A LIL and limit distributions for trimmed sums of random vectors attracted to operator semi-stable laws

Let . . .d be a unit vector and let X,X 1,X 2, . be a sequence of i.i.d. .d-valued random vectors attracted to operator semi-stable laws. For each integer n . 1, let X 1,n . . . X n,n denote the order statistics of X 1,X 2, ., X n according to priority of index, namely |.X 1,n , ..| . . . |.X n,n , ..|, where .·, ·. is an inner product on .d. For all integers r . 0, define by (r) S n = . n.ri=1 X i,n the trimmed sum. In this paper we investigate a law of the iterated logarithm and limit distributions for trimmed sums (r) S n . Our results give information about the maximal growth rate of sample paths for partial sums of X when r extreme terms are excluded. A stochastically compactness of (r) S n is obtained.