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Abstract:We examine the relationship between the state of the equatorial Indian Ocean, ENSO, and the Indian summer monsoon rainfall using data from 1881 to 1998. The zonal wind anomalies and SST anomaly gradient over the equatorial Indian Ocean are used as indices that represent the condition of the Indian Ocean. Although the index defined by the zonal wind anomalies correlates poorly with Indian summer monsoon rainfall, the linear reconstruction of Indian summer monsoon rainfall on the basis of a multiple regression from the NINO3 and this wind index better specifies the Indian summer monsoon rainfall than the regression with only NINO3. Using contingency tables, we find that the negative association between the categories of Indian summer monsoon rainfall and the wind index is significant during warm years (El Niño) but not during cold years (La Niña). Composite maps of land precipitation also indicate that this relationship is significant during El Niño events. We conclude that there is a significant negative association between Indian summer monsoon rainfall and the zonal wind anomalies over the equatorial Indian Ocean during El Niño events. A similar investigation of the relationship between the SST index and Indian summer monsoon rainfall does not reveal a significant association.

In this paper we present a decoupling inequality that shows that multivariate Ustatistics can be studied as sums of (conditionally) independent random variables. This result has important implications in several areas of probability and statistics including the study random graphs and multiple stochastic integration. More precisely, we get the following result: Theorem 1. Let {X j } be a sequence of independent random variables in a measurable 1,2

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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 − λ 2 B 2 t /2}. We develop inequalities for the moments of At/Bt or sup t≥0 At/{Bt(log log Bt) 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 Bt = M t, and sums of conditionally symmetric variables di with At = t i=1 di and Bt = 1. Introduction. A prototypical example of self-normalized random variables is Student's t-statistic which replaces the population standard devi-

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In this paper, we obtain general representations for the joint distributions and copulas of arbitrary dependent random variables absolutely continuous with respect to the product of given one-dimensional marginal distributions. The characterizations obtained in the paper represent joint distributions 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 corollary of the results, we obtain new representations for multivariate divergence measures as well as complete characterizations of important classes of dependent 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 independent 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 Pearson's correlation, the relative entropy or other multivariate divergence measures. A closely related result involves conditions for convergence in distribution 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 X ′ t s, 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 dependence of the random variables. Furthermore, we obtain new sharp complete decoupling moment and probability inequalities for dependent random variables in terms of their dependence characteristics.

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