arXiv subjects

Mikhail Lifshits

Publications and source records attributed to Mikhail Lifshits.

At least 37 records · Page 2Linked to original sources

arXiv2017

Adaptive Energy Saving Approximation for Random Stationary Processes

We consider a stationary process (with either discrete or continuous time) and find an adaptive approximating stationary process combining approximation quality and supplementary good properties that can be interpreted as additional smoothness or small expense of energy. The problem is solved in terms of the spectral characteristics of the approximated process by using classical analytic methods from prediction theory.

math.PR↗

arXiv2016

Some extensions of linear approximation and prediction problems for stationary processes

Let $(B(t))_{t\in Θ}$ with $Θ={\mathbb Z}$ or $Θ={\mathbb R}$ be a wide sense stationary process with discrete or continuous time. The classical linear prediction problem consists of finding an element in $\overline{span}\{B(s),s\le t\}$ providing the best possible mean square approximation to the variable $B(τ)$ with $τ>t$. In this article we investigate this and some other similar problems where, in addition to prediction quality, optimization takes into account other features of the objects we search for. One of the most motivating examples of this kind is an approximation of a stationary process $B$ by a stationary differentiable process $X$ taking into account the kinetic energy that $X$ spends in its approximation efforts.

math.PR↗

arXiv2016

Small deviations of sums of correlated stationary Gaussian sequences

We consider the small deviation probabilities (SDP) for sums of stationary Gaussian sequences. For the cases of constant boundaries and boundaries tending to zero, we obtain quite general results. For the case of the boundaries tending to infinity, we focus our attention on the discrete analogs of the fractional Brownian motion (FBM). It turns out that the lower bounds for the SDP can be transferred from the well studied FBM caseto the discrete time setting under the usual assumptions that imply weak convergence while the transfer of the corresponding upper bounds necessarily requires a deeper knowledge of the spectral structure of the underlying stationary sequence.

math.PR↗

arXiv2016

Small Deviations in $L_2$-norm for Gaussian Dependent Sequences

Let $U=(U_k)_{k\in\mathbb{Z}}$ be a centered Gaussian stationary sequence satisfying some minor regularity condition. We study the asymptotic behavior of its weighted $\ell_2$-norm small deviation probabilities. It is shown that \[ \ln \mathbb{P}\left( \sum_{k\in\mathbb{Z}} d_k^2 U_k^2 \leq \varepsilon^2\right) \sim - M \varepsilon^{-\frac{2}{2p-1}}, \qquad \textrm{ as } \varepsilon\to 0, \] whenever \[ d_k\sim d_{\pm} |k|^{-p}\quad \textrm{for some } p>\frac{1}{2} \, , \quad k\to \pm\infty, \] using the arguments based on the spectral theory of pseudo-differential operators by M. Birman and M. Solomyak. The constant $M$ reflects the dependence structure of $U$ in a non-trivial way, and marks the difference with the well-studied case of the i.i.d. sequences.

math.PR↗

arXiv2015

Bifractional Brownian motion: existence and border cases

Bifractional Brownian motion (bfBm) is a centered Gaussian process with covariance \[ R^{(H,K)}(s,t)= 2^{-K} \left( \left(|s|^{2H}+|t|^{2H} \right)^{K}-|t-s|^{2HK}\right), \qquad s,t\in R. \] We study the existence of bfBm for a given pair of parameters $(H,K)$ and encounter some related limiting processes.

math.PR↗

arXiv2015

Least Energy Approximation for Processes with Stationary Increments

A function $f=f_T$ is called least energy approximation to a function $B$ on the interval $[0,T]$ with penalty $Q$ if it solves the variational problem $$ \int_0^T \left[ f'(t)^2 + Q(f(t)-B(t)) \right] dt \searrow \min. $$ For quadratic penalty the least energy approximation can be found explicitly. If $B$ is a random process with stationary increments, then on large intervals $f_T$ also is close to a process of the same class and the relation between the corresponding spectral measures can be found. We show that in a long run (when $T\to \infty$) the expectation of energy of optimal approximation per unit of time converges to some limit which we compute explicitly. For Gaussian and Lévy processes we complete this result with almost sure and $L^1$ convergence. As an example, the asymptotic expression of approximation energy is found for fractional Brownian motion.

