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1.
Suppose K is a compact convex set in ℝ2 and X
i
, 1≤i≤n, is a random sample of points in the interior of K. Under general assumptions on K and the distribution of the X
i
we study the asymptotic properties of certain statistics of the convex hull of the sample.
Received: 24 July 1996/Revised version: 24 February 1998 相似文献
2.
Tobias Müller 《Combinatorica》2008,28(5):529-545
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. 相似文献
3.
Let n be an integer and A0,..., Ak random subsets of {1,..., n} of fixed sizes a0,..., ak, respectively chosen independently and uniformly. We provide an explicit and easily computable total variation bound between
the distance from the random variable
, the size of the intersection of the random sets, to a Poisson random variable Z with intensity λ = EW. In particular, the bound tends to zero when λ converges and
for all j = 0,..., k, showing that W has an asymptotic Poisson distribution in this regime.
Received February 24, 2005 相似文献
4.
Simeon M. Berman 《Annals of the Institute of Statistical Mathematics》1984,36(1):301-321
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=Xn,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=fn(Xj). 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. 相似文献
5.
J. B. Shearer 《Combinatorica》1985,5(3):241-245
LetX
1, ...,X
n
be events in a probability space. Let ϱi 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. 相似文献
6.
A convex labelling of a tree is an assignment of distinct non-negative integer labels to vertices such that wheneverx, y andz are the labels of vertices on a path of length 2 theny≦(x+z)/2. In addition if the tree is rooted, a convex labelling must assign 0 to the root. The convex label number of a treeT is the smallest integerm such thatT has a convex labelling with no label greater thanm. We prove that every rooted tree (and hence every tree) withn vertices has convex label number less than 4n. We also exhibitn-vertex trees with convex label number 4n/3+o(n), andn-vertex rooted trees with convex label number 2n +o(n).
The research by M. B. and A. W. was partly supported by NSF grant MCS—8311422. 相似文献
7.
8.
Marius Junge 《Positivity》2006,10(2):201-230
For n independent random variables f1, . . . ,fn and a symmetric norm || ||X on ℝn, we show that for 1≤ p < ∞
Here
is the disjoint sum of the fi's and h* is the non-increasing rearrangement. Similar results (where Lp is replaced by a more general rearrangement invariant function space) were obtained first by Litvak, Gordon, Schütt and Werner
for Orlicz spaces X and independently by S. Montgomery-Smith [22] for general X but without an explicit analysis of the order of growth for the constant in the upper estimate. The order is optimal and obtained from combinatorial estimates for doubly stochastic matrices. The result extends to Lorentz-norms
lf, q on ℝn under mild assumptions on f. We give applications to the theory of noncommutative Lp spaces. 相似文献
9.
We study the asymptotic behavior of a set of random vectors ξi, i = 1,..., m, whose coordinates are independent and identically distributed in a space of infinitely increasing dimension. We investigate
the asymptotics of the distribution of the random vectors, the consistency of the sets M
m(n) = ξ1,..., ξm and X
nλ = x ∈ X
n: ρ(x) ≤ λn, and the mutual location of pairs of vectors.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1706–1711, December, 1998. 相似文献
10.
Let Φ be a symmetric function, nondecreasing on [0,∞) and satisfying a Δ2 growth condition, (X
1,Y
1), (X
2,Y
2),…,(X
n
,Y
n
) be arbitrary independent random vectors such that for any given i either Y
i
=X
i
or Y
i
is independent of all the other variates. The purpose of this paper is to develop an approximation of
valid for any constants {a
ij
}1≤
i,j≤n
, {b
i
}
i
=1
n
, {c
j
}
j
=1
n
and d. Our approach relies primarily on a chain of successive extensions of Khintchin's inequality for decoupled random variables
and the result of Klass and Nowicki (1997) for non-negative bilinear forms of non-negative random variables. The decoupling
is achieved by a slight modification of a theorem of de la Pe?a and Montgomery–Smith (1995).
Received: 25 March 1997 / Revised version: 5 December 1997 相似文献
11.
Dr. Matthias Kriesell 《Combinatorica》2006,26(3):277-314
A non-complete graph G is called an (n,k)-graph if it is n-connected but G—X is not (n−|X|+1)-connected for any X ⊂V (G) with |X|≤k. Mader conjectured that for k≥3 the graph K2k+2−(1−factor) is the unique (2k,k)-graph(up to isomorphism).
Here we prove this conjecture. 相似文献
12.
André Adler 《Central European Journal of Mathematics》2006,4(1):1-4
Consider independent and identically distributed random variables {X
nk, 1 ≤ k ≤ m, n ≤ 1} from the Pareto distribution. We select two order statistics from each row, X
n(i) ≤ X
n(j), for 1 ≤ i < j ≤ = m. Then we test to see whether or not Laws of Large Numbers with nonzero limits exist for weighted sums of the random variables
R
ij = X
n(j)/X
n(i). 相似文献
13.
