Regular Banach Spaces and Large Deviations of Random Sums

1 Overview A typical result on large deviations of sums with random terms states that if ξ t are independent scalar random variables with zero means and such that " ξ t has as light tails as a Gaussian N (0, 4σ 2 t) random variable " , specifically, E exp{ξ 2 t /σ 2 t } ≤ O(1), (1) and S N = N t=1 ξ t , then Prob |S N | > t σ 2 1 + ... + σ 2 N ≤ O(1) exp{−O(1)t 2 } (2) (from now on, all O(1)'s are appropriate positive absolute constants). Our goal is to get similar results for the case when ξ t are independent random vectors with zero means in a finite-dimensional vector space E equipped with norm · , S N = N t=1 ξ t and the " light tail " condition (1) is stated as E exp{{ξ t 2 /σ 2 t } ≤ exp{1}. (3) Note that a straightforward guess E{ξ t } = 0∀t & (3) & {ξ t } are independent ⇒ Prob S N > t σ 2 1 + ... + σ 2 N ≤ O(1) exp{−O(1)t 2 } (4) is not true, as it is shown by the following example: • E = R n , x = x 1 ≡ j |x j |, • (ξ t) j = t , j = t(mod n) 0, otherwise , where 1 , 2 , ... are independent random variables taking values ±1 with probability 1/2, • σ t = 1, i ≥ 1.