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1.
In a rapidly growing population one expects that two individuals chosen at random from the nnth generation are unlikely to be closely related if nn is large. In this paper it is shown that for a broad class of rapidly growing populations this is not the case. For a Galton–Watson branching process with an offspring distribution {pj}{pj} such that p0=0p0=0 and ψ(x)=jpjI{jx}ψ(x)=jpjI{jx} is asymptotic to x−αL(x)xαL(x) as x→∞x where L(⋅)L() is slowly varying at ∞ and 0<α<10<α<1 (and hence the mean m=∑jpj=∞m=jpj=) it is shown that if XnXn is the generation number of the coalescence of the lines of descent backwards in time of two randomly chosen individuals from the nnth generation then n−XnnXn converges in distribution to a proper distribution supported by N={1,2,3,…}N={1,2,3,}. That is, in such a rapidly growing population coalescence occurs in the recent past rather than the remote past. We do show that if the offspring mean mm satisfies 1<m≡∑jpj<∞1<mjpj< and p0=0p0=0 then coalescence time XnXn does converge to a proper distribution as n→∞n, i.e., coalescence does take place in the remote past.  相似文献   

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We consider NN independent stochastic processes (Xj(t),t∈[0,T])(Xj(t),t[0,T]), j=1,…,Nj=1,,N, defined by a one-dimensional stochastic differential equation with coefficients depending on a random variable ?j?j and study the nonparametric estimation of the density of the random effect ?j?j in two kinds of mixed models. A multiplicative random effect and an additive random effect are successively considered. In each case, we build kernel and deconvolution estimators and study their L2L2-risk. Asymptotic properties are evaluated as NN tends to infinity for fixed TT or for T=T(N)T=T(N) tending to infinity with NN. For T(N)=N2T(N)=N2, adaptive estimators are built. Estimators are implemented on simulated data for several examples.  相似文献   

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We study the bounded regions in a generic slice of the hyperplane arrangement in RnRn consisting of the hyperplanes defined by xixi and xi+xjxi+xj. The bounded regions are in bijection with several classes of combinatorial objects, including the ordered partitions of [n][n] all of whose left-to-right minima occur at odd locations and the drawings of rooted plane trees with n+1n+1 vertices. These are sequences of rooted plane trees such that each tree in a sequence can be obtained from the next one by removing a leaf.  相似文献   

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Let K(n,1)K(n,1) denote the minimal cardinality of a binary code of length nn and covering radius one. Fundamental for the theory of lower bounds for K(n,1)K(n,1) is the covering excess method introduced by Johnson and van Wee. Let δiδi denote the covering excess on a sphere of radius ii, 0≤i≤n0in. Generalizing an earlier result of van Wee, Habsieger and Honkala showed δp1≥p−1δp1p1 whenever n≡−1n1 (mod pp) for an odd prime pp and δ0=δ1=?=δp2=0δ0=δ1=?=δp2=0 holds. In the present paper we give the new estimation δp1≥(p−2)p−1δp1(p2)p1 instead. This answers a question of Habsieger and yields a “general improvement of the general excess bound” for binary codes with covering radius one. The proof uses a classification theorem for certain subset systems as well as new congruence properties for the δδ-function, which were conjectured by Habsieger.  相似文献   

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Consider a graph GG with a minimal edge cut FF and let G1G1, G2G2 be the two (augmented) components of G−FGF. A long-open question asks under which conditions the crossing number of GG is (greater than or) equal to the sum of the crossing numbers of G1G1 and G2G2—which would allow us to consider those graphs separately. It is known that crossing number is additive for |F|∈{0,1,2}|F|{0,1,2} and that there exist graphs violating this property with |F|≥4|F|4. In this paper, we show that crossing number is additive for |F|=3|F|=3, thus closing the final gap in the question.  相似文献   

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It is proved that the cookie-cutter set in RR is structurally instable in C1C1 topology, that means for the invariant set EE of the IFS {fi}i{fi}i, we can always perturb {fi}i{fi}i arbitrarily small in C1C1 topology to provide an IFS {gi}i{gi}i with its invariant set FF, such that dimHE=dimHFdimHE=dimHF and E,FE,F are not Lipschitz equivalent.  相似文献   

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For s≥3s3 a graph is K1,sK1,s-free if it does not contain an induced subgraph isomorphic to K1,sK1,s. Cycles in K1,3K1,3-free graphs, called claw-free graphs, have been well studied. In this paper we extend results on disjoint cycles in claw-free graphs satisfying certain minimum degree conditions to K1,sK1,s-free graphs, normally called generalized claw-free graphs. In particular, we prove that if GG is K1,sK1,s-free of sufficiently large order n=3kn=3k with δ(G)≥n/2+cδ(G)n/2+c for some constant c=c(s)c=c(s), then GG contains kk disjoint triangles. Analogous results with the complete graph K3K3 replaced by a complete graph KmKm for m≥3m3 will be proved. Also, the existence of 22-factors for K1,sK1,s-free graphs with minimum degree conditions will be shown.  相似文献   

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In this note we study distance-regular graphs with a small number of vertices compared to the valency. We show that for a given α>2α>2, there are finitely many distance-regular graphs ΓΓ with valency kk, diameter D≥3D3 and vv vertices satisfying v≤αkvαk unless (D=3D=3 and ΓΓ is imprimitive) or (D=4D=4 and ΓΓ is antipodal and bipartite). We also show, as a consequence of this result, that there are finitely many distance-regular graphs with valency k≥3k3, diameter D≥3D3 and c2≥εkc2εk for a given 0<ε<10<ε<1 unless (D=3D=3 and ΓΓ is imprimitive) or (D=4D=4 and ΓΓ is antipodal and bipartite).  相似文献   

