共查询到20条相似文献,搜索用时 31 毫秒
1.
Let {S
n}
n0 be a random walk on the line. We give criteria for the existence of a nonrandom sequence n
i for which
respectively
We thereby obtain conditions for to be a strong limit point of {S
n} or {S
n
/n}. The first of these properties is shown to be equivalent to
for some sequence a
i , where T(a) is the exit time from the interval [–a,a]. We also obtain a general equivalence between
and
for an increasing function fand suitable sequences n
i and a
i. These sorts of properties are of interest in sequential analysis. Known conditions for
and
(divergence through the whole sequence n) are also simplified. 相似文献
2.
Let
, the parameter space, be an open subset ofR
k
,k1. For each
, let the r.v.'sX
n
,n=1, 2,... be defined on the probability space (X, P
) and take values in (S,S,L) whereS is a Borel subset of a Euclidean space andL is the -field of Borel subsets ofS. ForhR
k
and a sequence of p.d. normalizing matrices
n
=
n
k × k
(0 set
n
*
= * = 0 +
n
h, where 0 is the true value of , such that *,
. Let
n
(*, *)( be the log-likelihood ratio of the probability measure
with respect to the probability measure
, whereP
n
is the restriction ofP
over
n
= (X
1,X
2,...,X
n
. In this paper we, under a very general dependence setup obtain a rate of convergence of the normalized log-likelihood ratio statistic to Standard Normal Variable. Two examples are taken into account. 相似文献
3.
Allanus H. Tsoi 《Journal of Theoretical Probability》1993,6(4):693-698
LetG be a Lie group ofd×d matrices and be theLLie algebra ofG. We choose some Euclidean norm on , and an orthonormal basis (D
1,...D
m
) relative to it. Let
be the corresponding left invariant vector fields onG. In this paper we derive an integration by parts formula for aG-valued Brownian motion corresponding to the Laplacian
. 相似文献
4.
A compound Poisson process is of the form
where Z, Z
1, Z
2, are arbitrary i.i.d. random variables and N
is an independent Poisson random variable with parameter . This paper identifies the degree of precision that can be achieved when using exponential bounds together with a single truncation to approximate
. The truncation level introduced depends only on and Z and not on the overall exceedance level a. 相似文献
5.
Gerold Alsmeyer 《Journal of Theoretical Probability》2002,15(2):259-283
It is proved that for each random walk (S
n
)
n0 on
d
there exists a smallest measurable subgroup
of
d
, called minimal subgroup of (S
n
)
n0, such that P(S
n
)=1 for all n1.
can be defined as the set of all x
d
for which the difference of the time averages n
–1
n
k=1
P(S
k
) and n
–1
n
k=1
P(S
k
+x) converges to 0 in total variation norm as n. The related subgroup
* consisting of all x
d
for which lim
n P(S
n
)–P(S
n
+x)=0 is also considered and shown to be the minimal subgroup of the symmetrization of (S
n
)
n0. In the final section we consider quasi-invariance and admissible shifts of probability measures on
d
. The main result shows that, up to regular linear transformations, the only subgroups of
d
admitting a quasi-invariant measure are those of the form
1×...×
k
×
l–k
×{0}
d–l
, 0kld, with
1,...,
k
being countable subgroups of
. The proof is based on a result recently proved by Kharazishvili(3) which states no uncountable proper subgroup of
admits a quasi-invariant measure. 相似文献
6.
Let {B
t
,t[0,1]} be a fractional Brownian motion with Hurst parameter H > 1/2. Using the techniques of the Malliavin calculus we show that the trajectories of the indefinite divergence integral
t
0
u
s
B
s
belong to the Besov space
p,q
for all
, provided the integrand u belongs to the space
. Moreover, if u is bounded and belongs to
for some even integer p2 and for some large enough, then the trajectories of the indefinite divergence integral
t
0
u
s
B
s
belong to the Besov space
p,
H
. 相似文献
7.
Let * be the convolution on M(
+) associated with a second order singular differential operator L on ]0, +[. If is a probability measure on
+ with suitable moment conditions, we study how to normalize the measures *
n
; n
} (resp.
) in order to get vague convergence if n+ (resp. x+). The results depend on the asymptotic drift of the operator L and on a precise study of the asymptotic behaviour of its eigenfunctions. 相似文献
8.
Consider a double array
of i.i.d. random variables with mean and variance
and set
. Let
denote the empirical distribution function of Z1, n
,..., Z
N, n
and let be the standard normal distribution function. The main result establishes a functional law of the iterated logarithm for
, where n=n(N) as N. For the proof, some lemmas are derived which may be of independent interest. Some corollaries of the main result are also presented. 相似文献
9.
