Non-stationary log-periodogram regression

We study asymptotic properties of the log-periodogram semiparametric estimate of the memory parameter d for non-stationary (d>=1/2) time series with Gaussian increments, extending the results of Robinson (1995) for stationary and invertible Gaussian processes. We generalize the definition of the memory parameter d for non-stationary processes in terms of the (successively) differentiated series. We obtain that the log-periodogram estimate is asymptotically normal for dE[1/2, 3/4) and still consistent for dE[1/2, 1). We show that with adequate data tapers, a modified estimate is consistent and asymptotically normal distributed for any d, including both non-stationary and non-invertible processes. The estimates are invariant to the presence of certain deterministic trends, without any need of estimation.Publicad

Generalized Forward-Backward Splitting

This paper introduces the generalized forward-backward splitting algorithm for minimizing convex functions of the form $F + \sum_{i=1}^n G_i$, where $F$ has a Lipschitz-continuous gradient and the $G_i$'s are simple in the sense that their Moreau proximity operators are easy to compute. While the forward-backward algorithm cannot deal with more than $n = 1$ non-smooth function, our method generalizes it to the case of arbitrary $n$. Our method makes an explicit use of the regularity of $F$ in the forward step, and the proximity operators of the $G_i$'s are applied in parallel in the backward step. This allows the generalized forward backward to efficiently address an important class of convex problems. We prove its convergence in infinite dimension, and its robustness to errors on the computation of the proximity operators and of the gradient of $F$. Examples on inverse problems in imaging demonstrate the advantage of the proposed methods in comparison to other splitting algorithms.Comment: 24 pages, 4 figure

Krasnoselskii-Mann method for non-self mappings

AbstractLet H be a Hilbert space and let C be a closed, convex and nonempty subset of H. If $T:C\to H$ T : C → H is a non-self and non-expansive mapping, we can define a map $h:C\to\mathbb{R}$ h : C → R by $h(x):=\inf\{\lambda\geq 0:\lambda x+(1-\lambda)Tx\in C\}$ h ( x ) : = inf { λ ≥ 0 : λ x + ( 1 − λ ) T x ∈ C } . Then, for a fixed $x_{0}\in C$ x 0 ∈ C and for $\alpha_{0}:=\max\{1/2, h(x_{0})\}$ α 0 : = max { 1 / 2 , h ( x 0 ) } , we define the Krasnoselskii-Mann algorithm $x_{n+1}=\alpha _{n}x_{n}+(1-\alpha_{n})Tx_{n}$ x n + 1 = α n x n + ( 1 − α n ) T x n , where $\alpha_{n+1}=\max\{\alpha_{n},h(x_{n+1})\}$ α n + 1 = max { α n , h ( x n + 1 ) } . We will prove both weak and strong convergence results when C is a strictly convex set and T is an inward mapping