Limiting distribution of sums of nonnegative stationary random variables

Summary Let {X _n,j,−∞<j<∞∼,n≧1, be a sequence of stationary sequences on some probability space, with nonnegative random variables. Under appropriate mixing conditions, it is shown thatS _n=X_n,1+…+X _n,n has a limiting distribution of a general infinitely divisible form. The result is applied to sequences of functions {f _n(x)∼ defined on a stationary sequence {X _j∼, whereX _n.f=f_n(X_j). The results are illustrated by applications to Gaussian processes, Markov processes and some autoregressive processes of a general type. This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Sciences Foundation, Grant MCS 82-01119.     

Stick breaking process generated by virtual permutations with Ewens distribution

Given a sequence x of points in the unit interval, we associate with it a virtual permutation w=w(x) (that is, a sequence w of permutationsw _n such that for all n=1,2,..., w_n−1=w′_n is obtained from w_n by removing the last element n from its cycle). We introduce a detailed version of the well-known stick breaking process generating a random sequence x. It is proved that the associated random virtual permutation w(x) has a Ewens distribution. Up to subsets of zero measure, the space of virtual permutations is identified with the cube [0, 1]^∞. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 162–180.     

Recursive events in random sequences

Let ω be a Kolmogorov–Chaitin random sequence with ω_{1: n} denoting the first n digits of ω. Let P be a recursive predicate defined on all finite binary strings such that the Lebesgue measure of the set {ω|∃nP(ω_{1: n} )} is a computable real α. Roughly, P holds with computable probability for a random infinite sequence. Then there is an algorithm which on input indices for any such P and α finds an n such that P holds within the first n digits of ω or not in ω at all. We apply the result to the halting probability Ω and show that various generalizations of the result fail. Received: 1 December 1998 / Published online: 3 October 2001     

Self-Similar Tiling Systems,Topological Factors and Stretching Factors

In this paper we prove that if two self-similar tiling systems, with respective stretching factors λ ₁ and λ ₂, have a common factor which is a nonperiodic tiling system, then λ ₁ and λ ₂ are multiplicatively dependent.     

Record values and the exponential distribution

A sequence {X _n,n≧1} of independent and identically distributed random variables with continuous cumulative distribution functionF(x) is considered.X _j is a record value of this sequence ifX _j>max (X ₁, …,X _j−1). Let {X _L(n) n≧0} be the sequence of such record values. Some properties ofX _L(n) andX _L(n)−X_L(n−1) are studied when {X _n,n≧1} has the exponential distribution. Characterizations of the exponential distribution are given in terms of the sequence {X _L(n),n≧0} The work was partly completed when the author was at the Department of Statistics, University of Brasilia, Brazil.     

Random ergodic theorems and real cocycles

We study mean convergence of ergodic averages associated to a measure-preserving transformation or flow τ along the random sequence of times κ _n (ω) given by the Birkhoff sums of a measurable functionF for an ergodic measure-preserving transformationT. We prove that the sequence (k _n(ω)) is almost surely universally good for the mean ergodic theorem, i.e., that, for almost every, ω, the averages (*) converge for every choice of τ, if and only if the “cocycle”F satisfies a cohomological condition, equivalent to saying that the eigenvalue group of the “associated flow” ofF is countable. We show that this condition holds in many natural situations. When no assumption is made onF, the random sequence (k _n(ω)) is almost surely universally good for the mean ergodic theorem on the class of mildly mixing transformations τ. However, for any aperiodic transformationT, we are able to construct an integrable functionF for which the sequence (k _n(ω)) is not almost surely universally good for the class of weakly mixing transformations.     

Erdős measures,sofic measures,and Markov chains

We consider the random variable ζ = ξ₁ρ+ξ₂ρ²+…, where ξ₁, ξ₂, … are independent identically distibuted random variables taking the values 0 and 1 with probabilities P(ξ_i = 0) = p₀, P(ξ_i = 1) = p₁, 0 < p₀ < 1. Let β = 1/ρ be the golden number. The Fibonacci expansion for a random point ρζ from [0, 1] is of the form η₁ρ + η₂ρ² + … where the random variables η_k are {0, 1}-valued and η_kη_k+1 = 0. The infinite random word η = η₁η₂ … η_n … takes values in the Fibonacci compactum and determines the so-called Erdős measure μ(A) = P(η ∈ A) on it. The invariant Erdős measure is the shift-invariant measure with respect to which the Erdős measure is absolutely continuous. We show that the Erdős measures are sofic. Recall that a sofic system is a symbolic system that is a continuous factor of a topological Markov chain. A sofic measure is a one-block (or symbol-to-symbol) factor of the measure corresponding to a homogeneous Markov chain. For the Erdős measures, the corresponding regular Markov chain has 5 states. This gives ergodic properties of the invariant Erdős measure. We give a new ergodic theory proof of the singularity of the distribution of the random variable ζ. Our method is also applicable when ξ₁, ξ₂, … is a stationary Markov chain with values 0, 1. In particular, we prove that the distribution of ζ is singular and that the Erdős measures appear as the result of gluing together states in a regular Markov chain with 7 states. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 28–47.     

