Complex random energy model: zeros and fluctuations

The partition function of the random energy model at inverse temperature $\beta $ is a sum of random exponentials $ \mathcal{Z }_N(\beta )=\sum _{k=1}^N \exp (\beta \sqrt{n} X_k)$ , where $X_1,X_2,\ldots $ are independent real standard normal random variables (=random energies), and $n=\log N$ . We study the large N limit of the partition function viewed as an analytic function of the complex variable $\beta $ . We identify the asymptotic structure of complex zeros of the partition function confirming and extending predictions made in the theoretical physics literature. We prove limit theorems for the random partition function at complex $\beta $ , both on the logarithmic scale and on the level of limiting distributions. Our results cover also the case of the sums of independent identically distributed random exponentials with any given correlations between the real and imaginary parts of the random exponent.     

Limiting behavior of generalized delayed average of random sequence

Let (ξ _n ) _n∈N be a sequence of arbitrarily dependent random variables, and let \(T_{a_n ,k_n }\) be delayed sums. By virtue of the notion of asymptotic delayed log-likelihood ratio and Laplace transformation, the almost sure limiting behavior of the generalized delayed averages \(\frac{1} {{k_n }}T_{a_n ,k_n }\) is investigated, of which the results generalize and improve some work of Liu.     

Approximation of classes B_{p,\theta }^r of periodic functions of one and several variables

We obtain order-sharp estimates of best approximations to the classes $B_{p,\theta }^r$ of periodic functions of several variables in the space L _q , 1 ≤ p, q ≤ ∞ by trigonometric polynomials with “numbers” of harmonics from step hyperbolic crosses. In the one-dimensional case, we establish the order of deviation of Fourier partial sums of functions from the classes $ B_{1,\theta }^{r_1 } $ in the space L ₁.     

A general LIL for trimmed sums of random fields in banach spaces

Given a field of independent identically distributed (i.i.d.) random variables $ \left\{ {X_{\bar n} ;\bar n \in \aleph ^d } \right\} $ indexed by d-tuples of positive integers and taking values in a separable Banach space B, let $ X_{\bar n}^{(r)} = X_{\bar m} $ is the r-th maximum of $ \left\{ {\left\| {X_{\bar k} } \right\|;\bar k \leqq \bar n} \right\} $ and let $ ^{(r)} S_{\bar n} = S_{\bar n} - \left( {X_{\bar n}^{(1)} + \cdots + X_{\bar n}^{(r)} } \right) $ be the trimmed sums, where $ S_{\bar n} = \sum\nolimits_{\bar k \leqq \bar n} {X_{\bar k} } $ . This paper aims to obtain a general law of the iterated logarithm (LIL) for the trimmed sums which improves previous works.     

Uniform estimate for the tail probabilities of randomly weighted sums

Several authors have studied the uniform estimate for the tail probabilities of randomly weighted sumsa.ud their maxima. In this paper, we generalize their work to the situation thatis a sequence of upper tail asymptotically independent random variables with common distribution from the is a sequence of nonnegative random variables, independent of and satisfying some regular conditions. Moreover. no additional assumption is required on the dependence structureof {θi,i≥ 1）.     

Renewal reward process for T-related fuzzy random variables on (\mathbb {R}^{p}, \mathbb {R}^{q})

In this paper, following our previous studies, we investigate the renewal rewards process with respect to the necessity, credibility, chance measure and the expected value in which the random inter-arrival times and random rewards are characterized as weighted fuzzy numbers under \(t\) -norm-based fuzzy operations on \(\mathbb {R}^{p}\) and \(\mathbb {R}^{q}\,\,p,\,q \ge 1,\) respectively. Many versions of \(T\) -related fuzzy renewal rewards theorems are proved by using the law of large numbers for weighted fuzzy variables on \(\mathbb {R}^{p}\) . An application example is provided to illustrate the utility of the results.     

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??????.     

Average Best m-term Approximation

We introduce the concept of average best m-term approximation widths with respect to a probability measure on the unit ball or the unit sphere of $\ell_{p}^{n}$ . We estimate these quantities for the embedding $id:\ell_{p}^{n}\to\ell_{q}^{n}$ with 0<p??q??? for the normalized cone and surface measure. Furthermore, we consider certain tensor product weights and show that a typical vector with respect to such a measure exhibits a strong compressible (i.e., nearly sparse) structure. This measure may therefore be used as a random model for sparse signals.     

