A Generalization of Strassen’s Functional LIL

Let X ₁,X ₂,… be a sequence of i.i.d. mean zero random variables and let S _n denote the sum of the first n random variables. We show that whenever we have with probability one, lim?sup? _n→∞|S _n |/c _n =α ₀<∞ for a regular normalizing sequence {c _n }, the corresponding normalized partial sum process sequence is relatively compact in C[0,1] with canonical cluster set. Combining this result with some LIL type results in the infinite variance case, we obtain Strassen type results in this setting.     

Spectra of short monadic sentences about sparse random graphs

A random graph G(n, p) is said to obey the (monadic) zero–one k-law if, for any monadic formula of quantifier depth k, the probability that it is true for the random graph tends to either zero or one. In this paper, following J. Spencer and S. Shelah, we consider the case p = n ^?α. It is proved that the least k for which there are infinitely many α such that a random graph does not obey the zero–one k-law is equal to 4.     

Brochette percolation

We study bond percolation on the square lattice with one-dimensional inhomogeneities. Inhomogeneities are introduced in the following way: A vertical column on the square lattice is the set of vertical edges that project to the same vertex on Z. Select vertical columns at random independently with a given positive probability. Keep (respectively remove) vertical edges in the selected columns, with probability p (respectively 1?p). All horizontal edges and vertical edges lying in unselected columns are kept (respectively removed) with probability q (respectively 1 ? q). We show that, if p > p_c(Z²) (the critical point for homogeneous Bernoulli bond percolation), then q can be taken strictly smaller than p_c(Z²) in such a way that the probability that the origin percolates is still positive.     

Almost Sure Approximation of the Superposition of the Random Processes

This article presents sufficient conditions, which provide almost sure (a.s.) approximation of the superposition of the random processes S(N(t)), when càd-làg random processes S(t) and N(t) themselves admit a.s. approximation by a Wiener or stable Lévy processes. Such results serve as a source of numerous strong limit theorems for the random sums under various assumptions on counting process N(t) and summands. As a consequence we obtain a number of results concerning the a.s. approximation of the Kesten–Spitzer random walk, accumulated workload input into queuing system, risk processes in the classical and renewal risk models with small and large claims and use such results for investigation the growth rate and fluctuations of the mentioned processes.     

When does the zero-one <Emphasis Type="Italic">k</Emphasis>-law fail?

The limit probabilities of the first-order properties of a random graph in the Erd?s–Rényi model G(n, n^?α), α ∈ (0, 1), are studied. A random graph G(n, n^?α) is said to obey the zero-one k-law if, given any property expressed by a formula of quantifier depth at most k, the probability of this property tends to either 0 or 1. As is known, for α = 1? 1/(2^k?1 + a/b), where a > 2^k?1, the zero-one k-law holds. Moreover, this law does not hold for b = 1 and a ≤ 2^k?1 ? 2. It is proved that the k-law also fails for b > 1 and a ≤ 2^k?1 ? (b + 1)².     

Eigenfunctions of Composition Operators on Bloch-type Spaces

Suppose φ is a holomorphic self map of the unit disk and C_φ is a composition operator with symbol φ that fixes the origin and 0 < |φ'(0)| < 1. This paper explores sufficient conditions that ensure all the holomorphic solutions of Schröder equation for the composition operator C_φ to belong to a Bloch-type space B_α for some α > 0. In the second part of the paper, the results obtained for composition operators are extended to the case of weighted composition operators.     

The probability that a pair of group elements is autoconjugate

Let g and h be arbitrary elements of a given finite group G. Then g and h are said to be autoconjugate if there exists some automorphism α of G such that h = g^α. In this article, we construct some sharp bounds for the probability that two random elements of G are autoconjugate, denoted by \(\mathcal {P}_{a}(G)\). It is also shown that \(\mathcal {P}_{a}(G)|G|\) depends only on the autoisoclinism class of G.     