In this work we obtain an asymptotic estimate for the expected number of maxima of the random algebraic polynomial
, where a
j (j=0, 1,...,n–1) are independent, normally distributed random variables with mean and variance one. It is shown that for nonzero , the expected number of maxima is asymptotic to
log n, when n is large. 相似文献
10.
Rita Solomyak 《Journal of Theoretical Probability》1999,12(2):523-548
Let be a Cayley graph of a finitely generated group G. Subgraphs which contain all vertices of , have no cycles, and no finite connected components are called essential spanning forests. The set
of all such subgraphs can be endowed with a compact topology, and G acts on
by translations. We define a uniform G-invariant probability measure on
show that is mixing, and give a sufficient condition for directional tail triviality. For non-cocompact Fuchsian groups we show how can be computed on cylinder sets. We obtain as a corollary, that the tail -algebra is trivial, and that the rate of convergence to mixing is exponential. The transfer-current function (an analogue to the Green function), is computed explicitly for the Modular and Hecke groups. 相似文献
11.
Let (Z
n
)
n 0 be a supercritical Galton–Watson process with finite re-production mean and normalized limit W=lim
n –n
Z
n
. Let further : [0,) [0,) be a convex differentiable function with (0)=(0)=0 and such that (
) is convex with concave derivative for some n 0. By using convex function inequalities due to Topchii and Vatutin, and Burkholder, Davis and Gundy, we prove that 0 < E
(W) < if, and only if,
, where
We further show that functions (x)=x
L(x) which are regularly varying of order 1 at are covered by this result if {2
n
: n 0 } and under an additional condition also if =2
n
for some n0. This was obtained in a slightly weaker form and analytically by Bingham and Doney. If > 1, then
grows at the same order of magnitude as (x) so that
and E
(Z
1)< are equivalent. However, =1 implies
and hence that
is a strictly stronger condition than E
(Z
1) < . If (x)=x log
p
x for some p > 0 it can be shown that
grows like x log
p+1
x, as x. For this special case the result is due to Athreya. As a by-product we also provide a new proof of the Kesten–Stigum result that E
Z
1 log Z
1 < and EW > 0 are equivalent. 相似文献
12.
Andrzej Stachurski 《Mathematical Programming》1981,20(1):196-212
In this paper the so-called Broyden's bounded-class of methods is considered. It contains as a subclass Broyden's restricted-class of methods, in which the updating matrices retain symmetry and positive definiteness. These iteration methods are used for solving unconstrained minimization problems of the following form:
It is assumed that the step-size coefficient
k = 1 in each iteration and the functionalf : R
n R1 satisfies the standard assumptions, viz.f is twice continuously differentiable and the Hessian matrix is uniformly positive definite and bounded (there exist constantsm, M > 0 such that my2 y,
for ally R
n) and satisfies a Lipschitz-like condition at the optimal point
, the gradient vanishes at
Under these assumptions the local convergence of Broyden's methods is proved. Furthermore, the Q-superlinear convergence is shown. 相似文献
13.
Maher Mili 《Journal of Theoretical Probability》2000,13(3):717-731
Let K be respectively the parabolic biangle and the triangle in
and
be a sequence in [0, +[ such that limp (p)=+. According to Koornwinder and Schwartz,(7) for each
there exist a convolution structure (*(p)) such that (K, *(p)) is a commutative hypergroup. Consider now a random walk
on (K, *(p)), assume that this random walk is stopped after j(p) steps. Then under certain conditions given below we prove that the random variables
on K admit a selective limit theorems. The proofs depend on limit relations between the characters of these hypergroups and Laguerre polynomials that we give in this work. 相似文献
14.
Let
be a real separable Banach space and {X, X
n, m; (n, m) N
2} B-valued i.i.d. random variables. Set
. In this paper, the compact law of the iterated logarithm, CLIL(D), for B-valued random variables with two-dimensional indices ranging over a subset D of N
2 is studied. There is a gap between the moment conditions for CLIL(N
1) and those for CLIL(N
2). The main result of this paper fills this gap by presenting necessary and sufficient conditions for the sequence
to be almost surely conditionally compact in B, where, for 0, 1 r 2, N
r
(, ) = {(n, m) N
2; n
m n
exp{(log n)
r–1 (n)}} and (·) is any positive, continuous, nondecreasing function such that (t)/(log log t) is eventually decreasing as t , for some > 0. 相似文献
15.
