Search Results (28)

This paper develops a unified asymptotic theory for conditional single-index U-statistics and the associated conditional U-processes in the setting of locally stationary functional time series subject to missing-at-random response mechanisms. The proposed framework addresses, within a single nonparametric inferential architecture, three major sources of complexity in modern functional data analysis: infinite-dimensional covariates, smoothly time-varying stochastic dynamics, and incomplete response observations. The methodology is based on a class of kernel-type estimators combining temporal localization, functional single-index smoothing, and inverse-propensity correction. Temporal localization captures the gradual evolution of the underlying regression structure, the single-index projection provides an effective dimension-reduction mechanism for functional covariates, and the propensity adjustment restores the target conditional functional under the MAR sampling scheme. The principal contribution of the paper is the establishment of weak convergence, in a suitable space of bounded functions, for the resulting propensity-adjusted conditional U-process indexed by a general class of measurable kernels. Under absolute regularity conditions, local stationarity assumptions, small-ball probability requirements, entropy restrictions of VC type, and uniform consistency of the propensity-score estimator, the normalized process is shown to converge weakly to a tight centered Gaussian process. The limiting covariance structure explicitly reflects the interaction between temporal smoothing, functional concentration, dependence, and the random loss of responses. In parallel, uniform convergence rates are derived for the associated conditional single-index U-statistic estimators, thereby quantifying the respective contributions of smoothing bias, stochastic fluctuation, local-stationarity approximation error, and missingness-induced variance inflation. A substantial part of the analysis is devoted to the technical difficulties created by the simultaneous presence of dependence, nonstationarity, functional covariates, and incomplete observations. The proofs combine Hoeffding-type decompositions adapted to weighted incomplete data, blocking and coupling arguments for absolutely regular triangular arrays, refined entropy bounds for kernel-indexed function classes, and small-ball probability techniques for functional covariates. The MAR mechanism is incorporated via inverse-propensity weighting, and its effects on the effective sample size, asymptotic variance, and bias structure are made explicit. The theory also provides a rigorous foundation for bandwidth selection through blocked, propensity-adjusted cross-validation and clarifies its relation to the corresponding oracle risk. The proposed framework encompasses a broad class of statistical learning and inference problems involving pairwise or higher-order functionals of functional time series. In particular, it applies to conditional Kendall-type functionals, discrimination problems, metric learning with incomplete labels, and conditional independence testing under local stationarity. A simulation study illustrates the finite-sample behavior of the proposed estimators and supports the theoretical findings across varying regimes of temporal nonstationarity, serial dependence, functional concentration, and response missingness. Overall, the results provide a mathematically rigorous and methodologically flexible foundation for inference from evolving functional data when dependence, infinite dimensionality, and incomplete observation are present simultaneously. Full article

Let ${\mathcal{Z}}_2\left(s\right)$ denote the modified Mellin transforms of the modulus of the fourth power of the Riemann zeta-function. This paper is devoted to the probabilistic properties of generalized discrete shifts ${\mathcal{Z}}_2(s+i\psi \left(k\right))$, $k\in \mathbb{N}$, with a certain differentiable function $\psi \left(\tau \right)$ satisfying some estimate connected to the mean square of the function ${\mathcal{Z}}_2\left(s\right)$ and such that the sequence $\left\{\kappa \psi \right(k):k\in \mathbb{N}\}$ is uniformly distributed modulo 1 with every $\kappa \in \mathbb{R}\setminus \left\{0\right\}$. We propose the condition that ${\mathcal{Z}}_2(s+i\psi \left(k\right))$ in the space of analytic functions has a limit distribution concentrated at the point $g_0\left(s\right)\equiv 0$. Such a limit theorem is applied for the approximation of the function $g_0\left(s\right)$. Full article

