Mikhail Lifshits
Saint Petersburg State University
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Featured researches published by Mikhail Lifshits.
Archive | 2012
Mikhail Lifshits
Theory of random processes needs a kind of normal distribution. This is why Gaussian vectors and Gaussian distributions in infinite-dimensional spaces come into play. By simplicity, importance and wealth of results, theory of Gaussian processes occupies one of the leading places in modern Probability.
Annales De L Institut Henri Poincare-probabilites Et Statistiques | 2005
Mikhail Lifshits; Thomas Simon
Abstract Let { R t , 0 ⩽ t ⩽ 1 } be a symmetric α-stable Riemann–Liouville process with Hurst parameter H > 0 . Consider a translation invariant, β-self-similar, and p-pseudo-additive functional semi-norm ‖ ⋅ ‖ . We show that if H > β + 1 / p and γ = ( H − β − 1 / p ) −1 , then lim ɛ ↓ 0 ɛ γ log P [ ‖ R ‖ ⩽ ɛ ] = − K ∈ [ − ∞ , 0 ) , with K finite in the Gaussian case α = 2 . If α 2 , we prove that K is finite when R is continuous and H > β + 1 / p + 1 / α . We also show that under the above assumptions, lim ɛ ↓ 0 ɛ γ log P [ ‖ X ‖ ⩽ ɛ ] = − K ∈ ( − ∞ , 0 ) , where X is the linear α-stable fractional motion with Hurst parameter H ∈ ( 0 , 1 ) (if α = 2 , then X is the classical fractional Brownian motion). These general results cover many cases previously studied in the literature, and also prove the existence of new small deviation constants, both in Gaussian and non-Gaussian frameworks.
Transactions of the American Mathematical Society | 2012
Mikhail Lifshits
We provide upper bounds for entropy numbers for two types of operators: summation operators on binary trees and integral operators of Volterra type. Our efforts are concentrated on the critical cases where none of known methods works. Therefore, we develop a method which seems to be completely new and probably merits further applications.
Comptes Rendus De L Academie Des Sciences Serie I-mathematique | 1998
Thomas Dunker; Thomas Kühn; Mikhail Lifshits; Werner Linde
Abstract Let Td : L2([0, 1]d) → C([0, 1]d) be the d-dimensional integration operator. We show that its Kolmogorov and entropy numbers decrease with order at least k−1 (log k)d− 1/2. From this we derive that the small ball probabilities of the Brownian sheet on [0, 1]d under the C([0, 1]d)-norm can be estimated from below by exp(−Cɛ−2¦ log ɛ¦2d−1), which improves the best known lower bounds considerably. We also get similar results with respect to certain Orlicz norms.
Annals of Probability | 2005
Mikhail Lifshits; Zhan Shi
We consider one-dimensional systems of self-gravitating sticky particles with random initial data and describe the process of aggregation in terms of the largest cluster size L n at any fixed time prior to the critical time. The asymptotic behavior of L n is also analyzed for sequences of times tending to the critical time. A phenomenon of phase transition shows up, namely, for small initial particle speeds (cold gas) L n has logarithmic order of growth while higher speeds (warm gas) yield polynomial rates for L n .
arXiv: Probability | 2009
Mikhail Lifshits; Michael Weber
We study the behaviour of the Riemann zeta function ζ(1 2 + it), when t is sampled by the Cauchy random walk. More precisely, let X1, X2,. .. denote an infinite sequence of independent Cauchy-distributed random variables. Consider the sequence of partial sums Sn = X1 +. .. + Xn, n = 1, 2,. . .. We investigate the almost-sure asymptotic behaviour of the system ζ(1 2 + iSn), n = 1, 2,. .. . We develop a complete second-order theory for this system and show, by using a classical approximation formula of ζ(·), that it behaves almost like a system of non-correlated variables. Exploiting this fact in relation with the known criteria for almost-sure convergence allows us to prove the following almost-sure asymptotic behaviour: for any real b > 2 it is true that n k=1 ζ(1 2 + iS k) (a.s.) = n + O n 1/2 (log n) b .
Stochastic Processes and their Applications | 2008
Frank Aurzada; Mikhail Lifshits
We obtain several extensions of Talagrands lower bound for the small deviation probability using metric entropy. For Gaussian processes, our investigations are focused on processes with sub-polynomial and, respectively, exponential behaviour of covering numbers. The corresponding results are also proved for non-Gaussian symmetric stable processes, both for the cases of critically small and critically large entropy. The results extensively use the classical chaining technique; at the same time they are meant to explore the limits of this method.
Journal of Theoretical Probability | 1999
Natalia Gorn; Mikhail Lifshits
AbstractWe have found the limit
Journal of Complexity | 2008
Mikhail Lifshits; Marguerite Zani
Annals of Probability | 2005
Steffen Dereich; Mikhail Lifshits
L_h = \mathop {\lim \inf }\limits_{T \to \infty } (\log _2 T)^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} \left\| {\frac{{W(T \cdot )}}{{(2T\log _2 T)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }} - h} \right\|