Double roots of random littlewood polynomials

We consider random polynomials whose coefficients are independent and uniform on {-1, 1}. We prove that the probability that such a polynomial of degree n has a double root is o(n^-2) when n+1 is not divisible by 4 and asymptotic to \(1/\sqrt 3 \) otherwise. This result is a corollary of a more general theorem that we prove concerning random polynomials with independent, identically distributed coefficients having a distribution which is supported on {-1, 0, 1} and whose largest atom is strictly less than \(\frac{{8\sqrt 3 }}{{\pi {n^2}}}\). In this general case, we prove that the probability of having a double root equals the probability that either -1, 0 or 1 are double roots up to an o(n^-2) factor and we find the asymptotics of the latter probability.     

Membership of distributions of polynomials in the Nikolskii–Besov class

The main result of this paper asserts that the distribution density of any non-constant polynomial f(ξ₁,ξ₂,...) of degree d in independent standard Gaussian random variables ξ₁ (possibly, in infinitely many variables) always belongs to the Nikol’skii–Besov space B^1/d (R¹) of fractional order 1/d (see the definition below) depending only on the degree of the polynomial. A natural analog of this assertion is obtained for the density of the joint distribution of k polynomials of degree d, also with a fractional order that is independent of the number of variables, but depends only on the degree d and the number of polynomials. We also give a new simple sufficient condition for a measure on R^k to possess a density in the Nikol’skii–Besov class B^α(R)^k. This result is applied for obtaining an upper bound on the total variation distance between two probability measures on R^k via the Kantorovich distance between them and a certain Nikol’skii–Besov norm of their difference. Applications are given to estimates of distributions of polynomials in Gaussian random variables.     

Nikol’skii inequality between the uniform norm and <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>-norm with jacobi weight of algebraic polynomials on an interval

We study the Nikol’skii inequality for algebraic polynomials on the interval [?1, 1] between the uniform norm and the norm of the space L _q ^(α,β) , 1 ≤ q < ∞, with the Jacobi weight ?(α,β)(x) = (1 ? x) ^α (1 + x) ^β , α ≥ β > ?1. We prove that, in the case α > β ≥ ?1/2, the polynomial with unit leading coefficient that deviates least from zero in the space L _q ^(α+1,,β) with the Jacobi weight ? ^(α+1,β)(x) = (1?x) ^α+1(1+x) ^β is the unique extremal polynomial in the Nikol’skii inequality. To prove this result, we use the generalized translation operator associated with the Jacobi weight. We describe the set of all functions at which the norm of this operator in the space L _q ^(α,β) for 1 ≤ q < ∞ and α > β ≥ ?1/2 is attained.     

On perturbation theory for points of discrete spectrum of dissipative operator functions

We consider the operator function L(α, θ) = A(α) ? θR of two complex arguments, where A(α) is an analytic operator function defined in some neighborhood of a real point α ₀ ∈ ? and ranging in the space of bounded operators in a Hilbert space ?. We assume that A(α) is a dissipative operator for real α in a small neighborhood of the point α ₀ and R is a bounded positive operator; moreover, the point α ₀ is a normal eigenvalue of the operator function L(α, θ ₀) for some θ ₀ ∈ ?, and the number θ ₀ is a normal eigenvalue of the operator function L(α ₀ θ). We obtain analogs and generalizations of well-known results for self-adjoint operator functions A(α) on the decomposition of α- and θ-eigenvalues in neighborhoods of the points α ₀ and θ ₀, respectively, and on the representation of the corresponding eigenfunctions by series.     

Sharp Weak Type Inequalities for Fractional Integral Operators

Suppose that d≥1 is an integer, α∈(0,d) is a fixed parameter and let I _α be the fractional integral operator associated with d-dimensional Walsh-Fourier series on (0,1] ^d . Let p, q be arbitrary numbers satisfying the conditions 1≤p<d/α and 1/q=1/p?α/d. We determine the optimal constant K, which depends on α, d and p, such that for any f∈L ^p ((0,1] ^d ) we have

$$ ||I_{\alpha } f||_{L^{q,\infty }((0,1]^{d})}\leq K||f||_{L^{p}((0,1]^{d})}. $$

In fact, we shall prove this inequality in the more general context of probability spaces equipped with a regular tree-like structures. This allows us to obtain this result also for non-integer dimension. The proof exploits a certain modification of the so-called Bellman function method and appropriate interpolation-type arguments. We also present a sharp weighted weak-type bound for I _α , which can be regarded as a version of the Muckenhoupt-Wheeden conjecture for fractional integral operators.

