Limit theorems for the ratio of the Kaplan-Meier estimator or the Altshuler estimator to the true survival function

LetX ₁,X ₂, ...,X _n be a sequence of nonnegative independent random variables with a common continuous distribution functionF. LetY ₁,Y ₂, ...,Y _n be another sequence of nonnegative independent random variables with a common continuous distribution functionG, also independent of {X _i }. We can only observeZ _i =min(X _i ,Y _i ), and . LetH=1−(1−F)(1−G) be the distribution function ofZ. In this paper, the limit theorems for the ratio of the Kaplan-Meier estimator or the Altshuler estimator to the true survival functionS(t) are given. It is shown that (1)P(δ_(n)=1 i.o.)=0 ifF(τ _H ) < 1 andP(δ _n =0 i.o. )=0 ifG(τH) > 1 where δ_(n) is the corresponding indicator function of and have the same order a.s., where {T _n } is a sequence of constants such that 1−H(T _n )=n ^−α(logn)^β(log logn)^γ.     

Limit distributions of U-statistics resampled by symmetric stable laws

Summary If (Y _i) and (V _i) are independent random sequences such thatY _i are i.i.d. random variables belonging to the normal domain of attraction of a symmetric -stable law, 0<<2, andV _i are i.i.d. random variables, then the limit distributions of U-statistics , coincide with the probability laws of multiple stochastic integralsX ^d f = ... f (t ₁, ... ,t _d)dX(t _d) with respect to a symmetric -stable processX(t).The research was originated during author's visit at ORIE, Cornell University     

Privileged users in zero-error transmission over a noisy channel

The k-th power of a graph G is the graph whose vertex set is V(G) ^k , where two distinct k-tuples are adjacent iff they are equal or adjacent in G in each coordinate. The Shannon capacity of G, c(G), is lim _k→∞ α(G ^k )^1/k , where α(G) denotes the independence number of G. When G is the characteristic graph of a channel C, c(G) measures the effective alphabet size of C in a zero-error protocol. A sum of channels, C = Σ _i C _i , describes a setting when there are t ≥ 2 senders, each with his own channel C _i , and each letter in a word can be selected from any of the channels. This corresponds to a disjoint union of the characteristic graphs, G = Σ _i G _i . It is well known that c(G) ≥ Σ _i c(G _i ), and in [1] it is shown that in fact c(G) can be larger than any fixed power of the above sum. We extend the ideas of [1] and show that for every F, a family of subsets of [t], it is possible to assign a channel C _i to each sender i ∈ [t], such that the capacity of a group of senders X ⊂ [t] is high iff X contains some F ∈ F. This corresponds to a case where only privileged subsets of senders are allowed to transmit in a high rate. For instance, as an analogue to secret sharing, it is possible to ensure that whenever at least k senders combine their channels, they obtain a high capacity, however every group of k − 1 senders has a low capacity (and yet is not totally denied of service). In the process, we obtain an explicit Ramsey construction of an edge-coloring of the complete graph on n vertices by t colors, where every induced subgraph on exp vertices contains all t colors. Research supported in part by a USA-Israeli BSF grant, by the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. Research partially supported by a Charles Clore Foundation Fellowship.     

Two-point concentration in random geometric graphs

A random geometric graph G _n is constructed by taking vertices X ₁,…,X _n ∈ℝ ^d at random (i.i.d. according to some probability distribution ν with a bounded density function) and including an edge between X _i and X _j if ‖X _i -X _j ‖ < r where r = r(n) > 0. We prove a conjecture of Penrose ([14]) stating that when r=r(n) is chosen such that nr ^d = o(lnn) then the probability distribution of the clique number ω(G _n ) becomes concentrated on two consecutive integers and we show that the same holds for a number of other graph parameters including the chromatic number χ(G _n ). The author was partially supported by EPSRC, the Department of Statistics, Bekkerla-Bastide fonds, Dr. Hendrik Muller’s Vaderlandsch fonds, and Prins Bernhard Cultuurfonds.     

On a problem of spencer

LetX ₁, ...,X _n be events in a probability space. Let ϱ_i be the probabilityX _i occurs. Let ϱ be the probability that none of theX _i occur. LetG be a graph on [n] so that for 1 ≦i≦n X _i is independent of ≈X _j ‖(i, j)∉G≈. Letf(d) be the sup of thosex such that if ϱ₁, ..., ϱ _n ≦x andG has maximum degree ≦d then ϱ>0. We showf(1)=1/2,f(d)=(d−1) ^d−1 d ^−d ford≧2. Hence df(d)=1/e. This answers a question posed by Spencer in [2]. We also find a sharp bound for ϱ in terms of the ϱ _i andG.     

