Mikhail Lifshits

Lectures on Gaussian Processes

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.

Small deviations for fractional stable processes

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.

Bounds for entropy numbers for some critical operators

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.

Metric entropy of the integration operator and small ball probabilities for the Brownian sheet

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.

AGGREGATION RATES IN ONE-DIMENSIONAL STOCHASTIC SYSTEMS WITH ADHESION AND GRAVITATION

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 .

Sampling the Lindelöf Hypothesis with the Cauchy random walk

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 .

Small deviation probability via chaining

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.

Chung's Law and The Csáki Function

AbstractWe have found the limit

PROBABILITIES OF RANDOMLY CENTERED SMALL BALLS AND QUANTIZATION IN BANACH SPACES

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\|