共查询到20条相似文献,搜索用时 31 毫秒
1.
Abstract
Let Λ = {λ
k
} be an infinite increasing sequence of positive integers with λ
k
→∞. Let X = {X(t), t ∈? R
N
} be a multi-parameter fractional Brownian motion of index α(0 < α < 1) in R
d
. Subject to certain hypotheses, we prove that if N < αd, then there exist positive finite constants K
1 and K
2 such that, with unit probability,
if and only if there exists γ > 0 such that
where ϕ(s) = s
N/α
(log log 1/s)
N/(2α), ϕ-p
Λ(E) is the Packing-type measure of E,X([0, 1])
N
is the image and GrX([0, 1]
N
) = {(t,X(t)); ∈? [0, 1]
N
} is the graph of X, respectively. We also establish liminf type laws of the iterated logarithm for the sojourn measure of X.
Supported by the National Natural Science Foundation of China (No.10471148), Sci-tech Innovation Item for Excellent Young
and Middle-Aged University Teachers and Major Item of Educational Department of Hubei (No.2003A005) 相似文献
2.
We obtain asymptotic estimates for the quantity r = log P[Tf[rang]t] as t → ∞ where Tf = inf\s{s : |X(s)|[rang]f(s)\s} and X is a real diffusion in natural scale with generator a(x) d2(·)/dx2 and the ‘boundary’ f(s) is an increasing function. We impose regular variation on a and f and the result is expressed as r = ∫t0 λ1 (f(s) ds(1 + o(1)) where λ1(f) is the smallest eigenvalue for the process killed at ±f. 相似文献
3.
Philippe et al. [9], [10] introduced two distinct time-varying mutually invertible fractionally integrated filters A(d), B(d) depending on an arbitrary sequence d = (d
t
)
t∈ℤ of real numbers; if the parameter sequence is constant d
t
≡ d, then both filters A(d) and B(d) reduce to the usual fractional integration operator (1 − L)−d
. They also studied partial sums limits of filtered white noise nonstationary processes A(d)ε
t
and B(d)ε
t
for certain classes of deterministic sequences d. The present paper discusses the randomly fractionally integrated stationary processes X
t
A
= A(d)ε
t
and X
t
B
= B(d)ε
t
by assuming that d = (d
t
, t ∈ ℤ) is a random iid sequence, independent of the noise (ε
t
). In the case where the mean
, we show that large sample properties of X
A
and X
B
are similar to FARIMA(0,
, 0) process; in particular, their partial sums converge to a fractional Brownian motion with parameter
. The most technical part of the paper is the study and characterization of limit distributions of partial sums for nonlinear
functions h(X
t
A
) of a randomly fractionally integrated process X
t
A
with Gaussian noise. We prove that the limit distribution of those sums is determined by a conditional Hermite rank of h. For the special case of a constant deterministic sequence d
t
, this reduces to the standard Hermite rank used in Dobrushin and Major [2].
Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 3–28, January–March, 2007. 相似文献
4.
Michele Baldini 《Journal of Theoretical Probability》2007,20(2):327-337
Let X
t
be a reversible and positive recurrent diffusion in ℝd described by
Xt=x+s b(t)+ò0tm(Xs)ds,X_{t}=x+\sigma\,b(t)+\int_{0}^{t}m(X_{s})\mathrm {d}s, 相似文献
5.
Jerzy Szulga 《Probability Theory and Related Fields》1992,94(1):83-90
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 相似文献
6.
We study Karhunen-Loève expansions of the process(X
t
(α))
t∈[0,T) given by the stochastic differential equation $
dX_t^{(\alpha )} = - \frac{\alpha }
{{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T)
$
dX_t^{(\alpha )} = - \frac{\alpha }
{{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T)
, with the initial condition X
0(α) = 0, where α > 0, T ∈ (0, ∞), and (B
t
)t≥0 is a standard Wiener process. This process is called an α-Wiener bridge or a scaled Brownian bridge, and in the special case of α = 1 the usual Wiener bridge. We present weighted and unweighted Karhunen-Loève expansions of X
(α). As applications, we calculate the Laplace transform and the distribution function of the L
2[0, T]-norm square of X
(α) studying also its asymptotic behavior (large and small deviation). 相似文献
7.
Let X(t) be an N parameter generalized Lévy sheet taking values in ℝd 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. 相似文献
8.
We consider a general linear model
, where the innovations Zt belong to the domain of attraction of an α-stable law for α<2, so that neither Zt nor Xt have a finite variance. We do not assume that (Xt) is a standardARMA process of the form φ(B)Xt=ϕ(B)Zt, but we fit anARMA process of a given order to the data X1,...,Xn by estimating the coefficients of φ and ϕ. Given that (Xt) is anARMA process, it has been proved that the Whittle estimator is a consistent estimator of the true coefficients of ϕ and φ. Moreover,
it then has a heavytailed limit distribution and the rate of convergence is (n/logn)1/α, which compares favorably with the L2 situation with rate
. In this note we study the limit properties of the Whittle estimator when the underlying model is not necessarily anARMA process. Under general conditions we show that the Whittle estimate converges in probability. It converges weakly to a distribution
which does not have a finite moment of order a and the rate of convergence is again (n/logn)1/α. We also give an analytic expression for the limit distribution.
Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II, Eger, Hungary, 1994. 相似文献
9.