Regularity properties of solutions of some abstract parabolic equations

Consider the equation (i) (da/dt)—A(t)u(t)=f(t) where fort ∈ [a, b],A(t) is a densely defined and closed linear operator in a Banach spaceX. Assume the existence of bounded projectionsE _i(t),i=1, 2, such thatA(t) E ₁(t) and —A(t)E ₂(t) are infinitesimal generators of analytic semigroups andA(t) is completely reduced by the direct sum decompositionX = Σ _i ^{b = 1/2} ⊕E _i (t)X. We show that any solutionu(t) of (i) is inC ^∞(a, b) and satisfies the inequalities (1.2) provided thatf(t) andA(t) are infinitely differentiable in [a, b] in a suitable sense. In caseA(t) andf(t) are in a Gevrey class determined by the constants {M _n} we have (1. 3). Applications are given to the study of solution of (i) where fort ∈ [a, b]A(t) is the unbounded operator inH ^0,p (G) associated with an elliptic boundary value problem that satisfies Agmon’s conditions on the rays λ=±iτ, τ > 0. Research partially supported by an N.S.F. grant at Brandeis University.     

Strong law for the bootstrap

Let X₁, X₂, X₃, … be i.i.d. r.v. with E|X₁| < ∞, E X₁ = μ. Given a realization X = (X₁,X₂,…) and integers n and m, construct Y_n,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution $\text{P}{}^1\text{(Y}{}_{\mathrm{n,i}}\text{= X}{}_{\mathrm{j}}\text{) =}\text{1}\text{n}$ for 1 ? j ? n. ($\text{P}{}^1$ denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X₁ are given to ensure the almost sure convergence of $\text{(}\text{1}\text{m}\text{)Σ}{}^{\mathrm{m}}{}_{\mathrm{i=1}}\text{Y}{}_{\mathrm{n,i}}$ toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).     

Chaos caused by a strong-mixing measure-preserving transformation

The chaos caused by a strong-mixing preserving transformation is discussed and it is shown that for a topological spaceX satisfying the second axiom of countability and for an outer measurem onX satisfying the conditions: (i) every non-empty open set ofX ism-measurable with positivem-measure; (ii) the restriction ofm on Borel σ-algebra ℬ(X) ofX is a probability measure, and (iii) for everyY⊂X there exists a Borel setB⊂ℬ(X) such thatB⊃Y andm(B) =m(Y), iff:X→X is a strong-mixing measure-preserving transformation of the probability space (X, ℬ(X),m), and if {m}, is a strictly increasing sequence of positive integers, then there exists a subsetC⊂X withm (C) = 1, finitely chaotic with respect to the sequence {m _i}, i.e. for any finite subsetA ofC and for any mapF:A→X there is a subsequencer _i such that lim_i→∞ f ^r i(a) =F(a) for anya ∈A. There are some applications to maps of one dimension. the National Natural Science Foundation of China.     

The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion

Let {W(t),t∈R}, {B(t),t∈R } be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (Xi(t),... ,Xd(t)) and X1(t),... ,Xd(t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q (?) (0,∞), the exact Hausdorff measures of the image X(Q) = {X(t) : t∈Q} and the graph GrX(Q) = {(t, X(t)) :t∈Q}are established.     

Linear combination of concomitants of order statistics with application to testing and estimation

Summary Let (X ₁,Y ₁), (X ₂,Y ₂),…, (X _n,Y _n) be i.i.d. as (X, Y). TheY-variate paired with therth orderedX-variateX _rn is denoted byY _rn and terms the concomitant of therth order statistic. Statistics of the form are considered. The asymptotic normality ofT _n is established. The asymptotic results are used to test univariate and bivariate normality, to test independence and linearity ofX andY, and to estimate regression coefficient based on complete and censored samples.     

Sufficient conditions for the subexponential property of the convolution of two distributions

Conditions on the distributions of two independent nonnegative random variablesX andY are given for the sumX+Y to have a subexponential distribution, i.e., (1−F ^(2*)(t))/(1−F(t)) → 2 ast → +∞, whereF(t)=P{X+Y≤t} andF ^(2*)(t) is the convolution ofF(t) with itself. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 778–781, November, 1995.     

On some properties of solutions of second-order linear functional differential equations

The properties of solutions of the equationu″(t) =p ₁(t)u(τ₁(t)) +p ₂(t)u′(τ₂(t)) are investigated wherep _i :a, + ∞[→R (i=1,2) are locally summable functions τ₁ :a, + ∞[→R is a measurable function, and τ₂ :a, + ∞[→R is a nondecreasing locally absolutely continuous function. Moreover, τ _i (t) ≥t (i = 1,2),p ₁(t)≥0,p ₂ ² (t) ≤ (4 - ɛ)τ ₂ ^′ (t)p ₁(t), ɛ =const > 0 and . In particular, it is proved that solutions whose derivatives are square integrable on [α,+∞] form a one-dimensional linear space and for any such solution to vanish at infinity it is necessary and sufficient that .     

On the Similarity of Analytic Matrix Functions

Consider a domain , and two analytic matrix-valued functions functions . Consider also points and positive integers n ₁, n ₂, . . . , n _N . We are interested in the existence of an analytic function such that X(ω) is invertible, and G(ω) coincides with X(ω)F(ω)X(ω)⁻¹ up to order n _j at the point ω _j . We will see that such a function exists provided that F(ω _j ),G(ω _j ) have cyclic vectors, and the characteristic polynomials of F,G coincide up to order n _j at ω _j . This allows one to give a short proof to a result of Huang, Marcantognini and Young concerning spectral interpolation in the unit disk. The author was partially supported by a grant from the National Science Foundation. Received: September 8, 2006. Accepted: January 11, 2007.     

