On the nonexistence of a law of the iterated logarithm for weighted sums of identically distributed random variables.
Limit theorems for randomly selected adjacent order statistics from a Pareto distribution.
Laws of large numbers for asymmetrical Cauchy random variables.
Complete convergence for arrays of minimal order statistics.
On complete convergence of the sum of a random number of stable type random elements.
On the weak law of large numbers for normed weighted sums of i.i.d. random variables.
Exact laws for sums of ratios of order statistics from the Pareto distribution
Consider independent and identically distributed random variables {X nk, 1 ≤ k ≤ m, n ≤ 1} from the Pareto distribution. We select two order statistics from each row, X n(i) ≤ X n(j), for 1 ≤ i < j ≤ = m. Then we test to see whether or not Laws of Large Numbers with nonzero limits exist for weighted sums of the random variables R ij = X n(j)/X n(i).
Laws of large numbers for ratios of uniform random variables
Let {Xnn n ≥ 1} and {Yn, n ≥ 1} be two sequences of uniform random variables. We obtain various strong and weak laws of large numbers for the ratio of these two sequences. Even though these are uniform and naturally bounded random variables the ratios are not bounded and have an unusual behaviour creating Exact Strong Laws.
A note on the almost sure limiting behavior of the maximun of a sequence of partial sums.
The goal of this paper is to show that, in most strong laws of large numbers, the n partial sum can be replaced with the largest of the first n sums. Moreover it is shown that the usual assumptions of independence and common distribution are unnecessary and that these results apply also to strong laws for Banach valued random elements.
One sided strong laws for random variables with infinite mean
This paper establishes conditions that secure the almost sure upper and lower bounds for a particular normalized weighted sum of independent nonnegative random variables. These random variables do not possess a finite first moment so these results are not typical. These mild conditions allow us to show that the almost sure upper limit is infinity while the almost sure lower bound is one.
Strong laws of large numbers for weighted sums of random elements in normed linear spaces.
Page 1