René L. Schilling 《Probability Theory and Related Fields》1998,112(4):565-611
Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C
c
∞(ℝ
n
)⊂D(A) and A|C
c
∞(ℝ
n
) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|∞≤c(1+|ξ|2) and |Imp(x,ξ)|≤c
0Rep(x,ξ). We show that the associated Feller process {X
t
}
t
≥0 on ℝ
n
is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour
of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β∞
x
:={λ>0:lim
|ξ|→∞
|
x
−
y
|≤2/|ξ||p(y,ξ)|/|ξ|λ=0} or δ∞
x
:={λ>0:liminf
|ξ|→∞
|
x
−
y
|≤2/|ξ|
|ε|≤1|p(y,|ξ|ε)|/|ξ|λ=0}, and obtain a.s. (ℙ
x
) that lim
t
→0
t
−1/λ
s
≤
t
|X
s
−x|=0 or ∞ according to λ>β∞
x
or λ<δ∞
x
. Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27].
Received: 21 July 1997 / Revised version: 26 January 1998 相似文献
14.
Summary We give a survey of known results regarding Schur-convexity of probability distribution functions. Then we prove that the
functionF(p
1,...,pn;t)=P(X1+...+Xn≤t) is Schur-concave with respect to (p
1,...,pn) for every realt, whereX
i are independent geometric random variables with parametersp
i. A generalization to negative binomial random variables is also presented. 相似文献
15.
Complete convergence for arrays 总被引:4,自引:0,他引:4
A. Gut 《Periodica Mathematica Hungarica》1992,25(1):51-75
Let {(X
nk
, 1≤k≤n),n≥1}, be an array of rowwise independent random variables. We extend and generalize some recent results due to Hu, Móricz and
Taylor concerning complete convergence, in the sense of Hsu and Robbins, of the sequence of rowwise arithmetic means. 相似文献
16.
We consider a real random walk Sn=X1+...+Xn attracted (without centering) to the normal law: this means that for a suitable norming sequence an we have the weak convergence Sn/an⇒ϕ(x)dx, ϕ(x) being the standard normal density. A local refinement of this convergence is provided by Gnedenko's and Stone's Local Limit
Theorems, in the lattice and nonlattice case respectively. Now let denote the event (S1>0,...,Sn>0) and let Sn+ denote the random variable Sn conditioned on : it is known that Sn+/an ↠ ϕ+(x) dx, where ϕ+(x):=x exp (−x2/2)1(x≥0). What we establish in this paper is an equivalent of Gnedenko's and Stone's Local Limit Theorems for this weak convergence.
We also consider the particular case when X1 has an absolutely continuous law: in this case the uniform convergence of the density of Sn+/an towards ϕ+(x) holds under a standard additional hypothesis, in analogy to the classical case. We finally discuss an application of our
main results to the asymptotic behavior of the joint renewal measure of the ladder variables process. Unlike the classical
proofs of the LLT, we make no use of characteristic functions: our techniques are rather taken from the so–called Fluctuation
Theory for random walks. 相似文献
17.
Brice Franke 《Extremes》2011,14(1):127-152
We investigate the recursive sequence Z
n
: = max {Z
n − 1,λ(Z
n − 1)X
n
} where X
n
is a sequence of iid random variables with exponential distributions and λ is a periodic positive bounded measurable function. We prove that the Césaro mean of the sequence λ(Z
n
) converges toward the essential minimum of λ. Subsequently we apply this result and obtain a limit theorem for the distributions of the sequence Z
n
. The resulting limit is a Gumbel distribution. 相似文献
18.
We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1 − ε)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d ≥ 2 and 0 < ε < 1, there exists a constant c = c(d, ε) such that a random graph G(n, c/n) contains almost surely a copy of every tree T on (1 − ε)n vertices with maximum degree at most d. We also prove that if an (n, D, λ)-graph G (i.e., a D-regular graph on n vertices all of whose eigenvalues, except the first one, are at most λ in their absolute values) has large enough spectral gap D/λ as a function of d and ε, then G has a copy of every tree T as above.
Research supported in part by a USA-Israeli BSF grant, by NSF grant CCR-0324906, by a Wolfensohn fund and by the State of
New Jersey.
Research supported in part by USA-Israel BSF Grant 2002-133, and by grants 64/01 and 526/05 from the Israel Science Foundation.
Research supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant, and by an Alfred
P. Sloan fellowship. 相似文献
19.
M. Ivette Gomes 《Annals of the Institute of Statistical Mathematics》1984,36(1):71-85
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 {αn}n≧1(αn>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−bn)/an thanG(x), for moderaten. We thus consider differences of the formF
n(anx+bn)−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(anx+bn) than the ultimate itself. 相似文献
20.
Aurel Spătaru 《Probability Theory and Related Fields》2006,136(1):1-18
Let X1, X2, . . . be i.i.d. random variables, and set Sn=X1+ . . . +Xn. Several authors proved convergence of series of the type f(ɛ)=∑ncnP(|Sn|>ɛan),ɛ>α, under necessary and sufficient conditions. We show that under the same conditions, in fact i.e. the finiteness of ∑ncnP(|Sn|>ɛan),ɛ>α, is equivalent to the convergence of the double sum ∑k∑ncnP(|Sn|>kan). Two exceptional series required deriving necessary and sufficient conditions for E[supn|Sn|(logn)η/n]<∞,0≤η≤1. 相似文献