Bernstein-type large deviations inequalities for partial sums of strong mixing processes

Statistics ◽  
1993 ◽  
Vol 24 (1) ◽  
pp. 59-70 ◽  
Author(s):  
Denis Bosq
Author(s):  
Noureddine Rhomari

This article discusses some results on Bernstein type and maximal inequalities for partial sums of dependent random vectors taking their values in separable Hilbert or Banach spaces of finite or infinite dimension. Two types of measure of dependence are considered: strong mixing coefficients (α-mixing) and absolutely regular mixing coefficients (β-mixing). These inequalities, which are similar to those in the dependent real case, are used to derive the strong law of large numbers (SLLN) and the bounded law of the iterated logarithm (LIL) for absolutely regular Hilbert- or Banach-valued processes under minimal mixing conditions. The article first introduces the relevant notation and definitions before presenting the maximal inequalities in the strong mixing case, followed by the absolutely regular mixing case. It concludes with some applications to the SLLN, the bounded LIL for Hilbertian or Banachian absolutely regular processes, the recursive estimation of probability density, and the covariance operator estimations.


1995 ◽  
Vol 32 (3) ◽  
pp. 679-691 ◽  
Author(s):  
Tim Zajic

We extend large exceedence results for i.i.d. -valued random variables to a class of uniformly recurrent Markov-additive processes and stationary strong-mixing processes. As in the i.i.d. case, the results are proved via large deviations estimates.


1995 ◽  
Vol 32 (03) ◽  
pp. 679-691
Author(s):  
Tim Zajic

We extend large exceedence results for i.i.d. -valued random variables to a class of uniformly recurrent Markov-additive processes and stationary strong-mixing processes. As in the i.i.d. case, the results are proved via large deviations estimates.


1999 ◽  
Vol 15 (5) ◽  
pp. 643-663 ◽  
Author(s):  
Valentina Corradi

We construct properly scaled functions of Rp-valued partial sums of demeaned data and derive bounds via the functional law of the iterated logarithm for strong mixing processes. If we obtain a value below or equal to the bound we decide in favor of I(0); otherwise we decide in favor of I(1). This provides a consistent rule for classifying time series as being I(1) or I(0). The nice feature of the procedure lies in the almost sure nature of the bound, guaranteeing a lim sup–type result. We finally provide conditions for the strong consistency of estimators of the variance in the dependent and heterogeneous case.


2016 ◽  
Vol 05 (02) ◽  
pp. 1650006 ◽  
Author(s):  
Marwa Banna ◽  
Florence Merlevède ◽  
Pierre Youssef

In this paper, we obtain a Bernstein-type inequality for the sum of self-adjoint centered and geometrically absolutely regular random matrices with bounded largest eigenvalue. This inequality can be viewed as an extension to the matrix setting of the Bernstein-type inequality obtained by Merlevède et al. [Bernstein inequality and moderate deviations under strong mixing conditions, in High Dimensional Probability V: The Luminy Volume, Institute of Mathematical Statistics Collection, Vol. 5 (Institute of Mathematical Statistics, Beachwood, OH, 2009), pp. 273–292.] in the context of real-valued bounded random variables that are geometrically absolutely regular. The proofs rely on decoupling the Laplace transform of a sum on a Cantor-like set of random matrices.


2012 ◽  
Vol 12 (2) ◽  
pp. 1051-1066 ◽  
Author(s):  
V. F. Sofieva ◽  
N. Kalakoski ◽  
P. T. Verronen ◽  
S.-M. Päivärinta ◽  
E. Kyrölä ◽  
...  

Abstract. Sudden stratospheric warmings (SSW) are large-scale transient events, which have a profound effect on the Northern Hemisphere stratospheric circulation in winter. During the SSW events the temperature in stratosphere increases by several tens of Kelvins and zonal winds decelerate or reverse in direction. Changes in temperature and dynamics significantly affect the chemical composition of the middle atmosphere. In this paper, the response of the middle-atmosphere trace gases during several sudden stratospheric warmings in 2003–2008 is investigated using measurements from the GOMOS (Global Ozone Monitoring by Occultation of Stars) instrument on board the Envisat satellite. We have analyzed spatial and temporal changes of NO2 and NO3 in the stratosphere, and of ozone in the whole middle atmosphere. To facilitate our analyses, we have used the temperature profiles data from the MLS (Microwave Limb Sounder) instrument on board the Aura satellite, as well as simulations by the FinROSE chemistry-transport model and the Sodankylä Ion and Neutral Chemistry model (SIC). NO3 observations in the polar winter stratosphere during SSWs are reported for the first time. Changes in chemical composition are found not to be restricted to the stratosphere, but to extend to mesosphere and lower thermosphere. They often exhibit a complicated structure, because the distribution of trace gases is affected by changes in both chemistry and dynamics. The tertiary ozone maximum in the mesosphere often disappears with the onset of SSW, probably because of strong mixing processes. The strong horizontal mixing with outside-vortex air is well observed also in NO2 data, especially in cases of enhanced NO2 inside the polar vortex before SSW. Almost in all of the considered events, ozone near the secondary maximum decreases with onset of SSW. In both experimental data and FinROSE modelling, ozone changes are positively correlated with temperature changes in the lower stratosphere in the dynamically controlled region below ~35 km, and they are negatively correlated with temperature in the upper stratosphere (altitudes 35–50 km), where chemical processes play a significant role. Large enhancements of stratospheric NO3, which strongly correlate with temperature enhancements, are observed for all SSWs, as expected by the current understanding of temperature-dependence of NO3 concentrations and simulations with the CTM.


Sign in / Sign up

Export Citation Format

Share Document