For $j=1,\dots ,r$, let ${\mathcal{P}}_j$ be a system of generalized prime numbers, ${\mathcal{N}}_{{\mathcal{P}}_j}$ the corresponding system of generalized integers, and ${\zeta }_{{\mathcal{P}}_j}\left(s\right)$, $s=\sigma +it$, the Beurling zeta-function. In the paper, we consider simultaneous approximation of a collection of analytic functions by shifts $({\zeta }_{{\mathcal{P}}_1}(s_1+i\tau ),\dots ,{\zeta }_{{\mathcal{P}}_r}(s_r+i\tau ))$. For this, we require that the summatory functions satisfy the bounds ${\mathcal{M}}_{{\mathcal{P}}_j}\left(x\right)-a_{j}x{\ll }_{{\mathcal{P}}_j}{x}^{{\theta }_j}$ with $a_j>0$ and $0⩽{\theta }_j<1$, $j=1,\dots ,r$. Moreover, we suppose that the mean square of ${\zeta }_{{\mathcal{P}}_j}\left(s\right)$ is bounded for $\sigma \in ({\sigma }_{{\mathcal{P}}_j},1)$ with some ${\sigma }_{{\mathcal{P}}_j}$ depending on ${\mathcal{P}}_j$ and ${\theta }_j$. Then the main result of the paper asserts that there exists a closed non-empty set ${\mathcal{F}}_{{\mathcal{P}}_1,\dots ,{\mathcal{P}}_r}$ of analytic functions such that its functions $(f_1\left(s_1\right),\dots ,f_r\left(s_r\right))$ are approximated simultaneously by the above shifts of Beurling zeta-functions. It is proved that the set of approximating shifts has a positive lower density (or density with at most countably many exceptions of approximation accuracy). This shows a good joint approximation of Beurling zeta-functions. Full article

Let $\mathfrak{a}=\{a_m:m\in \mathbb{N}\}$ be a multiplicative periodic sequence of complex numbers. In this paper, we consider the approximation of analytic functions defined in the strip $\{s=\sigma +it:1/2<\sigma <1\}$ by shifts $\zeta (s+i\tau ;\mathfrak{a})$ of the zeta-function defined, for $\sigma >1$, by $\zeta (s;\mathfrak{a})={\sum }_{m=1}^{\infty }{a}_m{m}^{-s}$ and by analytic continuation elsewhere. Using stochastic techniques, we obtain that the set of the above shifts approximating a given analytic function has a positive lower density (or density with at most countably many exceptions) in the interval $[T,T+V]$ with $T^{23/70}⩽V⩽T^{1/2}$ as $T\to \infty$. The proofs are based on a limit theorem with an explicitly given limit probability measure in the space of analytic functions. Full article

The Riemann hypothesis (RH) says that all zeros of the Riemann zeta function $\zeta \left(s\right)$, $s=\sigma +it$, in the strip $\{s\in \mathbb{C}:0<\sigma <1\}$ lie on the line $\sigma =1/2$. There are many equivalents of RH in various terms. In this paper, we propose equivalents of RH in terms of self-approximation, i.e., of the approximation of $\zeta \left(s\right)$ by $\zeta (s+i\tau )$, $\tau \in \mathbb{R}$, in the interval $\tau \in [T,T+U]$ with $T^{\eta }⩽U⩽T$, $\eta =1273/4033$. We show that the RH is equivalent to the positivity of lower density and (with some exception for the accuracy of approximation) the density of the set of approximating shifts $\zeta (s+i\tau )$. For the proof, a probabilistic approach and mean square estimates for $\zeta \left(s\right)$ in short intervals are applied. Full article

Since 1975, it has been known that the Hurwitz zeta-function has a unique property to approximate by its shifts all analytic functions defined in the strip $\mathfrak{D}=\{s=\sigma +it:1/2<\sigma <1\}$. However, such an approximation causes efficiency problems, and applying short intervals is one of the measures to make that approximation more effective. In this paper, we consider the simultaneous approximation of a tuple of analytic functions in the strip $\mathfrak{D}$ by discrete shifts $(\zeta (s+ik{h}_1,{\alpha }_1),\dots ,\zeta (s+ik{h}_r,{\alpha }_r))$ with positive $h_1,\dots ,h_r$ of Hurwitz zeta-functions in the interval $[N,N+M]$ with $M={max}_{1⩽j⩽r}\left(h_j^{-1},{\left(N{h}_j\right)}^{23/70}\right)$. Two cases are considered: $1^{°}$ the set $\{(h_{j}log(m+{\alpha }_j),m\in {\mathbb{N}}_0,j=1,\dots ,r),2\pi \}$ is linearly independent over $\mathbb{Q}$; and $2^{°}$ a general case, where ${\alpha }_j$ and $h_j$ are arbitrary. In case $1^{°}$, we obtain that the set of approximating shifts has a positive lower density (and density) for every tuple of analytic functions. In case $2^{°}$, the set of approximated functions forms a certain closed set. For the proof, an approach based on new limit theorems on weakly convergent probability measures in the space of analytic functions in short intervals is applied. The power $\eta =23/70$ comes from a new mean square estimate for the Hurwitz zeta-function. Full article

