We study a particular class of moving average processes which possess a property called localisability. This means that, at any given point, they admit a “tangent process”, in a suitable sense. We give general conditions on the kernel g defining the moving average which ensures that the process is localisable and we characterize the nature of the associated tangent processes. Examples include t...

We show that the moving average process Xf (t) := ∫ t 0 f(t − s) dZ(s) t ∈ [0, T ] has a bounded version almost surely, when the kernel f has finite total 2– variation and Z is a symmetric Lévy process. We also obtain bounds for E| supt∈[0,T ] Xf (t)| and for uniform moduli of continuity of Xf ( · ) and for the largest jump of Xf ( · ) when it is not continuous. Similar results are obtained for...

This thesis introduces time-frequency-autoregressive-moving-average (TFARMA) models for underspread nonstationary stochastic processes (i.e., nonstationary processes with rapidly decaying TF correlations). TFARMAmodels are parsimonious as well as physically intuitive and meaningful because they are formulated in terms of time shifts (delays) and Doppler frequency shifts. They are a subclass of ...

In this paper we study the extremal behavior of a stationary continuoustime moving average process Y (t) = ∫∞ −∞ f(t−s) dL(s) for t ∈ R, where f is a deterministic function and L is a Lévy process whose increments, represented by L(1), are subexponential and in the maximum domain of attraction of the Gumbel distribution. We give necessary and sufficient conditions for Y to be a stationary, infi...

In this paper, we consider a continuous-time autoregressive fractionally integrated moving average (CARFIMA) model, which is defined as the stationary solution of a stochastic differential equation driven by a standard fractional Brownian motion. Like the discrete-time ARFIMA model, the CARFIMA model is useful for studying time series with short memory, long memory and antipersistence. We inves...

Let {Yi;-oc < i < c~} be a doubly infinite sequence of identically distributed and (b-mixing random variables, (ai; ~ < i < oc} an absolutely summable sequence of real numbers. In this paper, we prove the complete convergence of {Ek=xn ~io~=_¢xz ai+kYi/nt/,; n>~ 1} under some suitable conditions. AMS classification: 60G50; 60F15

In this paper we study the extremal behavior of stationary mixed moving average processes Y (t) = ∫ R+×R f(r, t − s) dΛ(r, s) for t ∈ R, where f is a deterministic function and Λ is an infinitely divisible independently scattered random measure, whose underlying driving Lévy process is regularly varying. We give sufficient conditions for the stationarity of Y and compute the tail behavior of ce...

(1-2) Pj(t) = Pj(t + d) , qt = qt+d and {E,} is a normal white noise with zero mean value and a variance a. Such process is natural analogy of the periodic autoregressive process (see e.g. [ l ] , [5], [6], [7]) and therefore the idea originates to use the estimation technique described in [3] for the treatment of the multiple moving average models (the same approach to the multiple autoregress...