Convergence of Point Processes with Weakly Dependent Points

For each n≥1, let {X _j,n }_1≤j≤n be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process $N_{n}=\sum_{j=1}^{n}\delta_{X_{j,n}}For each n≥1, let {X _j,n }_1≤j≤n be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process N_n=?_j=1ⁿd_Xj,nN_{n}=\sum_{j=1}^{n}\delta_{X_{j,n}} to an infinitely divisible point process. From the point process convergence we obtain the convergence in distribution of the partial sum sequence S _n =∑ _j=1 ⁿ X _j,n to an infinitely divisible random variable whose Lévy measure is related to the canonical measure of the limiting point process. As examples, we discuss the case of triangular arrays which possess known (row-wise) dependence structures, like the strong mixing property, the association, or the dependence structure of a stochastic volatility model.     

Left and Right Length of Paths in Binary Trees or on a Question of Knuth

We consider extended binary trees and study the joint right and left depth of leaf j, where the leaves are labelled from left to right by 0, 1, . . . , n, and the joint right and left external pathlength of binary trees of size n. Under the random tree model, i.e., the Catalan model, we characterize the joint limiting distribution of the suitably scaled left depth and the difference between the right and the left depth of leaf j in a random size-n binary tree when j ~ ρn with 0 < ρ > 1, as well as the joint limiting distribution of the suitably scaled left external pathlength and the difference between the right and the left external pathlength of a random size-n binary tree. This work was supported by the Austrian Science Foundation FWF, grant S9608-N13.     

On the Complete Convergence of Moving Average Process with Banach Space Valued Random Elements

Let {Y _i ;−∞<i<∞} be a doubly infinite sequence of independent random elements taking values in a separable real Banach space and stochastically dominated by a random variable X. Let {a _i ;−∞<i<∞} be an absolutely summable sequence of real numbers and set V _i =∑ _k=−∞^∞ a _i+k Y _i ,i≥1. In this paper, we derive that if and E|X| ^μ log  ^ρ |X|<0, for some μ (0<μ<2, μ≠1) and ρ>0 then for all ε>0. This work was partially supported by the Korean Research Foundation Grant funded by the Korean Government (KRF-2006-353-C00006, KRF-2006-251-C00026).     

Elementary Symmetric Polynomials in Random Variables

The subject of the paper is the probability-theoretic properties of elementary symmetric polynomials σ _k of arbitrary degree k in random variables X _i (i=1,2,…,m) defined on special subsets of commutative rings ℛ _m with identity of finite characteristic m. It is shown that the probability distributions of the random elements σ _k (X ₁,…,X _m ) tend to a limit when m→∞ if X ₁,…,X _m form a Markov chain of finite degree μ over a finite set of states V, V⊂ℛ _m , with positive conditional probabilities. Moreover, if all the conditional probabilities exceed a prescribed positive number α, the limit distributions do not depend on the choice of the chain.      

Strongly correlated random fields as observed by a random walker

Summary We are given a random walk S ₁, S ₂, ... on ℤ^ν, ν≧1, and a strongly correlated stationary random field ξ(x), xεℤ^ν, which is independent of the random walk. We consider the field as observed by a random walker and study partial sums of the form . It is assumed that the law corresponding to the random walk belongs to the domain of attraction of a non-degenerate stable law of index β, 0<β≦2. We further suppose that the field ξ satisfies the non-central limit theorem of Dobrushin and Major with a scaling factor . Under the assumption αk<β it is shown that converges weakly as n→∞ to a self-similar process {Δ _t , t≧0} with stationary increments, and Δ _t can be represented as a multiple Wiener-It? integral of a random function. This extends the noncentral limit theorem of Dobrushin and Major and yields a new example of a self-similar process with stationary increments.     

A limit theorem for the moments of sums of independent random variables

Forn≧1, letS _n=ΣX _n,i (1≦i≦r _n <∞), where the summands ofS _n are independent random variables having medians bounded in absolute value by a finite number which is independent ofn. Letf be a nonnegative function on (− ∞, ∞) which vanishes and is continuous at the origin, and which satisfies, for some for allt≧1 and all values ofx. Theorem.For centering constants c _n,let S _n − c _n converge in distribution to a random variable S. (A)In order that Ef(S_n − c_n) converge to a limit L, it is necessary and sufficient that there exist a common limit (B)If L exists, then L<∞ if and only if R<∞, and when L is finite, L=Ef(S)+R. Applications are given to infinite series of independent random variables, and to normed sums of independent, identically distributed random variables.     