In this paper, we prove the theorems on the simultaneous approximation of a pair of analytic functions by discrete shifts $(\zeta (s+ik{h}_1),\zeta (s+ik{h}_2,\alpha ))$, $h_1>0$, $h_2>0$ of the Riemann zeta function $\zeta \left(s\right)$ and Hurwitz zeta function $\zeta (s,\alpha )$. The lower density and density of the above approximating shifts are considered in short intervals $[N,N+M]$ as $N\to \infty$ with $M=o\left(N\right)$. If the set $\{(h_{1}logp:p\in \mathbb{P}),(h_{2}log(m+\alpha ):m\in {\mathbb{N}}_0),2\pi \}$ is linearly independent over $\mathbb{Q}$, the class of approximated pairs is explicitly given. If $\alpha$ and $h_1$, $h_2$ are arbitrary, then it is known that the set of approximated pairs is a certain non-empty closed subset of ${\mathbb{H}}^2(\Delta )$, where $\mathbb{H}(\Delta )$ is the space of analytic functions on the strip $\Delta =\{s\in \mathbb{C}:1/2<\mathrm{Re}s<1\}$. For the proof, limit theorems on weakly convergent probability measures in the space $H^2(\Delta )$ are applied. Full article

We prove a probabilistic limit theorem for the Epstein zeta-function $\zeta (s;Q)$ in the interval $[N,N+M]$ as $N\to \infty$, using discrete shifts $\zeta (\sigma +ikh;Q)$, where $h>0$ and $\sigma >{\displaystyle \frac{n-1}{2}}$ are fixed. Here, Q is a positive-definite $n\times n$ matrix, and the interval length M satisfies $h^{-1}{\left(Nh\right)}^{27/82}⩽M⩽h^{-1}{\left(Nh\right)}^{1/2}$. The limit measure is given explicitly. This theorem is the first result in short intervals for $\zeta (s;Q)$. The obtained theorem improves the known results established for the interval of length N. Since the considered probability measures are defined in terms of frequency, theorems in short intervals have a certain advantage in the detection of $\zeta (\sigma +ikh;Q)$ with a given property, as well as in the characterization of the asymptotic behaviour of $\zeta (s;Q)$ in general. Full article

In this paper, we obtain approximation theorems of classes of analytic functions by shifts $L(\lambda ,\alpha ,s+i\tau )$ of the Lerch zeta-function for $\tau \in [T,T+H]$ where $H\in [T^{27/82},T^{1/2}]$. The cases of all parameters, $\lambda ,\alpha \in (0,1]$, are considered. If the set $\{\log (m+\alpha ):m\in {\mathbb{N}}_0\}$ is linearly independent over $\mathbb{Q}$, then every analytic function in the strip $\{s=\sigma +it\in \mathbb{C}:\sigma \in (1/2,1\left)\right\}$ is approximated by the above shifts. Full article

For $j=1,\dots ,r$, let $Q_j$ be a positive definite $n_j\times n_j$ matrix, and $\zeta (s_j;Q_j)$ denote the corresponding Epstein zeta-function. In this paper, assuming that $n_j⩾4$ is even and ${\underline{x}}^{\mathrm{T}}{Q}_j\underline{x}\in \mathbb{Z}$,$\underline{x}\in {\mathbb{Z}}^r\setminus \left\{\underline{0}\right\}$, a joint limit theorem of Bohr–Jessen type for the functions $\zeta (s_1;Q_1),\dots ,\zeta (s_r;Q_r)$, by using generalizing shifts $\zeta ({\sigma }_1+i{\phi }_1\left(t\right);Q_1),\dots ,\zeta ({\sigma }_r+i{\phi }_r\left(t\right);Q_r)$, is proved. Here, the functions ${\phi }_1\left(t\right),\dots ,{\phi }_r\left(t\right)$ are increasing to $+\infty$, with monotonic derivatives ${\phi }_j^{\prime }\left(t\right)$ satisfying the asymptotic growth conditions: ${\phi }_j\left(t\right)\ll t{\phi }_j^{\prime }\left(t\right)$, and ${\phi }_j^{\prime }\left(t\right)=o\left({\phi }_{j+1}^{\prime }\left(t\right)\right)$ as $t\to \infty$. An explicit form of the limit measure is given. This theorem extends and generalizes the previous result on the joint value-distribution of Epstein zeta-functions. Full article