     

Harnack Inequality and Regularity for a Product of Symmetric Stable Process and Brownian Motion

In this paper, we consider a product of a symmetric stable process in ? ^d and a one-dimensional Brownian motion in ?^?+?. Then we define a class of harmonic functions with respect to this product process. We show that bounded non-negative harmonic functions in the upper-half space satisfy Harnack inequality and prove that they are locally Hölder continuous. We also argue a result on Littlewood–Paley functions which are obtained by the α-harmonic extension of an L ^p (? ^d ) function.     

Independence numbers of random subgraphs of distance graphs

We consider the distance graph G(n, r, s), whose vertices can be identified with r-element subsets of the set {1, 2,..., n}, two arbitrary vertices being joined by an edge if and only if the cardinality of the intersection of the corresponding subsets is s. For s = 0, such graphs are known as Kneser graphs. These graphs are closely related to the Erd?s–Ko–Rado problem and also play an important role in combinatorial geometry and coding theory. We study some properties of random subgraphs of G(n, r, s) in the Erd?s–Rényi model, in which every edge occurs in the subgraph with some given probability p independently of the other edges. We find the asymptotics of the independence number of a random subgraph of G(n, r, s) for the case of constant r and s. The independence number of a random subgraph is Θ(log₂n) times as large as that of the graph G(n, r, s) itself for r ≤ 2s + 1, while for r > 2s + 1 one has asymptotic stability: the two independence numbers asymptotically coincide.     

Zak transform and non-uniqueness in an extension of Pauli’s phase retrieval problem

The aim of this paper is to pursue the investigation of the phase retrieval problem for the fractional Fourier transform F _α started by the second author. We here extend a method of A. E. J. M. Janssen to show that there is a countable set Q such that for every finite subset A ? Q, there exist two functions f, g not multiple of one another such that |F _α f| = |F _α g| for every α ∈ A.This is done by constructing two functions ?, ψ such that F _α ? and F _α ψ have disjoint support for each α ∈ A. To do so, we establish a link between F _α [f], α ∈ Q and the Zak transform Z[f] generalizing the well known marginal properties of Z.     

Invariance principles for the law of the iterated logarithm under <Emphasis Type="Italic">G</Emphasis>-framework

We obtain a general invariance principle of G-Brownian motion for the law of the iterated logarithm (LIL for short). For continuous bounded independent and identically distributed random variables in G-expectation space, we also give an invariance principle for LIL. In some sense, this result is an extension of the classical Strassen’s invariance principle to the case where probability measure is no longer additive. Furthermore, we give some examples as applications.     

Fixed Points of Set Functors: How Many Iterations are Needed?

The initial algebra for a set functor can be constructed iteratively via a well-known transfinite chain, which converges after a regular infinite cardinal number of steps or at most three steps. We extend this result to the analogous construction of relatively initial algebras. For the dual construction of the terminal coalgebra Worrell proved that if a set functor is α-accessible, then convergence takes at most α + α steps. But until now an example demonstrating that fewer steps may be insufficient was missing. We prove that the functor of all α-small filters is such an example. We further prove that for β ≤ α the functor of all α-small β-generated filters requires precisely α + β steps and that a certain modified power-set functor requires precisely α steps. We also present an example showing that whether a terminal coalgebra exists at all does not depend solely on the object mapping of the given set functor. (This contrasts with the fact that existence of an initial algebra is equivalent to existence of a mere fixed point.)     

On the <Emphasis Type="Italic">α</Emphasis>-superposition of functions of a <Emphasis Type="Italic">k</Emphasis>-valued logic

We reveal a relation between the operations of α-completion and closure for the systems of functions of a k-valued logic. For k = 3, 4 we construct the α-bases consisting of two binary operations. We prove that the complete system T of functions of a 4-valued logic containing all permutations of the set E ₄ = {0, 1, 2, 3} and the operation of addition modulo 4 is not α-complete, whereas its α-completion [T _α] will be an α-complete system.