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1.
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   

2.
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   

3.
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   

4.
Packing Measure and Dimension of Random Fractals   总被引:1,自引:0,他引:1  
We consider random fractals generated by random recursive constructions. We prove that the box-counting and packing dimensions of these random fractals, K, equals , their almost sure Hausdorff dimension. We show that some almost deterministic conditions known to ensure that the Hausdorff measure satisfies also imply that the packing measure satisfies 0< . When these conditions are not satisfied, it is known . Correspondingly, we show that in this case , provided a random strong open set condition is satisfied. We also find gauge functions (t) so that the -packing measure is finite.  相似文献   

5.
Let (X t ) be a one dimensional diffusion corresponding to the operator , starting from x>0 and T 0 be the hitting time of 0. Consider the family of positive solutions of the equation with (0, ), where . We show that the distribution of the h-process induced by any such is , for a suitable sequence of stopping times (S M : M0) related to which converges to with M. We also give analytical conditions for , where is the smallest point of increase of the spectral measure associated to .  相似文献   

6.
Divergence of a Random Walk Through Deterministic and Random Subsequences   总被引:1,自引:0,他引: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   

7.
Let be the field , , or of real dimension . For each dimensiond2, we study isotropic random walks(Y 1)10 on the projective space with natural metricD where the random walk starts at some with jumps at each step of a size depending ond. Then the random variablesX 1 d :=cosD(Y 1 d ,x 0 d ) form a Markov chain on [–1, 1] whose transition probabilities are related to Jacobi convolutions on [–1, 1]. We prove that, ford, the random variables (vd/2)(X l(d) d +1) tend in distribution to a noncentral 2-distribution where the noncentrality parameter depends on relations between the numbers of steps and the jump sizes. We also derive another limit theorem for as well as thed-spheresS d ford.  相似文献   

8.
IfX is a locally compact abelian group, a probability measure onX and its Fourier transform, the mapping | | is obviously not injective. The aim of this article is to find conditions under which the identification of given | | is possible up to a shift and a central symmetry.Research partially supported by the Swiss National Science Foundation.  相似文献   

9.
Let u(x) xR q be a symmetric nonnegative definite function which is bounded outside of all neighborhoods of zero but which may have u(0)=. Let p x, (·) be the density of an R q valued canonical normal random variable with mean x and variance and let {G x, ; (x, )R q ×[0,1 ]} be the mean zero Gaussian process with covariance
A finite positive measure on R q is said to be in with respect to u, if
When , a multiple Wick product chaos is defined to be the limit in L 2, as 0, of
where
,
denotes the Wick product of the m j normal random variables .Consider also the associated decoupled chaos processes , defined as the limit in L 2, as 0, of
where are independent copies of G x,.Define
Note that a neighborhood of the diagonals of in is excluded, except those points on the diagonal which originate in the same Wick product in (i). Set
One of the main results of this paper is: Theorem A. If is continuous on (R q ) r for all then is continuous on .When u satisfies some regularity conditions simple sufficient conditions are obtained for the continuity of on (R q ) r . Also several variants of (i) are considered and related to different types of decoupled processes. These results have applications in the study of intersections of Lévy process and continuous additive functionals of several Lévy processes.  相似文献   

10.
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   

11.
Let be a probability space and a partition of . A necessary and sufficient condition is given for the existence of a -additive and measurable disintegration of P on . It is also shown that P admits a -additive (but not measurable) disintegration on whenever is a standard space and the set (1, 2):1 and 2 are in the same element of } is coanalytic in ×. Finally, sufficient statistics (in the classical Fisherian sense) are investigated by using -additive disintegrations as conditional probabilities.  相似文献   

12.
For an array {V nk ,k1,n1} of rowwise independent random elements in a real separable Banach space with almost surely convergent row sums , we provide criteria for S n A n to be stochastically bounded or for the weak law of large numbers to hold where {A n ,n1} is a (nonrandom) sequence in .  相似文献   

13.
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   

14.
We consider a Poisson point process on with intensity , and at each Poisson point we place a two sided mirror of random length and orientation. The length and orientation of a mirror is taken from a fixed distribution, and is independent of the lengths and orientations of the other mirrors. We ask if light shone from the origin will remain in a bounded region. We find that there exists a with 0 < < for which, if < , light leaving the origin in all but a countable number of directions will travel arbitrariliy far from the origin with positive probability. Also, if > , light from the origin will almost surely remain in a bounded region.  相似文献   

15.
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   

16.
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   

17.
Let X and be transient standard Markov processes in weak duality with respect to a -finite measure m. Let (Y, , ) be a second dual pair with the same state space E as (X, , m). Let Cap X and Cap Y be the 0-order capacities associated with (X, , m) and (Y, , ), and let V and denote the potential kernels for Y and . Assume that singletons are polar with respect to both X and Y, and that semipolar sets are of capacity zero for both dual pairs. We show that if Cap X (B)=Cap Y (B) for every Borel subset of E then there is a strictly increasing continuous additive functional D=(D t) t0 of (X, , m) such that
with the exception of a capacity-zero set of x's. Here U D (resp. Û D) is the potential kernel of the time-changed process (resp. , t0. In particular, if both X and Y are symmetric processes, then the equality of the capacities Cap X and Cap Y implies that X and Y are time changes of one another. This derivation rests on a generalization of a formula of Choquet concerning the differentiation of capacities. In the symmetric case, our main result extends a theorem of Glover et al.(23)  相似文献   

18.
Suppose that the d -valued random vector is strictly operator-stable in the sense that , the characteristic function of , satisfies for everyt<0, for some invertible linear operatorB on d . Suppose also that for the i.i.d. random vectors {X i } in d , . In the present paper, we study the rates of convergence of this operator-stable limit theorem in terms of several probability metrics. A new type of ideal metrics suitable for this rate-of-convergence problem is introduced.This research was partially supported by NSF, Grant DMS-9103452 and NATO, Grant CRG900798.  相似文献   

19.
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   

20.
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   

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