Fragmentation Processes with an Initial Mass Converging to Infinity

We consider a family of fragmentation processes where the rate at which a particle splits is proportional to a function of its mass. Let F ₁^(m)(t),F ₂^(m)(t),… denote the decreasing rearrangement of the masses present at time t in a such process, starting from an initial mass m. Let then m→∞. Under an assumption of regular variation type on the dynamics of the fragmentation, we prove that the sequence (F ₂^(m),F ₃^(m),…) converges in distribution, with respect to the Skorohod topology, to a fragmentation with immigration process. This holds jointly with the convergence of m−F ₁^(m) to a stable subordinator. A continuum random tree counterpart of this result is also given: the continuum random tree describing the genealogy of a self-similar fragmentation satisfying the required assumption and starting from a mass converging to ∞ will converge to a tree with a spine coding a fragmentation with immigration. Research supported in part by EPSRC GR/T26368.     

Orbits of subsystem stabilizers

Let Φ be a reduced irreducible root system. We consider pairs (S, X (S)), where S is a closed set of roots, X(S) is its stabilizer in the Weyl group W(Φ). We are interested in such pairs maximal with respect to the following order: (S₁, X (S₁)) ≤ (S₂, X (S₂)) if S₁ ⊆ S₂ and X(S₁) ≤ X(S₂). The main theorem asserts that if Δ is a root subsystem such that (Δ, X (Δ)) is maximal with respect to the above order, then X (Δ) acts transitively both on the long and short roots in Φ \ Δ. This result is a wide generalization of the transitivity of the Weyl group on roots of a given length. Bibliography: 23 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 98–124.     

Stability and localization of unbounded solutions of a nonlinear heat equation in a plane

The article investigates unbounded solutions of the equation u _t = div (u ^σgrad u) + u ^β in a plane. We numerically analyze the stability of two-dimensional self-similar solutions (structures) that increase with blowup. We confirm structural stability of the simple structure with a single maximum and metastability of complex structures. We prove structural stability of the radially symmetrical structure with a zero region at the center and investigate its attraction region. We study the effect of various perturbations of the initial function on the evolution of self-similar solutions. We further investigate how arbitrary compact-support initial distributions attain the self-similar mode, including distributions whose support is different from a disk. We show that the self-similar mode described by a simple radially symmetrical structure is achieved only in the central region, while the entire localization region does not have enough time to transform into a disk during blowup. We show for the first time that simple structures may merge into a complex structure, which evolves for a long time according to self-similar law.     

On the strong law of large numbers for dependent random variables

Let {X _n} _n ^=1/∞ be a sequence of random variables with partial sumsS _n, and let {ie241-1} be the σ-algebra generated byX ₁,…,X _n. Letf be a function fromR toR and suppose {ie241-2}. Under conditions off and moment conditions on theX' _ns, we show thatS _n/n converges a.e. (almost everywhere). We give several applications of this result. Research supported by N.S.F. Grant MCS 77-26809     

Branching random walk with a critical branching part

We consider the branching treeT(n) of the first (n+1) generations of a critical branching process, conditioned on survival till time βn for some fixed β>0 or on extinction occurring at timek _n withk _n /n→β. We attach to each vertexv of this tree a random variableX(v) and define , where π(0,v) is the unique path in the family tree from its root tov. FinallyM _n is the maximal displacement of the branching random walkS(·), that isM _n =max{S(v):v∈T(n)}. We show that if theX(v), v∈T(n), are i.i.d. with mean 0, then under some further moment conditionn ^−1/2 M _n converges in distribution. In particular {n ^−1/2 M _n } _n⩾1 is a tight family. This is closely related to recent results about Aldous' continuum tree and Le Gall's Brownian snake.     

The Erdös-Jacobson-Lehel conjecture on potentiallyP k-graphic sequence is true

A variation in the classical Turan extrernal problem is studied. A simple graphG of ordern is said to have propertyPk if it contains a clique of sizek+1 as its subgraph. Ann-term nonincreasing nonnegative integer sequence π=(d₁, d₂,⋯, d₂) is said to be graphic if it is the degree sequence of a simple graphG of ordern and such a graphG is referred to as a realization of π. A graphic sequence π is said to be potentiallyP _k-graphic if it has a realizationG having propertyP _k . The problem: determine the smallest positive even number σ(k, n) such that everyn-term graphic sequence π=(d₁, d₂,…, d₂) without zero terms and with degree sum σ(π)=(d ₁+d ₂+ …+d ₂) at least σ(k,n) is potentially P_k-graphic has been proved positive. Project supported by the National Natural Science Foundation of China (Grant No. 19671077) and the Doctoral Program Foundation of National Education Department of China.