## Limit Theorems, Regularity and Moments for Affine-normalized Sums of Independent, Identically Distributed Random Vectors |

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2LLnK a b n n Acosta affine-normalized assume Banach space Borel-Cantelli lemma Central Limit Theorem choose Chung cluster set constant denotes Distributed Random Vectors domain of attraction eigenvalue exists exponent finite Gaussian Griffin l983 Hahn and Klass implies indicator function Invariance Principle Iterated Logarithm Jain Kuelbs and Zinn L(T S large deviation theory lim inf max lim lr lim sup limit laws linear operators lLLnJ LLnK log(b Markov's inequality Math max lr n n n n n/LLn nK nK nk nk-l norm notation operator-stable laws orthonormal basis Ottaviani's inequality P(lA P(lT Partial Sums Pisier l975 PONB Preprint Prob proof proved relatively compact result Sharpe l969 shows stable stochastically compact subsequential limits summable Sums of Independent Suppose L(T S symmetric tight uniformly University of Wisconsin-Madison verw Wahrsch weakly convergent X G GDA(y