Hans M. Dietz 《Statistical Inference for Stochastic Processes》2001,4(3):249-258
Suppose one observes a path of a stochastic processX = (Xt)t≥0 driven by the equation
dXt=θ a(Xt)dt + dWt, t≥0, θ ≥ 0
with a(x) = x or a(x) = |x|α for some α ∈ [0,1) and given initial condition X
0. If the true but unknown parameter θ0 is positive then X is non-ergodic. It is shown that in this situation a trajectory fitting estimator for θ0 is strongly consistent and has the same limiting distribution as the maximum likelihood estimator, but converges of minor
order.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
10.
The Increment Ratio (IR) statistic (see (1.1) below) was introduced in Surgailis et al. [16]. The IR statistic can be used for testing nonparametric hypotheses for d-integrated (−1/2 < d < 5/4) behavior of time series, including short memory (d = 0), (stationary) long-memory (0 < d < 1/2), and unit roots (d = 1). For stationary/stationary increment Gaussian observations, in [16], a rate of decay of the bias of the IR statistic
and a central limit theorem are obtained. In this paper, we study the asymptotic distribution of the IR statistic under the
model X
t = X
t0 + g
N(t) (t = 1, …, N), where X
t0 is a stationary/stationary increment Gaussian process as in [16], and g
N(t) is a slowly varying deterministic trend. In particular, we obtain sufficient conditions on X
t0 and g
N(t) under which the IR test has the same asymptotic confidence intervals as in the absence of the trend. We also discuss the
asymptotic distribution of the IR statistic under change-points in mean and scale parameters.
Partially supported by the bilateral France-Lithuania scientific project Gilibert and Lithuanian State Science and Studies
Foundation, grant No. T-25/08. 相似文献
11.
Esa Järvenpää Maarit Järvenpää Antti Käenmäki Tapio Rajala Sari Rogovin Ville Suomala 《Mathematische Zeitschrift》2010,266(1):83-105
Let X be a metric measure space with an s-regular measure μ. We prove that if A ì X{A\subset X} is r{\varrho} -porous, then dimp(A) £ s-crs{{\rm {dim}_p}(A)\le s-c\varrho^s} where dimp is the packing dimension and c is a positive constant which depends on s and the structure constants of μ. This is an analogue of a well known asymptotically sharp result in Euclidean spaces. We illustrate by an example that the
corresponding result is not valid if μ is a doubling measure. However, in the doubling case we find a fixed N ì X{N\subset X} with μ(N) = 0 such that
dimp(A) £ dimp(X)-c(log\tfrac1r)-1rt{{\rm {dim}_p}(A)\le{\rm {dim}_p}(X)-c(\log \tfrac1\varrho)^{-1}\varrho^t} for all r{\varrho} -porous sets A ì X\ N{A \subset X{\setminus} N} . Here c and t are constants which depend on the structure constant of μ. Finally, we characterize uniformly porous sets in complete s-regular metric spaces in terms of regular sets by verifying that A is uniformly porous if and only if there is t < s and a t-regular set F such that A ì F{A\subset F} . 相似文献
12.
We investigate the convergence of distributions of partial sums of Appell polynomials
of a long-memory moving average process X
t
with i.i.d. innovations s in the case where the variance
, and the distribution of #x03BE;
0
m
belongs to the domain of attraction of an -stable law with 1<< 2. We prove that the limit distribution of partial sums of Appell polynomials is either an -stable Lévy process, or an mth order Hermite process, or the sum of two mutually independent processes depending on the values of , m, and d, where 0
13.
Litan Yan 《Mathematische Nachrichten》2003,259(1):84-98
Let X = (Xt, ?t) be a continuous local martingale with quadratic variation 〈X〉 and X0 = 0. Define iterated stochastic integrals In(X) = (In(t, X), ?t), n ≥ 0, inductively by $$ I_{n} (t, X) = \int ^{t} _{0} I_{n-1} (s, X)dX_{s} $$ with I0(t, X) = 1 and I1(t, X) = Xt. Let (??xt(X)) be the local time of a continuous local martingale X at x ∈ ?. Denote ??*t(X) = supx∈? ??xt(X) and X* = supt≥0 |Xt|. In this paper, we shall establish various ratio inequalities for In(X). In particular, we show that the inequalities $$ c_{n,p} \, \left\Vert (G ( \langle X \rangle _{\infty} )) ^{n/2} \right\Vert _{p} \; \le \; \left\Vert {\mathop \sup \limits _{t \ge 0}} \; {\left\vert I_{n} (t, X) \right\vert \over {(1+ \langle X \rangle _{t} ) ^{n/2}}} \right\Vert _{p} \; \le C_{n, p} \, \left\Vert (G ( \langle X \rangle _{\infty} )) ^{n/2} \right\Vert _{p} $$ hold for 0 < p < ∞ with some positive constants cn,p and Cn,p depending only on n and p, where G(t) = log(1+ log(1+ t)). Furthermore, we also show that for some γ ≥ 0 the inequality $$ E \left[ U ^{p}_{n} \exp \left( \gamma {U ^{1/n} _{n} \over {V}} \right) \right] \le C_{n, p, \gamma} E [V ^{n, p}] \quad (0 < p < \infty ) $$ holds with some positive constant Cn,p,γ depending only on n, p and γ, where Un is one of 〈In(X)〉1/2∞ and I*n(X), and V one of the three random variables X*, 〈X〉1/2∞ and ??*∞(X). (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
14.
Jonathan Luk 《Annales Henri Poincare》2010,11(5):805-880
We prove that sufficiently regular solutions to the wave equation ${\square_g\phi=0}
15.
Vladimir Varlamov 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,138(6):1017-1031
Riesz fractional derivatives are defined as fractional powers of the Laplacian, D α = (?Δ) α/2 for ${\alpha \in \mathbb{R}}
|