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In this paper, by employing the Euler–Maclaurin summation formula and real analysis techniques, an improved version of the parameterized Hardy–Hilbert inequality involving two partial sums is established. Based on the obtained inequality, the equivalent conditions of the best possible constant factor related to several parameters are discussed. Our results extend the classical Hardy–Hilbert inequality and improve certain existing results.

Recently, some researchers determined lower bounds for the normalized version of some special functions to its sequence of partial sums, e.g., Struve and Dini functions, Wright functions and Miller–Ross functions. In this paper, we determine lower bounds for the normalized Le Roy-type Mittag-Leffler function Fα,βγ(z)=z+∑n=1∞Anzn+1, where An=ΓβΓα(n−1)+βγ and its sequence of partial sums (Fα,βγ(z))m(z)=z+∑n=1mAnzn+1. Several examples of the main results are also considered.

By the use of the techniques of analysis and some useful formulas, we give a new extension of Hilbert-type inequality with two internal variables involving one partial sums, which is a refinement of a published inequality. We provide a few equivalent conditions of the best possible constant related to multi parameters. We obtain the equivalent inequalities, the operator expressions as well as a few inequalities with the particular parameters as applications.

Under an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-Gaussian stable random variable. A functional version of this is known to be true as well, the limit process being a stable Lévy process. The main result in the paper is that for a stationary, regularly varying sequence for which clusters of high-threshold excesses can be broken down into asymptotically independent blocks, the properly centered partial sum process still converges to a stable Lévy process. Due to clustering, the Lévy triple of the limit process can be different from the one in the independent case. The convergence takes place in the space of càdlàg functions endowed with Skorohod's M₁ topology, the more usual J₁ topology being inappropriate as the partial sum processes may exhibit rapid successions of jumps within temporal clusters of large values, collapsing in the limit to a single jump. The result rests on a new limit theorem for point processes which is of independent interest. The theory is applied to moving average processes, squared GARCH(1,1) processes and stochastic volatility models.

Let 𝑋,𝑋₁,𝑋₂, . . . be a stationary sequence of negatively associated random variables. A universal result in almost sure central limit theorem for the self-normalized partial sums 𝑆𝑛/𝑉𝑛is established, where: S n = ∑ i = 1 n X i , V n 2 = ∑ i = 1 n X i 2 .

The results of Komlós, Major and Tusnády give optimal Wiener approximation of partial sums of i.i.d. random variables and provide an extremely powerful tool in probability and statistical inference. Recently Wu [Ann. Probab. 35 (2007) 2294–2320] obtained Wiener approximation of a class of dependent stationary processes with finite pth moments, 2 < p ≤ 4, with error term o(n 1/p (log n) γ ), γ > 0, and Liu and Lin [Stochastic Process. Appl. 119 (2009) 249—280] removed the logarithmic factor, reaching the Komlós—Major—Tusnády bound o(n 1/p ). No similar results exist for p > 4, and in fact, no existing method for dependent approximation yields an a.s. rate better than o(n 1/4 ). In this paper we show that allowing a second Wiener component in the approximation, we can get rates near to o(n 1/p ) for arbitrary p > 2. This extends the scope of applications of the results essentially, as we illustrate it by proving new limit theorems for increments of stochastic processes and statistical tests for short term (epidemic) changes in stationary processes. Our method works under a general weak dependence condition covering wide classes of linear and nonlinear time series models and classical dynamical systems.

It is proved that $\\lim_{n\\rightarrow \\infty }inf(p_{n+1}-p_n)\\textless 7\\times 10^7$, where pn is the n-th prime. Our method is a refinement of the recent work of Goldston, Pintz and Yildirim on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime divisors only, but it is adequate for our purpose.

We consider the class of all analytic and locally univalent functionsfof the form f ( z ) = z + ∑ n = 2 ∞ a 2 n − 1 z 2 n − 1 , | z | 1 , satisfying the condition Re ( 1 + z f ″ ( z ) f ′ ( z ) ) > − 1 2 . We show that every section s 2 n − 1 ( z ) = z + ∑ k = 2 n a 2 k − 1 z 2 k − 1 , off, is convex in the disk | z | 2 / 3 . We also prove that the radius 2 / 3 is best possible, i.e. the number 2 / 3 cannot be replaced by a larger one.

This paper discusses two goodness-of-fit testing problems. The first problem pertains to fitting an error distribution to an assumed nonlinear parametric regression model, while the second pertains to fitting a parametric regression model when the error distribution is unknown. For the first problem the paper contains tests based on a certain martingale type transform of residual empirical processes. The advantage of this transform is that the corresponding tests are asymptotically distribution free. For the second problem the proposed asymptotically distribution free tests are based on innovation martingale transforms. A Monte Carlo study shows that the simulated level of the proposed tests is close to the asymptotic level for moderate sample sizes.

The modified Abel lemma on summation by parts is employed to investigate the partial sum of Dougall's ₅H₅-series. Several unusual transformation formulae into fast convergent series are established. They lead surprisingly to numerous infinite series identities involving π, ς(3) and the Catalan constant, including several important ones discovered by Ramanujan (1914) and recently by Guillera.