Marcinkiewicz laws of large numbers for φ-mixing strictly stationary sequences with r-th moment barely divergent, 0 < r < 2, are established. For this dependent analogs of the Lévy-Ottaviani-Etemadi and Hoffmann-Jørgensen inequalities are revisited.
In a one-parameter model for evolution of random trees strong law of large numbers and central limit theorem are proved for the number of vertices with low degree. The proof is based on elementary martingale theory.
An exponential inequality for the tail of the conditional expectation of sums of centered independent random variables is
obtained. This inequality is applied to prove analogues of the Law of the Iterated Logarithm and the Strong Law of Large Numbers
for conditional expectations. As corollaries we obtain certain strong theorems for the generalized allocation scheme and for
the nonuniformly distributed allocation scheme.
Franck, W. E.
and Hanson, D. L.
, Some results giving rates of convergence in the lawoflargenumbers for weighted sums of independent variables,.
Trans. Amer. Math. Soc.
for all & > 0, generalizing Baum and Katz's~(1965) generalization of the Hsu–Robbins–Erds (1947, 1949) law of large numbers,
also allowing us to characterize the convergence of the above series in the case where τn = n-1and
for n ≤ 2, thereby answering a question of Spătaru. Moreover, some results for non-identically distributed independent random variables
are obtained by a recent comparison inequality. Our basic method is to use a central limit theorem estimate of Nagaev (1965)
combined with the Hoffman-Jrgensen inequality~(1974).