The resolution and representation of time series in Banach space

We describe a systematic procedure to calculate the resolvent operator for a linear pencil on Banach space and thereby simplify, unify and extend known methods for resolution and representation of marginally stable time series. We pay particular attention to those time series commonly known as unit root processes. The new method uses infinite-length Jordan chains to find the key spectral separation projections which enable separation and solution of the fundamental equations for the Laurent series coefficients of the resolvent. It is then possible to define the desired Granger-Johansen representation for the time series. The method remains valid when the resolvent has an isolated essential singularity at unity.Comment: 45 pages, no figure

An inner-outer factorization in ℓp with applications to ARMA processes

The following inner-outer type factorization is obtained for the sequence space ℓp: if the complex sequence F = (F0, F1,F2,...) decays geometrically, then for an p sufficiently close to 2 there exists J and G in ℓp such that F = J * G; J is orthogonal in the Birkhoff-James sense to all of its forward shifts SJ, S2J, S3J, ...; J and F generate the same S-invariant subspace of ℓp; and G is a cyclic vector for S on ℓp. These ideas are used to show that and ARMA equation with characteristic roots inside and outside of the unit circle has Symmetric-α-Stable solution,s in which the process and the given white noise are expressed as causal moving averages of a related i.i.d. SαS white noise. An autregressive representation of the process is similarly obtained

Unendlichdimensionale und zeitstetige Moving-Average-Prozesse

This thesis consists of two parts both dealing with topics in time series analysis. In Chapter 2 we study necessary and sufficient conditions for the existence of strictly stationary solutions of ARMA equations in a separable complex Banach space. First, we obtain conditions for ARMA(1,q) equations by excluding zero and the unit circle from the spectrum of the operator of the AR part, where we use a decomposition similar to the Jordan decomposition of matrices. We then extend this to ARMA(p,q) equations by using a state space representation of an ARMA(p,q) process as an ARMA(1,q) process. We also show that many ARMA processes in Banach spaces possess a moving average process representation where the coefficients can be calculated as the coefficients of a Laurent series. Finally, we discuss various examples illustrating what may happen if one drops the assumptions we made. In Chapter 3 we study the asymptotic behaviour of the covariance estimator for a continuous-time moving average process with long memory. We choose the kernel function to be decaying polynomially slowly at infinity such that the continuous-time moving average process exhibits the long-memory property. We then show, depending on the speed of the polynomial decay of the kernel function and on the tail behaviour of the driving Levy process, that the covariance estimator is asymptotically Rosenblatt, stable or normal distributed.Diese Dissertation besteht aus zwei Teilen, die sich beide mit Fragestellungen aus der Zeitreihenanalyse beschäftigen. In Kapitel 2 studieren wir notwendige und hinreichende Bedingungen für die Existenz von strikt stationären Lösungen von ARMA-Gleichungen in einem separablen, komplexen Banachraum. Zuerst erhalten wir Bedingungen für ARMA(1,q)-Gleichungen, indem wir die Null und den Einheitskreis vom Spektrum des Operators des AR-Teils ausschließen, wobei wir eine Zerlegung benutzen, die ähnlich zur Jordanzerlegung von Matrizen ist. Wir erweitern dies dann auf ARMA(p,q)-Gleichungen, indem wir eine Zustandsraumdarstellung eines ARMA(p,q)-Prozesses als einen ARMA(1,q)-Prozess benutzen. Wir zeigen außerdem, dass viele ARMA-Prozesse in Banachräumen eine Moving-Average-Prozess-Darstellung besitzen, wobei die Koeffizienten als Koeffizienten einer Laurentreihe berechnet werden können. Schließlich diskutieren wir mehrere Beispiele, die illustrieren, was passieren kann, wenn man die Annahmen weglässt, die wir getroffen haben. In Kapitel 3 studieren wir das asymptotische Verhalten des Kovarianzschätzers für einen zeitstetigen Moving-Average-Prozess mit Long-Memory. Wir wählen die Kernelfunktion polynomiell langsam bei unendlich fallend, sodass der zeitstetige Moving-Average-Prozess die Long-Memory-Eigenschaft zeigt. Wir zeigen dann, dass der Kovarianzschätzer abhängig von der Geschwindigkeit des polynomiellen Abfalls der Kernelfunktion und vom Tailverhalten des treibenden Levy-Prozesses asymptotisch Rosenblatt-, stabil- oder normalverteilt ist

Weakly dependent chains with infinite memory

International audienceWe prove the existence of a weakly dependent strictly stationary solution of the equation $X_t=F(X_{t-1},X_{t-2},X_{t-3},\ldots;\xi_t)$ called {\em chain with infinite memory}. Here the {\em innovations} $\xi_t$ constitute an independent and identically distributed sequence of random variables. The function $F$ takes values in some Banach space and satisfies a Lipschitz-type condition. We also study the interplay between the existence of moments and the rate of decay of the Lipschitz coefficients of the function $F$. With the help of the weak dependence properties, we derive Strong Laws of Large Number, a Central Limit Theorem and a Strong Invariance Principle