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5000 iterations affine transformations artificial joint pdf asymptotically equivalent Cell Frequencies Cell Numbers central limit theorem covariance matrix Cramer-von Mises type denote Empirical and x2 empirical characteristic function empirical characteristic process empirical distribution function Empirical Powers following Lemma Gaussian process hn/n i j=i i=i i=i i=l i=l Independence against Alternatives integer Jn*PijZij joint distribution JR(n kn(mn kn+l mn l j=l l)hn Lemma logn lognormal marginal distribution measurable region Multivariate Normality Test n j=i n*Pij nonrandom Normal Distribution null hypothesis number of cells number of observations op(l p-variate pdf for Nij power power power powers in percentage proof Define random variables random vectors Section selected order statistics squared radii stochastic process Table Test of Independence test statistic underlying distribution univariate weak convergence Weiss x2 Percentiles Xt(r