math.PR↗

arXiv2014

Energy of taut strings accompanying Wiener process

Let $W$ be a Wiener process. The function $h(\cdot)$ minmizing energy $\int_0^T h'(t)^2\, dt$ among all functions satisfying $W(t)-r \le h(t) \le W(t)+ r$ on an interval $[0,T]$ is called taut string. This is a classical object well known in Variational Calculus, Mathematical Statistics, etc. We show that the energy of this taut string on large intervals is equivalent to $C^2 T\, /\, r^2$ where $C$ is some finite positive constant. While the precise value of $C$ remains unknown, we give various theoretical bounds for it as well as rather precise results of computer simulation. While the taut string clearly depends on entire trajectory of $W$, we also consider an adaptive version of the problem by giving a construction (Markovian pursuit) of a random function based only on the past values of $W$ and having minimal asymptotic energy. The solution, an optimal pursuit strategy, quite surprisingly turns out to be related with a classical minimization problem for Fisher information on the bounded interval.

math.PR↗

arXiv2014

Approximation of additive random fields based on standard information: average case and probabilistic settings

We consider approximation problems for tensor product and additive random fields based on standard information in the average case setting. We also study the probabilistic setting of the mentioned problem for tensor products. The main question we are concerned with in this paper is ``How much do we loose by considering standard information algorithms against those using general linear information?'' For both types of the fields, the error of linear algorithms has been studied in great detail. However, the power of standard information for them was not addressed so far, which we do here. Our main conclusion is that in most interesting cases there is no more than a logarithmic loss in approximation error when information is being restricted to the standard one. The results are obtained by randomization techniques.

math.PR↗

arXiv2012

Fractional integration operators of variable order: continuity and compactness properties

Let a:[0,1] -> R be a Lebesgue-almost everywhere positive function. We consider the Riemann-Liouville operator R^a of variable order a(.) as an operator from L_p[0,1] to L_q[0,1]. Our first aim is to study its continuity properties. For example, we show that R^a is always continuous in L_p[0,1] if p>1. Surprisingly, this becomes false for p=1. In order R^a to be continuous in L_1[0,1], the function a(.) has to satisfy some additional assumptions. In the second, central part of this paper we investigate compactness properties of R^a. We characterize functions a(.) for which R^a is a compact operator and for certain classes of functions a(.) we provide order-optimal bounds for the dyadic entropy numbers e_n(R^a).

math.FA↗

arXiv2012

Approximation of a random process with variable smoothness

We consider the rate of piecewise constant approximation to a locally stationary process $X(t),t\in [0,1]$, having a variable smoothness index $α(t)$. Assuming that $α(\cdot)$ attains its unique minimum at zero and satisfies the regularity condition, we propose a method for construction of observation points (composite dilated design) and find an asymptotics for the integrated mean square error, where a piecewise constant approximation $X_n$ is based on $N(n)\sim n$ observations of $X$. Further, we prove that the suggested approximation rate is optimal, and then show how to find an optimal constant.

math.PR↗

arXiv2012

Persistence probabilities for an integrated random walk bridge

We prove that an integrated simple random walk, where random walk and integrated random walk are conditioned to return to zero, has asymptotic probability $n^{-1/2}$ to stay positive. This question is motivated by so-called random polymer models and proves a conjecture by Caravenna and Deuschel.

math.PR↗

arXiv2011

An Inequality Related to Bifractional Brownian Motion

We prove that for any pair of i.i.d. random variables $X,Y$ with finite moment of order $a \in (0,2]$ it is true that $E |X-Y|^a \leq E |X+Y|^a$. Surprisingly, this inequality turns out to be related with bifractional Brownian motion. We extend this result to Bernstein functions and provide some counter-examples.

