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1.
R. A. Doney 《Probability Theory and Related Fields》1997,107(4):451-465
Summary. If {S
n
,n≧0} is an integer-valued random walk such that S
n
/a
n
converges in distribution to a stable law of index α∈ (0,1) as n→∞, then Gnedenko’s local limit theorem provides a useful estimate for P{S
n
=r} for values of r such that r/a
n
is bounded. The main point of this paper is to show that, under certain circumstances, there is another estimate which is
valid when r/a
n
→ +∞, in other words to establish a large deviation local limit theorem. We also give an asymptotic bound for P{S
n
=r} which is valid under weaker assumptions. This last result is then used in establishing some local versions of generalized
renewal theorems.
Received: 9 August 1995 / In revised form: 29 September 1996 相似文献
2.
We establish an integral test involving only the distribution of the increments of a random walk S which determines whether limsup n→∞(Sn/nκ) is almost surely zero, finite or infinite when 1/2<κ<1 and a typical step in the random walk has zero mean. This completes the results of Kesten and Maller [9] concerning finiteness of one-sided passage times over power law boundaries, so that we now have quite explicit criteria for all values of κ≥0. The results, and those of [9], are also extended to Lévy processes.This work is partially supported by ARC Grant DP0210572. 相似文献
3.
Let B be the Brownian motion on a noncompact non Euclidean rank one symmetric space H. A typical examples is an hyperbolic space H
n
, n > 2. For ν > 0, the Brownian bridge B
(ν) of length ν on H is the process B
t
, 0 ≤t≤ν, conditioned by B
0 = B
ν = o, where o is an origin in H. It is proved that the process converges weakly to the Brownian excursion when ν→ + ∞ (the Brownian excursion is the radial part of the Brownian Bridge
on ℝ3). The same result holds for the simple random walk on an homogeneous tree.
Received: 4 December 1998 / Revised version: 22 January 1999 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.
In this article we study the exponential behavior of the continuous stochastic Anderson model, i.e. the solution of the stochastic
partial differential equation u(t,x)=1+∫0tκΔxu (s,x) ds+∫0t W(ds,x) u (s,x), when the spatial parameter x is continuous, specifically x∈R, and W is a Gaussian field on R+×R that is Brownian in time, but whose spatial distribution is widely unrestricted. We give a partial existence result of the
Lyapunov exponent defined as limt→∞t−1 log u(t,x). Furthermore, we find upper and lower bounds for lim supt→∞t−1 log u(t,x) and lim inft→∞t−1 log u(t,x) respectively, as functions of the diffusion constant κ which depend on the regularity of W in x. Our bounds are sharper, work for a wider range of regularity scales, and are significantly easier to prove than all previously
known results. When the uniform modulus of continuity of the process W is in the logarithmic scale, our bounds are optimal.
This author's research partially supported by NSF grant no. : 0204999 相似文献
6.
E. A. Rovba 《Mathematical Notes》1997,61(2):221-226
Summation rational positive operatorsD
4n−n(x; f) of the Jackson type are constructed on the real axis. The corresponding approximations of continuous functionsf onℝwith coinciding finite limits limx→−∞
f(x) and limx→+∞
f(x) are estimated.
Translated fromMatematischeskie Zametki, Vol. 61, No. 2, pp. 270–277, February, 1997.
Translated by N. K. Kulman 相似文献
7.
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
8.
A.A. Borovkov 《Probability Theory and Related Fields》2003,125(3):421-446
Let be independent identically distributed random variables with regularly varying distribution tails:
where α≤ min (1,β), and L and L
W
are slowly varying functions as t→∞. Set S
n
=X
1
+⋯+X
n
, ˉS
n
= max
0≤ k ≤ n
S
k
. We find the asymptotic behavior of P
(S
n
> x)→0 and P
(ˉS
n
> x)→0 as x→∞, give a criterion for ˉS
∞
<∞ a.s. and, under broad conditions, prove that P (ˉS
∞
> x)˜c V(x)/W(x).
In case when distribution tails of X
j
admit regularly varying majorants or minorants we find sharp estimates for the mentioned above probabilities under study.
We also establish a joint distributional representation for the global maximum ˉS
∞
and the time η when it was attained in the form of a compound Poisson random vector.
Received: 4 June 2001 / Revised version: 10 September 2002 / Published online: 21 February 2003
Research supported by INTAS (grant 00265) and the Russian Foundation for Basic Research (grant 02-01-00902)
Mathematics Subject Classification (2000): 60F99, 60F10, 60G50
Key words or phrases: Attraction domain of a stable law – Maximum of sums of random variables – Criterion for the maximum of sums – Large deviations 相似文献
9.
K. F. Cheng 《Annals of the Institute of Statistical Mathematics》1982,34(1):479-489
Summary Letf
n
(p)
be a recursive kernel estimate off
(p) thepth order derivative of the probability density functionf, based on a random sample of sizen. In this paper, we provide bounds for the moments of
and show that the rate of almost sure convergence of
to zero isO(n
−α), α<(r−p)/(2r+1), iff
(r),r>p≧0, is a continuousL
2(−∞, ∞) function. Similar rate-factor is also obtained for the almost sure convergence of
to zero under different conditions onf.
