Olav Kallenberg

Some time change representations of stable integrals, via predictable transformations of local martingales

From the predictable reduction of a marked point process to Poisson, we derive a similar reduction theorem for purely discontinuous martingales to processes with independent increments. Both results are then used to examine the existence of stochastic integrals with respect to stable Levy processes, and to prove a variety of time change representations for such integrals. The Knight phenomenon, where possibly dependent but orthogonal processes become independent after individual time changes, emerges as a general principle.

Symmetries on random arrays and set-indexed processes

AbstractA processX on the setÑ of all finite subsetsJ ofN is said to be spreadable, if

On the representation theorem for exchangeable arrays

\left( {X_{pJ} } \right)\mathop = \limits^d \left( {X_J } \right)

Characterizations and embedding properties in exchangeability

for all subsequencesp=(p1,p2,...) ofN, wherepJ={pj;j∈J}. Spreadable processes are characterized in this paper by a representation formula, similar to those obtained by Aldous and Hoover for exchangeable arrays of r.v.s. Our representation is equivalent to the statement that a process onÑ is spreadable, iff it can be extended to an exchangeable process indexed by all finite sequences of distinct elements fromN. The latter result may be regarded as a multivariate extension of a theorem by Ryll-Nardzewski, stating that, for infinite sequences of r.v.s, the notions of exchangeability and spreadability are equivalent.

Exchangeable Random Measures in the Plane

SummaryGiven any local maringaleM inRd orl2, there exists a local martingaleN inR2, such that |M|=|N|, [M]=[N], and «M»=«N». It follows in particular that any inequality for martingales inR2 which involves only the processes |M|, [M] and «M» remains true in arbitrary dimension. WhenM is continuous, the processes |M|2 and |M| satisfy certain SDEs which are independent of dimension and yield information about the growth rate ofM. This leads in particular to tail estimates of the same order as in one dimension. The paper concludes with some new maximal inequalities in continuous time.

Multivariate Sampling and the Estimation Problem for Exchangeable Arrays

Aldous and Hoover have proved independently that an array X = (Xij, i, j [set membership, variant] ) of random variables is exchangeable under separate or joint permutations of rows and columns, iff a.s. Xij[reverse not equivalent]f([alpha], [xi]i, [eta]j, [xi]ij) or Xij[reverse not equivalent]f([alpha], [xi]i, [xi]j, [xi]ij), respectively, for some measurable function f: 4--> and some i.i.d. random variables [alpha], [xi]i, [eta]j, [xi]ij, i, j[set membership, variant], or [alpha], [xi]i, [xi]ij=[xi]ji, 1

On the structure of stationary flat processes

SummaryNecessary and sufficient conditions are given for the existence of a multiple stochastic integral of the form ∫...∫fdX1...dXd, where X1, ..., Xd are components of a positive or symmetric pure jump type Lévy process in ℝd. Conditions are also given for a sequence of integrals of this type to converge in probability to zero or infinity, or to be tight. All arguments proceed via reduction to the special case of Poisson integrals.

Some New Representations in Bivariate Exchangeability.

SummaryOur key result is the characterization of exchangeable sequences as being strongly stationary, i.e. invariant in distribution under stopping time shifts. From this we prove homogeneity characterizations of pure and mixed Markov chains. These results carry over to continuous time processes and random sets, and on a whole, our theory provides a unified approach to exchangeability. The key result above is closely related to Dacunha-Castelles embedding characterization of exchangeability, which is partially extended here to processes on [0, 1]. In the other direction, we prove that a previsible sample from a finite or infinite exchangeable sequence X may be embedded into a copy of X. We finally establish some uniqueness results for exponentially and uniformly killed exchangeable random sets.