The famous Riemann hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function $\zeta \left(s\right)$ (zeros different from $s=-2m$, $m\in \mathbb{N}$) lie on the critical line $\sigma =1/2$. In this paper, combining the universality property of $\zeta \left(s\right)$ with probabilistic limit theorems, we prove that the RH is equivalent to the positivity of the density of the set of shifts $\zeta (s+i{t}_{\tau })$ approximating the function $\zeta \left(s\right)$. Here, $t_{\tau }$ denotes the Gram function, which is a continuous extension of the Gram points. Full article

In this paper, the asymptotic behavior of the modified Mellin transform ${\mathcal{Z}}_2\left(s\right)$, $s=\sigma +it$, of the fourth power of the Riemann zeta function is characterized by weak convergence of probability measures in the space of analytic functions. The main results are devoted to probability measures defined by generalized shifts ${\mathcal{Z}}_2(s+i\phi \left(\tau \right))$ with a real increasing to $+\infty$ differentiable functions connected to the growth of the second moment of ${\mathcal{Z}}_2\left(s\right)$. It is proven that the mass of the limit measure is concentrated at the point expressed as $h\left(s\right)\equiv 0$. This is used for approximation of $h\left(s\right)$ by ${\mathcal{Z}}_2(s+i\phi \left(\tau \right))$. Full article

We consider the Beurling zeta function ${\zeta }_{\mathbb{P}}\left(s\right)$, $s=\sigma +it$, of the system of generalized prime numbers $\mathbb{P}$ with generalized integers m satisfying the condition ${\sum }_{m⩽x}1=ax+O\left(x^{\delta }\right)$, $a>0$, $0⩽\delta <1$, and suppose that ${\zeta }_{\mathbb{P}}\left(s\right)$ has a bounded mean square for $\sigma >{\sigma }_{\mathbb{P}}$ with some ${\sigma }_{\mathbb{P}}<1$. Then, we prove that, for every $h>0$, there exists a closed non-empty set of analytic functions that are approximated by discrete shifts ${\zeta }_{\mathbb{P}}(s+ilh)$. This set shifts has a positive density. For the proof, a weak convergence of probability measures in the space of analytic functions is applied. Full article

In this study, the approximation of a pair of analytic functions defined on the strip $\{s=\sigma +it\in \mathbb{C}:1/2<\sigma <1\}$ by shifts $\left(\zeta \right(s+i\tau ),\zeta (s+i\tau ,\alpha \left)\right)$, $\tau \in \mathbb{R}$, of the Riemann and Hurwitz zeta-functions with transcendental $\alpha$ in the interval $[T,T+H]$ with $T^{27/82}⩽H⩽T^{1/2}$ was considered. It was proven that the set of such shifts has a positive density. The main result was an extension of the Mishou theorem proved for the interval $[0,T]$, and the first theorem on the joint mixed universality in short intervals. For proof, the probability approach was applied. Full article

U-statistics are fundamental in modeling statistical measures that involve responses from multiple subjects. They generalize the concept of the empirical mean of a random variable X to include summations over each m-tuple of distinct observations of X. W. Stute introduced conditional U-statistics, extending the Nadaraya–Watson estimates for regression functions. Stute demonstrated their strong pointwise consistency with the conditional expectation $r^{\left(m\right)}(\phi ,\mathbf{t})$, defined as $\mathbb{E}\left[\phi (Y_1,\dots ,Y_m)\right|(X_1,\dots ,X_m)=\mathbf{t}]$ for $\mathbf{t}\in {\mathcal{X}}^m$. This paper focuses on estimating functional single index (FSI) conditional U-processes for regular time series data. We propose a novel, automatic, and location-adaptive procedure for estimating these processes based on k-Nearest Neighbor (kNN) principles. Our asymptotic analysis includes data-driven neighbor selection, making the method highly practical. The local nature of the kNN approach improves predictive power compared to traditional kernel estimates. Additionally, we establish new uniform results in bandwidth selection for kernel estimates in FSI conditional U-processes, including almost complete convergence rates and weak convergence under general conditions. These results apply to both bounded and unbounded function classes, satisfying certain moment conditions, and are proven under standard Vapnik–Chervonenkis structural conditions and mild model assumptions. Furthermore, we demonstrate uniform consistency for the nonparametric inverse probability of censoring weighted (I.P.C.W.) estimators of the regression function under random censorship. This result is independently valuable and has potential applications in areas such as set-indexed conditional U-statistics, the Kendall rank correlation coefficient, and discrimination problems. Full article