Recurrent statistical experiments with persistent linear regression

Statistical experiments with persistent linear regression are considered as normalized sums $ {S_N}(k)=\frac{1}{N}\sum\nolimits_{r=1}^N {{\delta_r}(k),\;\;k\geq 0} $ , of jointly independent and identically distributed, for every $ r\geq 1 $ , random variables $ {\delta_r}(k),\;\;1\leq r\leq N,\;\;k\geq 0 $ , which can take two values ±1. The persistent linear regression means that the recurrence relation $$ E\left[ {{S_N}\left( {k+1} \right)|{S_N}(k)} \right]=c-2a{S_N}(k),\;k\geq 0, $$ holds. We investigate the stable mode (equilibrium) (Theorem 2.1) and propose a stochastic approximation of statistical tests as a discrete-time autoregression process with normal perturbations (Theorem 3.1).     

On the Isotropic Constant of Random Polytopes with Vertices on an $$\ell _p$$-Sphere

The symmetric convex hull of random points that are independent and distributed according to the cone probability measure on the \(\ell _p\)-unit sphere of \({{\mathbb {R}}}^n\) for some \(1\le p < \infty \) is considered. We prove that these random polytopes have uniformly absolutely bounded isotropic constants with overwhelming probability. This generalizes the result for the Euclidean sphere \((p=2)\) obtained by Alonso-Gutiérrez. The proof requires several different tools including a probabilistic representation of the cone measure due to Schechtman and Zinn and moment estimates for sums of independent random variables with log-concave tails originating in a paper of Gluskin and Kwapień.     

Asymptotic behaviors of higher order nonlinear dynamic equations on time scales

In this paper, we study asymptotic behaviour of solutions of the following higher order nonlinear dynamic equations $$y^{\triangle^n}(t)+\delta p(t)f(y(g(t)))=0$$ and $$y^{\triangle^n}(t)+\delta p(t)f(y(h(t)))=0$$ on an arbitrary time scale $\mathbb{T}$ with $\sup {\mathbb{T}}=\infty$ , where n is a positive integer and δ=1 or ?1. We obtain some sufficient conditions for the equivalence of the oscillation of the above equations.     

An Invariance Principle for Fractional Brownian Sheets

We establish a central limit theorem for partial sums of stationary linear random fields with dependent innovations, and an invariance principle for anisotropic fractional Brownian sheets. Our result is a generalization of the invariance principle for fractional Brownian motions by Dedecker et al. (Bernoulli 17:88–113, 2011) to high dimensions. A key ingredient of their argument, the martingale approximation, is replaced by an \(m\) -approximation argument. An important tool of our approach is a moment inequality for stationary random fields recently established by El Machkouri et al. (Stoch. Process. Appl. 123:1–14, 2013).     

Asymptotics for the Tail Probability of Random Sums with a Heavy-Tailed Random Number and Extended Negatively Dependent Summands

Let {X,X_k:k≥1} be a sequence of extended negatively dependent random variables with a common distribution F satisfying EX 0.Let r be a nonnegative integer-valued random variable,independent of {X,X_k:k≥1}.In this paper,the authors obtain the necessary and sufficient conditions for the random sums S_r =(?)X_n to have a consistently varying tail when the random number t has a heavier tail than the summands,i.e.,(P(Xx))/(P(r x))→0as x→∞     

{\mathcal{C}^{\alpha}}-regularity for a class of non-diagonal elliptic systems with p-growth

We consider weak solutions to nonlinear elliptic systems in a W ^1,p -setting which arise as Euler equations to certain variational problems. The solutions are assumed to be stationary in the sense that the differential of the variational integral vanishes with respect to variations of the dependent and independent variables. We impose new structure conditions on the coefficients which yield everywhere ${\mathcal{C}^{\alpha}}$ -regularity and global ${\mathcal{C}^{\alpha}}$ -estimates for the solutions. These structure conditions cover variational integrals like ${\int F(\nabla u)\; dx}$ with potential ${F(\nabla u):=\tilde F (Q_1(\nabla u),\ldots, Q_N(\nabla u))}$ and positively definite quadratic forms in ${\nabla u}$ defined as ${Q_i(\nabla u)=\sum_{\alpha \beta} a_i^{\alpha \beta} \nabla u^\alpha \cdot \nabla u^\beta}$ . A simple example consists in ${\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{2}} + |\xi_2|^{\frac{p}{2}}}$ or ${\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{4}}|\xi_2|^{\frac{p}{4}}}$ . Since the Q _i need not to be linearly dependent our result covers a class of nondiagonal, possibly nonmonotone elliptic systems. The proof uses a new weighted norm technique with singular weights in an L ^p -setting.     

Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

We consider the Markov chain ${\{X_n^x\}_{n=0}^\infty}$ on ${\mathbb{R}^d}$ defined by the stochastic recursion ${X_{n}^{x}= \psi_{\theta_{n}} (X_{n-1}^{x})}$ , starting at ${x\in\mathbb{R}^d}$ , where ?? ₁, ?? ₂, . . . are i.i.d. random variables taking their values in a metric space ${(\Theta, \mathfrak{r})}$ , and ${\psi_{\theta_{n}} :\mathbb{R}^d\mapsto\mathbb{R}^d}$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure ??. Under appropriate assumptions on ${\psi_{\theta_n}}$ , we will show that the measure ?? has a heavy tail with the exponent ???>?0 i.e. ${\nu(\{x\in\mathbb{R}^d: |x| > t\})\asymp t^{-\alpha}}$ . Using this result we show that properly normalized Birkhoff sums ${S_n^x=\sum_{k=1}^n X_k^x}$ , converge in law to an ??-stable law for ${\alpha\in(0, 2]}$ .     

Character sums and primitive roots in algebraic number fields

Let ν denote a totally positive integer of an algebraic number fieldK such that ν is a least primitive root modulo a prime ideal \(\mathfrak{p}\) ofK, least in the sense that its normNν is minimal. One of the simplest questions that presents itself is that of the order of magnitude ofNν in comparison toN \(\mathfrak{p}\) . In the present paper the following bound is shown: $$N_\nu<< N\mathfrak{p}^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4} + a} for fixed a > 0.$$ The proof of this result is based on deep estimates for certain character sums inK.     

Boundedness of Singular Integrals on Multiparameter Weighted Hardy Spaces \text{{\textit{H}}}^\text{{\textit{p}}}_{\text{{\textit{w}}}}\ (\mathbb{R}^{\text{{\textit{n}}}}\times \mathbb{R}^{\text{{\textit{m}}}})

We apply the discrete version of Calderón??s identity and Littlewood?CPaley?CStein theory with weights to derive the $(H^p_w, H^p_w)$ and $(H^p_w, L^p_w) (0<p\le 1)$ boundedness for multiparameter singular integral operators. It turns out that even in the one-parameter case, our results substantially improve the known ones in the literature where w????A ₁ was needed. Our results in the multiparameter setting can be regarded as a natural extension of $L^p_w$ boundedness for p?>?1 for w????A _p to the case of weighted Hardy spaces $H^p_w$ for p????1, but under a weaker assumption that w belongs to the class of product A _???? weights with respect to rectangles in product spaces.     

Central limit theorems for moving average processes*

Let $ {{\left( {{\xi_n}} \right)}_{{n\in \mathbb{Z}}}} $ be a stationary sequence of real random variables with E ξ ₀ = 0 and infinite variance. Furthermore, assume that $ {{\left( {{c_n}} \right)}_{{n\in \mathbb{Z}}}} $ is a sequence of real numbers and $ {X_n}=\sum {_{{j\in \mathbb{Z}}}{c_j}{\xi_{n-j }}} $ is a moving average processes driven by $ {{\left( {{\xi_n}} \right)}_{{n\in \mathbb{Z}}}} $ . By using a decomposition of the moving average processes, a central limit theorem for the partial sums $ \sum\nolimits_{k=1}^n {{X_k}} $ is established. As applications, we obtain some central limit theorems for stationary dependent sequences $ {{\left( {{\xi_n}} \right)}_{{n\in \mathbb{Z}}}} $ , such as associated sequence, martingale difference, and so on.     

Moment Functions and Central Limit Theorem for Jacobi Hypergroups on [0,\infty [

In this paper, we derive sharp estimates and asymptotic results for moment functions on Jacobi type hypergroups. Moreover, we use these estimates to prove a central limit theorem (CLT) for random walks on Jacobi hypergroups with growing parameters $\alpha ,\beta \rightarrow \infty $ . As a special case, we obtain a CLT for random walks on the hyperbolic spaces ${H}_d(\mathbb F )$ with growing dimensions $d$ over the fields $\mathbb F =\mathbb R ,\ \mathbb C $ or the quaternions $\mathbb H $ .