A Limit Result for U-Statistics of Binary Variables

Define , where is a symmetric U-type statistic, H _k() is the Hermite polynomial of degree k, and {X, X _n, n1} are independent identically distributed binary random variables with Pr(X{–1, 1}})=1. We show that according as EX=0 or EX0, respectively.     

Compact Laws of the Iterated Logarithm for B-Valued Random Variables with Two-Dimensional Indices

Let be a real separable Banach space and {X, X _{n, m}; (n, m) N ²} B-valued i.i.d. random variables. Set . In this paper, the compact law of the iterated logarithm, CLIL(D), for B-valued random variables with two-dimensional indices ranging over a subset D of N ² is studied. There is a gap between the moment conditions for CLIL(N ¹) and those for CLIL(N ²). The main result of this paper fills this gap by presenting necessary and sufficient conditions for the sequence to be almost surely conditionally compact in B, where, for 0, 1 r 2, N ^r (, ) = {(n, m) N ²; n m n exp{(log n) ^r–1 (n)}} and (·) is any positive, continuous, nondecreasing function such that (t)/(log log t) is eventually decreasing as t , for some > 0.     

Approximation of the Distribution of the Supremum of a Centered Random Walk. Application to the Local Score

Let (X _n ) _{n 0} be a real random walk starting at 0, with centered increments bounded by a constant K. The main result of this study is: |P(S _n n x)–P( sup_{0 u 1} B _u x)| C(n,K) n/n, where x 0, ² is the variance of the increments, S _n is the supremum at time n of the random walk, (B _u ,u 0) is a standard linear Brownian motion and C(n,K) is an explicit constant. We also prove that in the previous inequality S _n can be replaced by the local score and sup₀ u 1 B _u by sup₀ u 1|B _u |.     

The Halász Inequality for Additive Function of Rational Argument

We obtain an upper bound for the quantity . Here I is an interval, is the set of rational numbers q=m/n I such that nx, and f(q) is an arbitrary real-valued additive function of rational argument. The interval I and function f may depend on x3.     

Limiting Laws for Entrance Times of Critical Mappings of a Circle

A renormalization group transformation R ₁ has a single stable point in the space of the analytic circle homeomorphisms with a single cubic critical point and with the rotation number (the golden mean). Let a homeomorphism T be the C ¹-conjugate of . We let denote the sequence of distribution functions of the time of the kth entrance to the nth renormalization interval for the homeomorphism T. We prove that for any , the sequence has a finite limiting distribution function , which is continuous in , and singular on the interval [0,1]. We also study the sequence for k>1.     

On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence

Let M _f(r) and _f(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let be a continuously differentiable function convex on (–, +) and such that x = o((x)) as x +. We establish that, in order that the equality be true for any entire function f, it is necessary and sufficient that ln (x) = o((x)) as x +.     

On the asymptotic behaviour of first passage times for transient random walk

Summary Let _x denote the time at which a random walk with finite positive mean first passes into (x, ), wherex0. This paper establishes the asymptotic behaviour of Pr { _x >n} asn for fixedx in two cases. In the first case the left hand tail of the step-distribution is regularly varying, and in the second the step-distribution satisfies a one-sided Cramér type condition. As a corollary, it follows that in the first case Pr { _x >n}/Pr{ ₀ >n} coincides with the limit of the same quantity for recurrent random walk satisfying Spitzer's condition, but in the second case the limit is more complicated.     

Large Deviations of the Finite Cluster Shape for Two-Dimensional Percolation in the Hausdorff and L1 Metric

We consider supercritical two-dimensional Bernoulli percolation. Conditionally on the event that the open cluster C containing the origin is finite, we prove that: the laws of C/N satisfy a large deviations principle with respect to the Hausdorff metric; let f(N) be a function from to such that f(N)/ln N+ and f(N)/N0 as N goes to the laws of {x ² : d(x, C)f(N)}/N satisfy a large deviations principle with respect to the L ¹ metric associated to the planer Lebesgue measure. We link the second large deviations principle with the Wulff construction.     

Generalization of some classical inequalities in the theory of orthogonal series

Let {X_i} _– be a sequence of random variables, E(X_i) 0. If 1, estimates for the -th moments can be derived from known estimates of the -th moment. Here we generalized the Men'shov-Rademacher inequality for =2 for orthonormal X_i, to the case 1 and dependent random variables. The Men'shov-Payley inequality >2 for orthonormal X_i) is generalized for >2 to general random variables. A theorem is also proved that contains both the Erdös -Stechkin theorem and Serfling's theorem withv > 2 for dependent random variables.Translated from Matematicheskie Zametki, Vol. 17, No. 2, pp. 219–230, February, 1975.This article was written while the author was working in the V. A. Steklov Mathematics Institute, Academy of Sciences of the USSR.     

On the Boundedness of a Recurrence Sequence in a Banach Space

We investigate the problem of the boundedness of the following recurrence sequence in a Banach space B: where |y _n} and | _n } are sequences bounded in B, and A _k, k 1, are linear bounded operators. We prove that if, for any > 0, the condition is satisfied, then the sequence |x _n} is bounded for arbitrary bounded sequences |y _n} and | _n } if and only if the operator has the continuous inverse for every z C, |z| 1.     