T. Simon 《Journal of Theoretical Probability》2004,17(4):979-1002
Let Z
t
, t 0 be a strictly stable process on
with index (0, 2]. We prove that for every p > , there exists = , p
and
such that
where || Z||
p
stands for the strong p-variation of Z on [0,1]. The critical exponent p
, takes a different shape according as | Z| is a subordinator and p > 1, or not. The small ball constant
is explicitly computed when p > 1, and a lower bound on
is easily obtained in the general case. In the symmetric case and when p > 2, we can also give an upper bound on
in terms of the Brownian small ball constant under the (1/p)-Höder semi-norm. Along the way, we remark that the positive random variable
is not necessarily stable when p > 1, which gives a negative answer to an old question of P. E. Greenwood.10 相似文献
16.
Let
be a continuous semimartingale and let
be a continuous function of bounded variation. Setting
and
suppose that a continuous function
is given such that F is C1,2 on
and F is
on
. Then the following change-of-variable formula holds:
where
is the local time of X at the curve b given by
and
refers to the integration with respect to
. A version of the same formula derived for an Itô diffusion X under weaker conditions on F has found applications in free-boundary problems of optimal stopping. 相似文献
17.
Let X be a real-valued random variable and
a -algebra. We show that the minimum
-distance between X and a random variable distributed as X and independant of
can be viewed as a dependence coefficient (
,X) whose definition is comparable (but different) to that of the usual -mixing coefficient between
and (X). We compare this new coefficient to other well known measures of dependence, and we show that it can be easily computed in various situations, such as causal Bernoulli shifts or stable Markov chains defined via iterative random maps. Next, we use coupling techniques to obtain Bennett and Rosenthal-type inequalities for partial sums of -dependent sequences. The former is used to prove a strong invariance principle for partial sums. 相似文献
18.
Goran Peskir 《Journal of Theoretical Probability》2001,14(3):845-855
Let X=(X
t
)
t0 be a one-dimensional time-homogeneous diffusion process associated with the infinitesimal generator
where x(x) and x(x)>0 are continuous. We show how the question of finding a function xH(x) such that
holds for all stopping times of X relates to solutions of the equation:
Explicit expressions for H are derived in terms of and . The method of proof relies upon a domination principle established by Lenglart and Itô calculus. 相似文献
19.
In this paper, a general technique for solving nonlinear, two-point boundary-value problems is presented; it is assumed that the differential system has ordern and is subject top initial conditions andq final conditions, wherep+q=n. First, the differential equations and the boundary conditions are linearized about a nominal functionx(t) satisfying thep initial conditions. Next, the linearized system is imbedded into a more general system by means of a scaling factor , 01, applied to each forcing term. Then, themethod of particular solutions is employed in order to obtain the perturbation x(t)=A(t) leading from the nominal functionx(t) to the varied function
(t); this method differs from the adjoint method and the complementary function method in that it employs only one differential system, namely, the nonhomogeneous, linearized system.The scaling factor (or stepsize) is determined by a one-dimensional search starting from =1 so as to ensure the decrease of the performance indexP (the cumulative error in the differential equations and the boundary conditions). It is shown that the performance index has a descent property; therefore, if is sufficiently small, it is guaranteed that
<P. Convergence to the desired solution is achieved when the inequalityP is met, where is a small, preselected number.In the present technique, the entire functionx(t) is updated according to
(t)=x(t)+A(t). This updating procedure is called Scheme (a). For comparison purposes, an alternate procedure, called Scheme (b), is considered: the initial pointx(0) is updated according to
(0)=x(0)+A(0), and the new nominal function
(t) is obtained by forward integration of the nonlinear differential system. In this connection, five numerical examples are presented; they illustrate (i) the simplicity as well as the rapidity of convergence of the algorithm, (ii) the importance of stepsize control, and (iii) the desirability of updating the functionx(t) according to Scheme (a) rather than Scheme (b).This research, supported by the National Science Foundation, Grant No. GP-18522, is based on Ref. 1. The authors are indebted to Mr. A. V. Levy for computational assistance. 相似文献
20.
Yimin Xiao 《Journal of Theoretical Probability》1997,10(4):849-866
Let X(t) (tR) be a real-valued centered Gaussian process with stationary increments. We assume that there exist positive constants
0, C
1, and c
2 such that for any tR and hR with |h|0
and for any 0r<min{|t|, 0}
where
is regularly varying at zero of order (0 < < 1). Let be an inverse function of near zero such that (s)=(s) log log(1/s) is increasing near zero. We obtain exact estimates for the weak -variation of X(t) on [0,a]. 相似文献