Hypothesis testing for the common mean of two normal distributions in the presence of an indifference zone

Summary LetX ₁,...,X _m andY _t,...,Y be independent, random samples from populations which are N(θ,σ _x ² ) and N(θ,σ _y ² ), respectively, with all parameters unknown. In testingH ₀:θ=0 againstH ₁:θ≠0, thet-test based upon either sample is known to be admissible in the two-sample setting. If, however, one testsH ₀ againstH ₁:|θ|≧ε>0, with ε arbitrary, our main results show: (i) the construction of a test which is better than the particulart-test chosen, (ii) eacht-test is admissible under the invariance principle with respect to the group of scale changes, and (iii) there does not exist a test which simultaneously is better than botht-tests.     

Penultimate limiting forms in extreme value theory

Summary Let {X _n}_n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM _n=max (X ₁,…,X _n), suitably normalized with attraction coefficients {α_n}_n≧1(α_n>0) and {b _n}_n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s which better approximate the d.f. of(M _n−b_n)/a_n thanG(x), for moderaten. We thus consider differences of the formF ⁿ(a_nx+b_n)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф _α(x)=exp (−x^−α), x>0] or a type III [Ψ _α(x)= exp (−(−x)^α), x<0] d.f. of largest values, much closer toF ⁿ(a_nx+b_n) than the ultimate itself.     

Some path properties of generalized Lévy sheet

Let X(t) be an N parameter generalized Lévy sheet taking values in ℝ^d with a lower index α, ℜ = {(s, t] = ∏ _i=1 ^N (s _i, t _i], s _i < t _i}, E(x, Q) = {t ∈ Q: X(t) = x}, Q ∈ ℜ be the level set of X at x and X(Q) = {x: ∃t ∈ Q such that X(t) = x} be the image of X on Q. In this paper, the problems of the existence and increment size of the local times for X(t) are studied. In addition, the Hausdorff dimension of E(x, Q) and the upper bound of a uniform dimension for X(Q) are also established.     

Asymptotic Behavior of the Maximum of Sums of I.I.D. Random Variables along Monotone Blocks

Let {X_i, Y_i}_i=1,2,... be an i.i.d. sequence of bivariate random vectors with P(Y₁ = y) = 0 for all y. Put M_n(j) = max_0≤k≤n-j (X_k+1 + ... X_k+j)I_k,j, where I_k,k+j = I{Y_k+1 < ⋯ < Y_k+j} denotes the indicator function for the event in brackets, 1 ≤ j ≤ n. Let L_n be the largest index l ≤ n for which I_k,k+l = 1 for some k = 0, 1, ..., n - l. The strong law of large numbers for “the maximal gain over the longest increasing runs,” i.e., for M_n(L_n) has been recently derived for the case where X₁ has a finite moment of order 3 + ε, ε > 0. Assuming that X₁ has a finite mean, we prove for any a = 0, 1, ..., that the s.l.l.n. for M(L_n - a) is equivalent to EX ₁ ^3+a I{X₁ > 0} < ∞. We derive also some new results for the a.s. asymptotics of L_n. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 179–189.     

A preferential attachment model with random initial degrees

In this paper, a random graph process {G(t)} _t≥1 is studied and its degree sequence is analyzed. Let {W _t } _t≥1 be an i.i.d. sequence. The graph process is defined so that, at each integer time t, a new vertex with W _t edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on G(t-1), the probability that a given edge of vertex t is connected to vertex i is proportional to d _i (t-1)+δ, where d _i (t-1) is the degree of vertex i at time t-1, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent τ=min{τ_W,τ_P}, where τ_W is the power-law exponent of the initial degrees {W _t } _t≥1 and τ_P the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze.     

Two Paths Joining Given Vertices in 2k-Edge-Connected Graphs

Let k≥2 be an integer and G = (V(G), E(G)) be a k-edge-connected graph. For X⊆V(G), e(X) denotes the number of edges between X and V(G) − X. Let {s_i, t_i}⊆X_i⊆V(G) (i=1,2) and X₁∩X₂=∅. We here prove that if k is even and e(X_i)≤2k−1 (i=1,2), then there exist paths P₁ and P₂ such that P_i joins s_i and t_i, V(P_i)⊆X_i (i=1,2) and G − E(P₁∪P₂) is (k−2)-edge-connected (for odd k, if e(X₁)≤2k−2 and e(X₂)≤2k−1, then the same result holds [10]), and we give a generalization of this result and some other results about paths not containing given edges.     

Functional characterization of Vasil’ev invariants

A cover of a manifold X is called an r-cover if any r points of X belong to a set in the cover. Let X and Y be two smooth manifolds, let Emb(X, Y) be the family of smooth embeddings X → Y, let M be an Abelian group, and let F: Emb(X, Y) → M be a functional. One says that the degree of F does not exceed r if for each finite open r-cover {U _i } _i∈I ; of X there exist functionals F _i : Emb(U _i , Y) → M, i ∈ I, such that for each a ∈ Emb(X, Y) one has