Victor H. de la Peña

Self-normalized processes: exponential inequalities, moment bounds and iterated logarithm laws

Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, self-normalization often eliminates or weakens moment assumptions. In this paper we present several exponential and moment inequalities, particularly those related to laws of the iterated logarithm, for self-normalized random variables including martingales. Tail probability bounds are also derived. For random variables Bt>0 and At, let Yt(λ)=exp{λAt−λ2Bt2/2}. We develop inequalities for the moments of At/Bt or supt≥0At/{Bt(log logBt)1/2} and variants thereof, when EYt(λ)≤1 or when Yt(λ) is a supermartingale, for all λ belonging to some interval. Our results are valid for a wide class of random processes including continuous martingales with At=Mt and

Characterizations of joint distributions, copulas, information, dependence and decoupling, with applications to time series

B_{t}=\sqrt {\langle M\rangle _{t}}

Climate Change over the Equatorial Indo-Pacific in Global Warming*

, and sums of conditionally symmetric variables di with At=∑i=1tdi and

EXPONENTIAL BURKHOLDER DAVIS GUNDY INEQUALITIES

B_{t}=\sqrt{\sum_{i=1}^{t}d_{i}^{2}}

On sharp Burkholder-Rosenthal-type inequalities for infinite-degree U-statistics

. A sharp maximal inequality for conditionally symmetric random variables and for continuous local martingales with values in Rm, m≥1, is also established. Another development in this paper is a bounded law of the iterated logarithm for general adapted sequences that are centered at certain truncated conditional expectations and self-normalized by the square root of the sum of squares. The key ingredient in this development is a new exponential supermartingale involving ∑i=1tdi and ∑i=1tdi2. A compact law of the iterated logarithm for self-normalized martingales is also derived in this connection.

Inequalities for tails of adapted processes with an application to Wald's lemma

In this paper, we obtain general representations for the joint dis- tributions and copulas of arbitrary dependent random variables absolutely continuous with respect to the product of given one-dimensional marginal dis- tributions. The characterizations obtained in the paper represent joint distrib- utions of dependent random variables and their copulas as sums of U -statistics in independent random variables. We show that similar results also hold for expectations of arbitrary statistics in dependent random variables. As a corol- lary of the results, we obtain new representations for multivariate divergence measures as well as complete characterizations of important classes of depen- dent random variables that give, in particular, methods for constructing new copulas and modeling different dependence structures. The results obtained in the paper provide a device for reducing the analysis of convergence in distribution of a sum of a double array of dependent random variables to the study of weak convergence for a double array of their inde- pendent copies. Weak convergence in the dependent case is implied by similar asymptotic results under independence together with convergence to zero of one of a series of dependence measures including the multivariate extension of Pearsons correlation, the relative entropy or other multivariate divergence measures. A closely related result involves conditions for convergence in dis- tribution of m-dimensional statistics h(Xt ,X t+1 ,...,X t+m−1) of time series {Xt} in terms of weak convergence of h(ξt ,ξ t+1 ,...,ξ t+m−1), where {ξt} is a sequence of independent copies of Xs, and convergence to zero of measures of intertemporal dependence in {Xt}. The tools used include new sharp estimates for the distance between the distribution function of an arbitrary statistic in dependent random variables and the distribution function of the statistic in independent copies of the random variables in terms of the measures of depen- dence of the random variables. Furthermore, we obtain new sharp complete decoupling moment and probability inequalities for dependent random vari- ables in terms of their dependence characteristics.

A note on second moment of a randomly stopped sum of independent variables

Abstract The response of the equatorial Indian Ocean climate to global warming is investigated using model outputs submitted to the Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report. In all of the analyzed climate models, the SSTs in the western equatorial Indian Ocean warm more than the SSTs in the eastern equatorial Indian Ocean under global warming; the mean SST gradient across the equatorial Indian Ocean is anomalously positive to the west in a warmer twenty-first-century climate compared to the twentieth-century climate, and it is dynamically consistent with the anomalous westward zonal wind stress and anomalous positive zonal sea level pressure (SLP) gradient to the east at the equator. This change in the zonal SST gradient in the equatorial Indian Ocean is detected even in the lowest-emission scenario, and the size of the change is not necessarily larger in the higher-emission scenario. With respect to the change over the equatorial Pacific in climate projections, the subsur...

FROM BOUNDARY CROSSING OF NON-RANDOM FUNCTIONS TO BOUNDARY CROSSING OF STOCHASTIC PROCESSES

Klass has established decoupling inequalities for discrete time processes with independent increments. We extend his result to continuous time processes with independent increments. This extension enables us to obtain BDG inequalities for exponential functions instead of moderate functions.

Moment Bounds for Self-Normalized Martingales

In this paper, we present a method that allows one to obtain a number of sharp inequalities for expectations of functions of infinite-degree U -statistics. Using the approach, we prove, in particular, the following result: Let D be the class of functions f : R+ → R+ such that the function f( x+ z) − f( x)is concave in x ∈ R+ for all z ∈ R+. Then the following estimate holds: Ef m

Decoupling and domination inequalities with application to Wald's identity for martingales

AbstractIn this paper we introduce a new tail probability version of Walds lemma for expectations of randomly stopped sums of independent random variables. We also make a connection between the works of Klass(18, 19) and Gundy(11) on Walds lemma. In making the connection, we develop new Lenglart and Good Lambda inequalities comparing the tails of various types of adapted processes. As a consequence of our Good Lambda inequalities we include the following result. Let {di}, {ei} be two sequences of variables adapted to the same increasing sequence of σ-fields ℱn↗ℱ, (e.g., ℱn=σ({di}i=1n, {Ei}i=1n), and letN⩽∞ be a stopping time adapted to {ℱn}. Then for allp>0, there exists a constant 0<Cp<∞ depending onp only, such that