General Blowings-up of ℙ2 and Injectivity of Gauss Maps

We consider the blowing-up Y _k of the projective plane along k general points P ₁,...,P _k . Let _k : Y _k ² be the projection map and E _i the exceptional divisor corresponding to P _i for 1ik. For m2 and km(m+3)/2–4 let _k be the invertible sheaf _k ^*( ²(m)) _{Y k} (–E ₁–···–E _k ) on Y _k , and let _k: Y _k ^N be the morphism corresponding to _k . As _k is a local embedding, the Gauss map _k corresponding to _k is defined on Y _k by _k (x)=(d _{x k} )(T _x (Y _k )) for all xY _k . We prove that this Gauss map _k is injective.     

On Universal Representation of Random Graphs

It is shown that every probability measure on the interval [0, 1] gives rise to a unique infinite random graph g on vertices {v₁, v₂, . . .} and a sequence of random graphs gn on vertices {v₁, . . . , v_n} such that . In particular, for Bernoulli graphs with stable property Q, can be strengthened to: probability space (, F, P), set of infinite graphs G(Q) , F with property Q such that .AMS Subject Classification: 05C80, 05C62.     

Fourier inversion for multidimensional characteristic functions

A density functionf(x),xR ⁿ is said to bepiecewise smooth if for eachxR ⁿ , the mean value function is piecewiseC with compact support. (d is normalized surface measure on the unit sphere). The Fourier transform is with spherical partial sum . Theorem. For suchf, lim _r f _R (x)=M ₀+f(x) if and only ifrM _r f(x) hask=[(n–3)/2] continuous derivatives. ([]=integer part). Otherwise we have lim where 0 is uniquely determined.     

Cohomology of solvable lie algebras and solvmanifolds

The cohomology H^* (G/,) of the de Rham complex *(G/) of a compact solvmanifold G/ with deformed differential d = d + , where is a closed 1 -form, is studied. Such cohomologies naturally arise in Morse-Novikov theory. It is shown that, for any completely solvable Lie group G containing a cocompact lattice G, the cohomology H^*(G/, ) is isomorphic to the cohomology H^*( ) of the tangent Lie algebra of the group G with coefficients in the one-dimensional representation : defined by () = (). Moreover, the cohomology H ^*(G/,) is nontrivial if and only if -[] belongs to a finite subset of H ¹(G/,) defined in terms of the Lie algebra .Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 67–79.Original Russian Text Copyright © 2005 by D. V. Millionshchikov.This revised version was published online in April 2005 with a corrected issue number.     

The Minimal Subgroup of a Random Walk

It is proved that for each random walk (S _n ) _n0 on ^d there exists a smallest measurable subgroup of ^d , called minimal subgroup of (S _n ) _n0, such that P(S _n )=1 for all n1. can be defined as the set of all x ^d for which the difference of the time averages n ^{–1 n} _k=1 P(S _k ) and n ^{–1 n} _k=1 P(S _k +x) converges to 0 in total variation norm as n. The related subgroup * consisting of all x ^d for which lim _n P(S _n )–P(S _n +x)=0 is also considered and shown to be the minimal subgroup of the symmetrization of (S _n ) _n0. In the final section we consider quasi-invariance and admissible shifts of probability measures on ^d . The main result shows that, up to regular linear transformations, the only subgroups of ^d admitting a quasi-invariant measure are those of the form ₁×...× _k × ^l–k ×{0} ^d–l , 0kld, with ₁,..., _k being countable subgroups of . The proof is based on a result recently proved by Kharazishvili⁽³⁾ which states no uncountable proper subgroup of admits a quasi-invariant measure.     

An application of Shoenfield's absoluteness theorem to the theory of uniform distribution

IfC is a Polish probability space, a Borel set whose sectionsW _x ( have measure one and are decreasing , then we show that the set _x W _x has measure one. We give two proofs of this theorem—one in the language of set theory, the other in the language of probability theory, and we apply the theorem to a question on completely uniformly distributed sequences.Supported by DFG grant Ko 490/7-1.     

Interpolation Sequences for the Class of Functions of Finite η-Type Analytic in the Unit Disk

We establish conditions for the existence of a solution of the interpolation problem f( _n ) = b _n in the class of functions f analytic in the unit disk and such that

Absolute Continuity of the Spectrum of the Periodic Maxwell Operator in a Layer

The Maxwell operator in a layer is studied. It is assumed that the electric permittivity (x) and the magnetic permeability (x)are periodic along the layer. On the boundary of the layer, conditions of ideal conductivity are imposed. Under wide assumptions on (x) and (x), it is shown that the spectrum of the Maxwell operator is absolutely continuous. Bibliography: 10 titles.     

On the solution of singular linear systems of algebraic equations by semiiterative methods

Summary For a square matrixT ^n,n , where (I–T) is possibly singular, we investigate the solution of the linear fixed point problemx=T x+c by applying semiiterative methods (SIM's) to the basic iterationx ₀ ⁿ ,x _k T c _k–1+c(k1). Such problems arise if one splits the coefficient matrix of a linear systemA x=b of algebraic equations according toA=M–N (M nonsingular) which leads tox=M ^–1 N x+M ^–1 bT x+c. Even ifx=T x+c is consistent there are cases where the basic iteration fails to converge, namely ifT possesses eigenvalues 1 with ||1, or if =1 is an eigenvalue ofT with nonlinear elementary divisors. In these cases — and also ifx=T x+c is incompatible — we derive necessary and sufficient conditions implying that a SIM tends to a vector which can be described in terms of the Drazin inverse of (I–T). We further give conditions under which is a solution or a least squares solution of (I–T)x=c.Research supported in part by the Alexander von Humboldt-Stiftung