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This paper considers the short- and long-memory linear processes with GARCH (1,1) noises. The functional limit distributions of the partial sum and the sample autocovariances are derived when the tail index αα is in (0,2)(0,2), equal to 2, and in (2,∞)(2,), respectively. The partial sum weakly converges to a functional of αα-stable process when α<2α<2 and converges to a functional of Brownian motion when α≥2α2. When the process is of short-memory and α<4α<4, the autocovariances converge to functionals of α/2α/2-stable processes; and if α≥4α4, they converge to functionals of Brownian motions. In contrast, when the process is of long-memory, depending on αα and ββ (the parameter that characterizes the long-memory), the autocovariances converge to either (i) functionals of α/2α/2-stable processes; (ii) Rosenblatt processes (indexed by ββ, 1/2<β<3/41/2<β<3/4); or (iii) functionals of Brownian motions. The rates of convergence in these limits depend on both the tail index αα and whether or not the linear process is short- or long-memory. Our weak convergence is established on the space of càdlàg functions on [0,1][0,1] with either (i) the J1J1 or the M1M1 topology (Skorokhod, 1956); or (ii) the weaker form SS topology (Jakubowski, 1997). Some statistical applications are also discussed.  相似文献   

12.
Given k   pairs of vertices (si,ti)(si,ti)(1≤i≤k)(1ik) of a digraph G, how can we test whether there exist k   vertex-disjoint directed paths from sisi to titi for 1≤i≤k1ik? This is NP-complete in general digraphs, even for k=2k=2 [2], but for k=2k=2 there is a polynomial-time algorithm when G is a tournament (or more generally, a semicomplete digraph), due to Bang-Jensen and Thomassen [1]. Here we prove that for all fixed k there is a polynomial-time algorithm to solve the problem when G is semicomplete.  相似文献   

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We prove that if GG is a finite simple group which is the unit group of a ring, then GG is isomorphic to: (a) a cyclic group of order 2; or (b) a cyclic group of prime order 2k−12k1 for some kk; or (c) a projective special linear group PSLn(F2)PSLn(F2) for some n≥3n3. Moreover, these groups do all occur as unit groups. We deduce this classification from a more general result, which holds for groups GG with no non-trivial normal 2-subgroup.  相似文献   

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We prove that, unless assuming additional set theoretical axioms, there are no reflexive spaces without unconditional sequences of the density continuum. We show that for every integer nn there are normalized weakly-null sequences of length ωnωn without unconditional subsequences. This together with a result of Dodos et al. (2011) [7] shows that ωωωω is the minimal cardinal κκ that could possibly have the property that every weakly null κκ-sequence has an infinite unconditional basic subsequence. We also prove that for every cardinal number κκ which is smaller than the first ωω-Erd?s cardinal there is a normalized weakly-null sequence without subsymmetric subsequences. Finally, we prove that mixed Tsirelson spaces of uncountable densities must always contain isomorphic copies of either c0c0 or ?p?p, with p≥1p1.  相似文献   

15.
A hypergraph is called an r×rr×rgrid   if it is isomorphic to a pattern of rr horizontal and rr vertical lines, i.e., a family of sets {A1,…,Ar,B1,…,Br}{A1,,Ar,B1,,Br} such that AiAj=BiBj=0?AiAj=BiBj=0? for 1≤i<j≤r1i<jr and |AiBj|=1|AiBj|=1 for 1≤i,j≤r1i,jr. Three sets C1,C2,C3C1,C2,C3 form a triangle   if they pairwise intersect in three distinct singletons, |C1C2|=|C2C3|=|C3C1|=1|C1C2|=|C2C3|=|C3C1|=1, C1C2C1C3C1C2C1C3. A hypergraph is linear  , if |E∩F|≤1|EF|1 holds for every pair of edges E≠FEF.  相似文献   

16.
Let M=(Mt)t0M=(Mt)t0 be any continuous real-valued stochastic process. We prove that if there exists a sequence (an)n1(an)n1 of real numbers which converges to 0 and such that MM satisfies the reflection property at all levels anan and 2an2an with n≥1n1, then MM is an Ocone local martingale with respect to its natural filtration. We state the subsequent open question: is this result still true when the property only holds at levels anan? We prove that this question is equivalent to the fact that for Brownian motion, the σσ-field of the invariant events by all reflections at levels anan, n≥1n1 is trivial. We establish similar results for skip free ZZ-valued processes and use them for the proof in continuous time, via a discretization in space.  相似文献   

17.
A plane partition   is a p×qp×q matrix A=(aij)A=(aij), where 1?i?p1?i?p and 1?j?q1?j?q, with non-negative integer entries, and whose rows and columns are weakly decreasing. From a geometric point of view plane partitions are equivalent to pyramids  , subsets of the integer lattice Z3Z3 which play an important role in Discrete Tomography. As a consequence, some typical problems concerning the tomography of discrete lattice sets can be rephrased and considered via plane partitions. In this paper we focus on some of them. In particular, we get a necessary and sufficient condition for additivity, a canonical procedure for checking the existence of (weakly) bad configurations, and an algorithm which constructs minimal pyramids (with respect to the number of levels) with assigned projection of a bad configurations.  相似文献   

18.
In this paper, we study first the problem of nonparametric estimation of the stationary density ff of a discrete-time Markov chain (Xi)(Xi). We consider a collection of projection estimators on finite dimensional linear spaces. We select an estimator among the collection by minimizing a penalized contrast. The same technique enables us to estimate the density gg of (Xi,Xi+1)(Xi,Xi+1) and so to provide an adaptive estimator of the transition density π=g/fπ=g/f. We give bounds in L2L2 norm for these estimators and we show that they are adaptive in the minimax sense over a large class of Besov spaces. Some examples and simulations are also provided.  相似文献   

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