Orthogonal exponentials of self-affine measures on ℝn

Let the self-affine measure [Formula: see text] be generated by an expanding matrix [Formula: see text] and a finite integer digit set [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text]. In this paper, we show that if [Formula: see text] for an integer [Formula: see text], then [Formula: see text] admits an infinite orthogonal set of exponential functions if and only if there exists [Formula: see text] such that [Formula: see text] for some [Formula: see text] with [Formula: see text] and [Formula: see text].

COINTEGRATION IN FUNCTIONAL AUTOREGRESSIVE PROCESSES

This article defines the class of ${\cal H}$-valued autoregressive (AR) processes with a unit root of finite type, where ${\cal H}$ is a possibly infinite-dimensional separable Hilbert space, and derives a generalization of the Granger–Johansen Representation Theorem valid for any integration order $d = 1,2, \ldots$. An existence theorem shows that the solution of an AR process with a unit root of finite type is necessarily integrated of some finite integer order d, displays a common trends representation with a finite number of common stochastic trends, and it possesses an infinite-dimensional cointegrating space when ${\rm{dim}}{\cal H} = \infty$. A characterization theorem clarifies the connections between the structure of the AR operators and (i) the order of integration, (ii) the structure of the attractor space and the cointegrating space, (iii) the expression of the cointegrating relations, and (iv) the triangular representation of the process. Except for the fact that the dimension of the cointegrating space is infinite when ${\rm{dim}}{\cal H} = \infty$, the representation of AR processes with a unit root of finite type coincides with the one of finite-dimensional VARs, which can be obtained setting ${\cal H} = ^p $ in the present results.

ON THE WEAK-HASH METRIC FOR BOUNDEDLY FINITE INTEGER-VALUED MEASURES

It is known that the space of boundedly finite integer-valued measures on a complete separable metric space becomes a complete separable metric space when endowed with the weak-hash metric. It is also known that convergence under this topology can be characterised in a way that is similar to the weak convergence of totally finite measures. However, the original proofs of these two fundamental results assume that a certain term is monotonic, which is not the case as we show by a counterexample. We clarify these original proofs by addressing the parts that rely on this assumption and finding alternative arguments.