135 research outputs found
Phantom distribution functions for some stationary sequences
The notion of a phantom distribution function (phdf) was introduced by
O'Brien (1987). We show that the existence of a phdf is a quite common
phenomenon for stationary weakly dependent sequences. It is proved that any
-mixing stationary sequence with continuous marginals admits a
continuous phdf. Sufficient conditions are given for stationary sequences
exhibiting weak dependence, what allows the use of attractive models beyond
mixing. The case of discontinuous marginals is also discussed for
-mixing.
Special attention is paid to examples of processes which admit a continuous
phantom distribution function while their extremal index is zero. We show that
Asmussen (1998) and Roberts et al. (2006) provide natural examples of such
processes. We also construct a non-ergodic stationary process of this type
Functional Convergence of Linear Sequences in a non-Skorokhod Topology
In this article, we prove a new functional limit theorem for the partial sum
sequence corresponding to a linear sequence of
the form X_i=\sum_{j \in \bZ}c_j \xi_{i-j} with i.i.d. innovations
(\xi_i)_{i \in \bZ} and real-valued coefficients (c_j)_{j \in \bZ}. This
weak convergence result is obtained in space \bD[0,1] endowed with the
-topology introduced in Jakubowski (1992), and the limit process is a linear
fractional stable motion (LFSM). One of our result provides an extension of the
results of Avram and Taqqu (1992) to the case when the coefficients (c_j)_{j
\in \bZ} may not have the same sign. The proof of our result relies on the
recent criteria for convergence in Skorokhod's -topology (due to Louhichi
and Rio (2011)), and a result which connects the weak -convergence of the
sum of two processes with the weak -convergence of the two individual
processes. Finally, we illustrate our results using some examples and computer
simulations.Comment: 30 pages, 6 figure
Functional Convergence of Linear Processes with Heavy-Tailed Innovations
We study convergence in law of partial sums of linear processes with
heavy-tailed innovations. In the case of summable coefficients necessary and
sufficient conditions for the finite dimensional convergence to an
-stable L\'evy Motion are given. The conditions lead to new, tractable
sufficient conditions in the case . In the functional setting we
complement the existing results on -convergence, obtained for linear
processes with nonnegative coefficients by Avram and Taqqu (1992) and improved
by Louhichi and Rio (2011), by proving that in the general setting partial sums
of linear processes are convergent on the Skorokhod space equipped with the
topology, introduced by Jakubowski (1997).Comment: 39 pages; revised version of arxiv 1209.114
Existence of weak solutions to stochastic evolution inclusions
We consider the Cauchy problem for a semilinear stochastic differential
inclusion in a Hilbert space. The linear operator generates a strongly
continuous semigroup and the nonlinear term is multivalued and satisfies a
condition which is more heneral than the Lipschitz condition. We prove the
existence of a mild solution to this problem. This solution is not "strong" in
the probabilistic sense, that is, it is not defined on the underlying
probability space, but on a larger one, which provides a "very good extension"
in the sense of Jacod and Memin. Actually, we construct this solution as a
Young measure, limit of approximated solutions provided by the Euler scheme.
The compactness in the space of Young measures of this sequence of approximated
solutions is obtained by proving that some measure of noncompactness equals
zero
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