On the spectral representation of symmetric stable processes

The so-called spectral representation theorem for stable processes linearly imbeds each symmetric stable process of index p into L^p (0 < p ≤ 2). We use the theory of L^p isometries for 0 < p < 2 to study the uniqueness of this representation for the non-Gaussian stable processes. We also determine the form of this representation for stationary processes and for substable processes. Complex stable processes are defined, and a complex version of the spectral representation theorem is proved. As a corollary to the complex theory we exhibit an imbedding of complex L^q into real or complex L^p for 0 < p < q ≤ 2.     

Two-dimensional convexities are join-hull commutative

It is shown that two-dimensional convex structures with certain natural properties are join-hull commutative. The main intermediate step is the computation of the so-called exchange number. We also give two examples of three-dimensional convexities which are not join-hull commutative. The second one has certain additional properties showing that the main theorem is sharp in many other respects. These properties are obtained from a study of convex hyperspaces.     

Types of laws of random vectors and best estimators

Optimality of estimators of a vector parameter in terms of the probability of the estimators being contained in suitable region(s) around the parameter point is defined. Conditions under which optimal estimators in the usual senses are also optimal in the above sense are investigated.     

The sextet construction for cubic graphs

We show how to construct cubic graphs which have automorphism groups acting regularly on thes-arcs,s=4 or 5.     

Conditions for the convergence in distribution of maxima of stationary normal processes

The asymptotic distribution of the maximum M_n=max_1?t?nξ_t in a stationary normal sequence ξ₁,ξ,… depends on the correlation r_t between ξ₀ and ξ_t. It is well known that if r_t log t → 0 as t → ∞ or if Σr²_t<∞, then the limiting distribution is the same as for a sequence of independent normal variables. Here it is shown that this also follows from a weaker condition, which only puts a restriction on the number of t-values for which rt log t islarge. The condition gives some insight into what is essential for this asymptotic behaviour of maxima. Similar results are obtained for a stationary normal process in continuous time.     

Explicit constructions of graphs without short cycles and low density codes

We give an explicit construction of regular graphs of degree 2r withn vertices and girth ≧c logn/logr. We use Cayley graphs of factor groups of free subgroups of the modular group. An application to low density codes is given.     

The diameter of random regular graphs

We give asymptotic upper and lower bounds for the diameter of almost everyr-regular graph onn vertices (n → ∞).