math.PR↗

arXiv2010

Random Gaussian sums on trees

Let $T$ be a tree with induced partial order $\preceq$. We investigate centered Gaussian processes $X=(X_t)_{t\in T}$ represented as $$ X_t=σ(t)\sum_{v \preceq t}α(v)ξ_v $$ for given weight functions $α$ and $σ$ on $T$ and with $(ξ_v)_{v\in T}$ i.i.d. standard normal. In a first part we treat general trees and weights and derive necessary and sufficient conditions for the a.s. boundedness of $X$ in terms of compactness properties of $(T,d)$. Here $d$ is a special metric defined via $α$ and $σ$, which, in general, is not comparable with the Dudley metric generated by $X$. In a second part we investigate the boundedness of $X$ for the binary tree and for homogeneous weights. Assuming some mild regularity assumptions about $α$ we completely characterize weights $α$ and $σ$ with $X$ being a.s. bounded.

math.PR↗

arXiv2010

Compactness properties of weighted summation operators on trees - the critical case

The aim of this paper is to provide upper bounds for the entropy numbers of summation operators on trees in a critical case. In a recent paper [10] we elaborated a framework of weighted summation operators on general trees where we related the entropy of the operator with those of the underlying tree equipped with an appropriate metric. However, the results were left incomplete in a critical case of the entropy behavior, because this case requires much more involved techniques. In the present article we fill the gap left open in [10]. To this end we develop a method, working in the context of general trees and general weighted summation operators, which was recently proposed in [9] for a particular critical operator on the binary tree. Those problems appeared in natural way during the study of compactness properties of certain Volterra integral operators in a critical case.

math.FA↗

arXiv2010

Compactness Properties of Weighted Summation Operators on Trees

We investigate compactness properties of weighted summation operators $V_{α,σ}$ as mapping from $\ell_1(T)$ into $\ell_q(T)$ for some $q\in (1,\infty)$. Those operators are defined by $$ (V_{α,σ} x)(t) :=α(t)\sum_{s\succeq t}σ(s) x(s)\,,\quad t\in T\;, $$ where $T$ is a tree with induced partial order $t \preceq s$ (or $s \succeq t$) for $t,s\in T$. Here $α$ and $σ$ are given weights on $T$. We introduce a metric $d$ on $T$ such that compactness properties of $(T,d)$ imply two--sided estimates for $e_n(V_{α,σ})$, the (dyadic) entropy numbers of $V_{α,σ}$. The results are applied for concrete trees as e.g. moderate increasing, biased or binary trees and for weights with $α(t)σ(t)$ decreasing either polynomially or exponentially. We also give some probabilistic applications for Gaussian summation schemes on trees.

math.FA↗

arXiv2010

Maximal Intersection Queries in Randomized Input Models

Consider a family of sets and a single set, called the query set. How can one quickly find a member of the family which has a maximal intersection with the query set? Time constraints on the query and on a possible preprocessing of the set family make this problem challenging. Such maximal intersection queries arise in a wide range of applications, including web search, recommendation systems, and distributing on-line advertisements. In general, maximal intersection queries are computationally expensive. We investigate two well-motivated distributions over all families of sets and propose an algorithm for each of them. We show that with very high probability an almost optimal solution is found in time which is logarithmic in the size of the family. Moreover, we point out a threshold phenomenon on the probabilities of intersecting sets in each of our two input models which leads to the efficient algorithms mentioned above.

cs.IR↗

arXiv2010

Small deviations of stable processes and entropy of the associated random operators

We investigate the relation between the small deviation problem for a symmetric $α$-stable random vector in a Banach space and the metric entropy properties of the operator generating it. This generalizes former results due to Li and Linde and to Aurzada. It is shown that this problem is related to the study of the entropy numbers of a certain random operator. In some cases, an interesting gap appears between the entropy of the original operator and that of the random operator generated by it. This phenomenon is studied thoroughly for diagonal operators. Basic ingredients here are techniques related to random partitions of the integers. The main result concerning metric entropy and small deviations allows us to determine or provide new estimates for the small deviation rate for several symmetric $α$-stable random processes, including unbounded Riemann--Liouville processes, weighted Riemann--Liouville processes and the ($d$-dimensional) $α$-stable sheet.

math.PR↗

arXiv2009

On the Supremum of Some Random Dirichlet Polynomials

We study the supremum of some random Dirichlet polynomials with independent coefficients and obtain sharp upper and lower bounds for supremum expectation thus extending the results from our previous work (see http://arXiv.org/abs/math/0703691). Our approach in proving these results is entirely based on methods of stochastic processes, in particular the metric entropy method.

math.PR↗