Stochastic Wiener Filter in the White Noise Space

In this paper we introduce a new approach to the study of filtering theory by allowing the system's parameters to have a random character. We use Hida's white noise space theory to give an alternative characterization and a proper generalization to the Wiener filter over a suitable space of stochastic distributions introduced by Kondratiev. The main idea throughout this paper is to use the nuclearity of this spaces in order to view the random variables as bounded multiplication operators (with respect to the Wick product) between Hilbert spaces of stochastic distributions. This allows us to use operator theory tools and properties of Wiener algebras over Banach spaces to proceed and characterize the Wiener filter equations under the underlying randomness assumptions

On the relationship between variable Wiener index and variable Szeged index

We resolve two conjectures of Hriňáková et al. (2019) concerning the relationship between the variable Wiener index and variable Szeged index for a connected, non-complete graph, one of which would imply the other. The strong conjecture is that for any such graph there is a critical exponent in (0,1], below which the variable Wiener index is larger and above which the variable Szeged index is larger. The weak conjecture is that the variable Szeged index is always larger for any exponent exceeding 1. They proved the weak conjecture for bipartite graphs, and the strong conjecture for trees. In this note we disprove the strong conjecture, although we show that it is true for almost all graphs, and for bipartite and block graphs. We also show that the weak conjecture holds for all graphs by proving a majorization relationship

Iterated limits for aggregation of randomized INAR(1) processes with Poisson innovations

We discuss joint temporal and contemporaneous aggregation of $N$ independent copies of strictly stationary INteger-valued AutoRegressive processes of order 1 (INAR(1)) with random coefficient $\alpha\in(0,1)$ and with idiosyncratic Poisson innovations. Assuming that $\alpha$ has a density function of the form $\psi(x)(1 - x)^\beta$, $x\in(0,1)$, with $\lim_{x\uparrow 1}\psi(x) = \psi_1 \in(0,\infty)$, different limits of appropriately centered and scaled aggregated partial sums are shown to exist for $\beta\in(-1,0)$, $\beta = 0$, $\beta\in(0,1)$ or $\beta\in(1,\infty)$, when taking first the limit as $N\to\infty$ and then the time scale $n\to\infty$, or vice versa. In fact, we give a partial solution to an open problem of Pilipauskaite and Surgailis (2014) by replacing the random-coefficient AR(1) process with a certain randomized INAR(1) process.Comment: 49 pages. Results on centralization by the empirical mean are adde

Stationarity and ergodicity for an affine two factor model

We study the existence of a unique stationary distribution and ergodicity for a 2-dimensional affine process. The first coordinate is supposed to be a so-called alpha-root process with \alpha\in(1,2]. The existence of a unique stationary distribution for the affine process is proved in case of \alpha\in(1,2]; further, in case of \alpha=2, the ergodicity is also shown.Comment: 28 pages; the title has been changed; a mistake in the proof of Theorem 4.1 has been correcte

Asymptotic behavior of unstable INAR(p) processes

In this paper the asymptotic behavior of an unstable integer-valued autoregressive model of order p (INAR(p)) is described. Under a natural assumption it is proved that the sequence of appropriately scaled random step functions formed from an unstable INAR(p) process converges weakly towards a squared Bessel process. We note that this limit behavior is quite different from that of familiar unstable autoregressive processes of order p. An application for Boston armed robberies data set is presented.Comment: 35 pages; corrected and extended version: a new section on an application for Boston armed robberies data set is adde