This work was supported in part by the Research Foundation of SUNY. 相似文献
10.
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
11.
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
12.
We establish conditions under which, for a Dirichlet series F(z) = Σ
n = 1
∞
d
n
exp(λ
n
z), the inequality ⋎F(x)⋎≤y(x),x≥x
o, implies the relation Σ
n = 1
∞ |d
n
exp(λ
n
z)| ⪯ γ((1 + o(1))x) as x→+∞, where γ is a nondecreasing function on (−∞,+∞).
Franko Drohobych State Pedagogic Institute, Drohobych. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12,
pp. 1610–1616. December, 1997 相似文献
13.
Giuliana Palmieri 《Milan Journal of Mathematics》1995,65(1):335-356
Given a setS and a function σ:S x S→[0, +∞[ such that σ(x, x)=0, we define the generalizedp-energy of a curve γ: [a, b]→S, so that, ifS is a Hilbert space and σ(x, y)=‖x−y‖ we obtain
. Sufficient conditions for the equality of the defined energies are also given. Moreover the case in whichS is a set of measurable parts of ℝn and σr is a family of functions used in order to study the generalized minimizing motions, is discussed.
Conferenza tenuta il 30 ottobre 1995 相似文献
Conferenza tenuta il 30 ottobre 1995 相似文献
14.
M.S. Bernabei 《Probability Theory and Related Fields》2001,119(3):410-432
The Central Limit Theorem for a model of discrete-time random walks on the lattice ℤν in a fluctuating random environment was proved for almost-all realizations of the space-time nvironment, for all ν > 1 in
[BMP1] and for all ν≥ 1 in [BBMP]. In [BMP1] it was proved that the random correction to the average of the random walk for
ν≥ 3 is finite. In the present paper we consider the cases ν = 1,2 and prove the Central Limit Theorem as T→∞ for the random correction to the first two cumulants. The rescaling factor for theaverage is for ν = 1 and (ln T), for ν=2; for the covariance it is , ν = 1,2.
Received: 25 November 1999 / Revised version: 7 June 2000 / Published online: 15 February 2001 相似文献
15.
Adrian Constantin 《Annali dell'Universita di Ferrara》1995,41(1):1-4
LetH be a complex Hilbert space and letB be the space of all bounded linear operators fromH intoH with the strong operator topology. We will give a boundedness result for the solutions of the differential equationx′=A(t)x+f(t,x) whereA: I=[t
0, ∞)→B is continuous,f: I×H→H is also continuous and for every bounded setS⊂I×H there exists a constantM(S)>0 such that |f(t,x)−f(t,y)|≤M(S)|x−y|,(t,x), (t,y)∈S.
Sunto SiaH uno spazio di Hilbert complesso e siaB lo spazio degli operatori lineari limitati daH inH, con la topologia forte. In questo lavoro si prova un risultato di limitatezza per le soluzioni dell'equazione differenzialex′=A(t)x+f(t,x), doveA: I=[t 0, ∞)→B è continua,f: I×H→H è continua e per ogni insieme limitatoS⊂I×H esiste una costanteM(S)>0 tale che |f(t,x)−f(t,y)|≤M(S)|x−y| per ogni(t,x), (t,y)∈S.相似文献
16.
StrongwayShi 《高校应用数学学报(英文版)》2000,15(1):45-54
Abstract. Let {Xn,n≥1} be a stationary strongly mixing random sequence satisfying EX1=u, 相似文献
17.
18.
负相依随机变量的自正则化中心极限定理与部分和的方差估计 总被引:1,自引:0,他引:1
§ 1 IntroductionWe firstintroduce some concepts.Random variables X and Y are called negative dependent ( ND) if for any pair ofmonotonically non-decresing functions f and g,Cov{ f( X) ,g( Y) }≤ 0 .Clearly itis equivalenttoP( X≤ x,Y≤ y)≤ P( X≤ x) P( Y≤ y)for all x,y∈R.A random sequence{ Xi,i≥ 1 } is said to be negative quadrant dependent( NQD) if any pairof variables Xi,Xj( i≠j) are ND.A sequence of random variables{ Xi,i≥ 1 } is said to be linear negative quadrand depend… 相似文献
19.
In a uniform random recursive k-directed acyclic graph, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If S
n
is the shortest path distance from node n to the root, then we determine the constant σ such that S
n
/log n→σ in probability as n→∞. We also show that max 1≤i≤n
S
i
/log n→σ in probability. 相似文献
20.
Allan Gut 《Stochastic Processes and their Applications》1974,2(1):115-126
Let Sn,n = 1, 2, …, denote the partial sums of integrable random variables. No assumptions about independence are made. Conditions for the finiteness of the moments of the first passage times N(c) = min {n: Sn>ca(n)}, where c ≥ 0and a(y) is a positive continuous function on [0, ∞), such that a(y) = o(y)as y → ∞, are given. With the further assumption that a(y) = yP,0 ≤ p < 1, a law of large numbers and the asymptotic behaviour of the moments when c → ∞ are obtained. The corresponding stopped sums are also studied. 相似文献