative continuously differentiable functions in [0, ∞), Aj are bounded linear operators, under conditions on the resolvent and on the right hand sidef(t), we have obtained an asymptotic formula for any solution u(t) from L₂ in terms of the exponential solutions u_k(t), k=1, ..., n, of the equation $$\frac{1}{i}\frac{{du}}{{dt}} - A_0 u - \sum\nolimits_{j = 0}^m {A_j u} (t - h_j^0 ) = 0,$$ connected with the poles λ_k, k=1, ..., n, of the resolvent R_λ in a certain strip.     

On the growth order of the rectangular partial sums of multiple non-orthogonal series

Пустьd-натуральное ч исло,Z ^d — множество на боров k=(k ₁, ...,k _d ), состоящих из неотрицательных цел ыхk _j ,Z ₊ ^d =k∈Z ^d :k≧1. Предположи м, что системаf _k (x):k∈Z ₊ ^d ? ?L₂(X,A, μ) и последовател ьностьa _k :k∈Z ₊ ^d . таковы, чт о для всех b∈Z^d и m∈Z ₊ ^d выполн ены неравенства (2) $$\left\| {\sum\limits_{b + 1 \leqq k \leqq b + m} {a_k f_k (x)} } \right\|_2^2 \leqq w^2 (m)\sum\limits_{b + 1 \leqq k \leqq b + m} {a_k^2 } $$ где последовательно сть {w(m): m∈Z ₊ ^d положительн а и не убывает. Например, есл иf _k (х) — квазистационарная система, то для соотве тствующей последовательности {ω(m) (2) имeeт Меcтo ДЛЯ ЛЮбОЙ ПОС ЛеДОВатеЛЬНОСТИ {a_k}. В работе получены оце нки порядка роста пря моугольных частных суммS _m (x)= =∑ a_kf_k(x) при maxmj→∞ как в случ ае {a_k}∈l₂, таки для {a_k}l₂. Эти оценки явля1≦k≦m 1≦j≦d ются новыми даже для о ртогональных кратны х рядов. Показано, что упомяну тые оценки в общем слу чае являются точными.     

Asymptotics of the variance of the number of real roots of random trigonometric polynomials

Let an,n 1 be a sequence of independent standard normal random variables.Consider the random trigonometric polynomial Tn(θ)=∑nj=1 aj cos(jθ),0≤θ≤2π and let Nn be the number of real roots of Tn(θ) in(0,2π).In this paper it is proved that limn →∞ Var(Nn)/n=c0,where 0相似文献   

Limit ultraspherical series and their approximation properties

We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$ , k = 0, 1, …, is obtained by passing to the limit as α→?1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> ?1, an orthonormal system with weight (1 ? x ₂)α on [?1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ . In particular, it is shown that the operator S _n ^?1 (f) = S _n ^?1 (f, x) is the projection onto the subspace of algebraic polynomials p _n = p _n (x) of degree at most n, i.e., S _n (p _n ) = p _n ; in addition, S _n ^?1 (f, x) coincides with f(x) at the endpoints ±1, i.e., S _n ^?1 (f,±1) = f(±1). It is proved that the Lebesgue function Λ _n (x) of the partial sums S _n ^?1 (f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ .     

Solution of the Szili type conjugate problem on the roots of the laguerre polynomials

Let ?1<α≤0 and let $$L_n^{(\alpha )} (x) = \frac{1}{{n!}}x^{ - \alpha } e^x \frac{{d^n }}{{dx^n }}(x^{\alpha + n} e^{ - x} )$$ be the generalizednth Laguerre polynomial,n=1,2,… Letx ₁,x ₂,…,x _n andx*₁,x*₂,…,x* _n?1 denote the roots ofL _n ^(α) (x) andL _n ^(α)′ (x) respectively and putx*₀=0. In this paper we prove the following theorem: Ify ₀,y ₁,…,y _n ?1 andy ₁ ^′ ,…,y _n ^′ are two systems of arbitrary real numbers, then there exists a unique polynomialP(x) of degree 2n?1 satisfying the conditions $$\begin{gathered} P\left( {x_k^* } \right) = y_k (k = 0,...,n - 1) \hfill \\ P'\left( {x_k } \right) = y_k^\prime (k = 1,...,n). \hfill \\ \end{gathered} $$ .     

