Representation and Characterization of Nonstationary Processes by Dilation Operators and Induced Shape Space Manifolds

We proposed in this work the introduction of a new vision of stochastic processes through geometry induced by dilation. The dilation matrices of a given process are obtained by a composition of rotation matrices built in with respect to partial correlation coefficients. Particularly interesting is the fact that the obtention of dilation matrices is regardless of the stationarity of the underlying process. When the process is stationary, only one dilation matrix is obtained and it corresponds therefore to Naimark dilation. When the process is nonstationary, a set of dilation matrices is obtained. They correspond to Kolmogorov decomposition. In this work, the nonstationary class of periodically correlated processes was of interest. The underlying periodicity of correlation coefficients is then transmitted to the set of dilation matrices. Because this set lives on the Lie group of rotation matrices, we can see them as points of a closed curve on the Lie group. Geometrical aspects can then be investigated through the shape of the obtained curves, and to give a complete insight into the space of curves, a metric and the derived geodesic equations are provided. The general results are adapted to the more specific case where the base manifold is the Lie group of rotation matrices, and because the metric in the space of curve naturally extends to the space of shapes; this enables a comparison between curves’ shapes and allows then the classification of random processes’ measures.

Representation and Characterization of Nonstationary Processes by Dilation Operators and Induced Shape Space Manifolds

We proposed in this work the introduction of a new vision of stochastic processes through geometry induced by dilation. The dilation matrices of a given process are obtained by a composition of rotation matrices built in with respect to partial correlation coefficients. Particularly interesting is the fact that the obtention of dilation matrices is regardless of the stationarity of the underlying process. When the process is stationary, only one dilation matrix is obtained and it corresponds therefore to Naimark dilation. When the process is nonstationary, a set of dilation matrices is obtained. They correspond to Kolmogorov decomposition. In this work, the nonstationary class of periodically correlated processes was of interest. The underlying periodicity of correlation coefficients is then transmitted to the set of dilation matrices. Because this set lives on the Lie group of rotation matrices, we can see them as points of a closed curve on the Lie group. Geometrical aspects can then be investigated through the shape of the obtained curves, and to give a complete insight into the space of curves, a metric and the derived geodesic equations are provided. The general results are adapted to the more specific case where the base manifold is the Lie group of rotation matrices, and because the metric in the space of curve naturally extends to the space of shapes, this enables a comparison between curves’ shapes and allows then the classification of processes’ measure.

Positive-Defininte Functions on Free Semigroups

An extension of the Naimark dilation theorem [N], [SzF2] to positive-definite functions on free semigroups is given. This is used to extend the operatorial trigonometric moment problem [A] to a non-commutative setting and to characterize the classes 𝐶ρ (ρ > 0) of all n-tuples of operators that have a p-isometric dilation (see [SzF2] for the case n = 1). It is also shown that 𝐶ρ ⊂ 𝐶ρ′ and 𝐶ρ ≠ 𝐶ρ′ for 0 < ρ < ρ′ < ∞.The von Neumann inequality [vN], [Po2] is extended to the classes 𝐶p. This is used to prove that any element in 𝐶ρ is simultaneously similar to an element in 𝐶1.