Asymptotic Behavior of Semistable Lévy Exponents and Applications to Fractal Path Properties

Abstract

This paper proves sharp bounds on the tails of the Lévy exponent of an operator semistable law on R d . These bounds are then applied to explicitly compute the Hausdorff and packing dimensions of the range, graph, and other random sets describing the sample paths of the corresponding operator semi-selfsimilar Lévy processes. The proofs are elementary, using only the properties of the Lévy exponent, and certain index formulae.E n , and ε u denotes the Dirac measure at the point u ∈ R d . Operator semistable distributions were introduced by Jajte [12];

Multiple Points of Operator Semistable Lévy Processes

“…Theorem 1 is more general than [28,Theorem 1], where B is assumed to be a diagonal matrix with entries on the diagonal α j ∈ (1, 2) (1 ≤ j ≤ d). Note also that the dimension formula for double points in R 2 and R 3 is given in [14,Corollary 3.8]. Since M 3 = ∅ a.s. for d = 3 and M 2 = ∅ a.s. for d ≥ 4 (see the beginning of Section 4 for the proof), Theorem 1 completes the solution of the Hausdorff dimension problem for M k in the setting of symmetric operator semistable Lévy processes.…”

“…almost surely, where ξ = (ξ 1 , ..., ξ k ) for ξ j ∈ R d and · denotes the usual Euclidean norm in R d . Here we use the convention inf ∅ = d. Furthermore, in the case when X is a symmetric operator semistable Lévy process, [14,Corollary 2.2] gives the following estimate for its characteristic exponent Ψ: for every ε > 0, there exists a constant τ > 1 such that for all ξ ∈ R d with ξ ≥ τ , we have…”

“…Assume (14) holds for some k ≥ 1 and set k ′ := k + 1. Consider first the integration over A 1,1 k ′ (q, r).…”

On the carrying dimension of occupation measures for self-affine random fields

“…, m p , then recently Wedrich [47] (cf. also [27]) has shown that for any operator semistable Lévy process X almost surely…”

On exact Hausdorff measure functions of operator semistable Lévy processes

“…Lastly, we will also derive exact Hausdorff measure functions for type B which turned out to be more challenging than asserted in Remark 1 of [8]. The Hausdorff dimension for the range and the graph of operator semistable Lévy processes have recently been determined in [11] and [23], respectively; see also [10] for an alternative derivation based on an index formula presented in [14]. The special case of the limit process in subsequent cointossing games of the famous St. Petersburg paradox has been studied in [12] in detail.The methods applied in this paper are similar to the ones used in [22] and [8] with complementary work necessary to handle the weaker semistable scaling (1.1) or the non-diagonality of the operator E. The paper is structured as follows.…”