Limiting Spectral Distribution of Random k-Circulants

Consider random k-circulants A _k,n with n????,k=k(n) and whose input sequence {a _l } _l??0 is independent with mean zero and variance one and $\sup_{n}n^{-1}\sum_{l=1}^{n}\mathbb{E}|a_{l}|^{2+\delta}<\infty$ for some ??>0. Under suitable restrictions on the sequence {k(n)} _n??1, we show that the limiting spectral distribution (LSD) of the empirical distribution of suitably scaled eigenvalues exists, and we identify the limits. In particular, we prove the following: Suppose g??1 is fixed and p ₁ is the smallest prime divisor of g. Suppose $P_{g}=\prod_{j=1}^{g}E_{j}$ where {E _j }_1??j??g are i.i.d. exponential random variables with mean one. (i) If k ^g =?1+sn where s=1 if g=1 and $s=o(n^{p_{1}-1})$ if g>1, then the empirical spectral distribution of n ^?1/2 A _k,n converges weakly in probability to $U_{1}P_{g}^{1/(2g)}$ where U ₁ is uniformly distributed over the (2g)th roots of unity, independent of P _g . (ii) If g??2 and k ^g =1+sn with $s=o(n^{p_{1}-1})$ , then the empirical spectral distribution of n ^?1/2 A _k,n converges weakly in probability to $U_{2}P_{g}^{1/(2g)}$ where U ₂ is uniformly distributed over the unit circle in ?², independent of P _g . On the other hand, if k??2, k=n ^o(1) with gcd?(n,k)=1, and the input is i.i.d. standard normal variables, then $F_{n^{-1/2}A_{k,n}}$ converges weakly in probability to the uniform distribution over the circle with center at (0,0) and radius $r=\exp(\mathbb{E}[\log\sqrt{E}_{1}])$ .     

Exponential sums over primes and the prime twin problem

The purpose of this paper is to obtain an effective estimate of the exponential sum $\sum_{n\le x}\Lambda(n)e\left(\left(\frac{a}{q}+\beta\right)n\right)$ (where e(??)=e ^{2?? i ??} , ??,?????, (a,q)=1 and ?? is the von Mangoldt function) in the range ${(\log x)}^{1/2+\varepsilon}\le q\le \frac{x}{{(\log\log\log x)}^{1+\varepsilon}}$ and $|\beta|<\frac{1}{q{(\log\log\log x)}^{1+\varepsilon}}$ . It improves Daboussi??s estimate [2, Theorem 1] in the range q??(log?x) ^D and x(log?x)^?D ??q, D>0 and is valid in a wider range for ??.     

Padé approximants of special functions

Let $ {f_{\gamma }}(x) = \sum\nolimits_{{k = 0}}^{\infty } {{{{T_k (x)}} \left/ {{{{\left( \gamma \right)}_k}}} \right.}} $ , where (??) _k =??(??+1) ? (??+k?1) and T _k (x)=cos (k arccos x) are Padé?CChebyshev polynomials. For such functions and their Padé?CChebyshev approximations $ \pi_{n,m}^{ch}\left( {x;{f_{\gamma }}} \right) $ , we find the asymptotics of decreasing the difference $ {f_{\gamma }}(x) - \pi_{n,m}^{ch}\left( {x;{f_{\gamma }}} \right) $ in the case where 0 ? m ? m(n), m(n) = o (n), as n???? for all x ?? [?1, 1]. Particularly, we determine that, under the same assumption, the Padé?CChebyshev approximations converge to f _?? uniformly on the segment [?1, 1] with the asymptotically best rate.     

Convolutions of random measures on compact groups

LetG be a compact group andM ₁(G) be the convolution semigroup of all Borel probability measures onG with the weak topology. We consider a stationary sequence {μ _n } _n=?∞ ^+∞ of random measures μ _n =μ _n (ω) inM ₁(G) and the convolutions $$v_{m,n} (\omega ) = \mu _m (\omega )* \cdots *\mu _{n - 1} (\omega ), m< n$$ and $$\alpha _n^{( + k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n,n + i} (\omega ),} \alpha _n^{( - k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n - i,n} (\omega )} $$ We describe the setsA _m ⁺ (ω) andA _n ⁺ (ω) of all limit points ofv _m,n(ω) asm→?∞ orn→+∞ and the setA _∞(ω) of its two-sided limit points for typical realizations of {μ _n (ω)} _n=?∞ ^+∞ . Using an appropriate random ergodic theorem we study the limit random measures ρ _n ^(±) (ω)=lim _k→∞ α _n ^(±k) (ω).     

On the relation of generalized Valiron summability to Cesàro summability

A family (V _a ^k ) of summability methods, called generalized Valiron summability, is defined. The well-known summability methods (B_α,γ), (E _ρ, (T_α), (S _β) and (V _a) are members of this family. In §3 some properties of the (B _α,γ) and (V _a ^k ) transforms are established. Following Satz II of Faulhaber (1956) it is proved that the members of the (V _a ^k ) family are all equivalent for sequences of finite order. This paper is a good illustration of the use of generalized Boral summability. The following theorem is established: Theorem.If s _n (n ≥ 0) isa real sequence satisfying $$\mathop {lim}\limits_{ \in \to 0 + } \mathop {lim inf}\limits_{m \to \infty } \mathop {min}\limits_{m \leqslant n \leqslant m \in \sqrt m } \left( {\frac{{S_n - S_m }}{{m^p }}} \right) \geqslant 0(\rho \geqslant 0)$$ , and if s_n →s (V _a ^k ) thens _n → s (C, 2ρ).