N. Etemadi

An elementary proof of the strong law of large numbers

SummaryIn the following note we present a proof for the strong law of large numbers which is not only elementary, in the sense that it does not use Kolmogorovs inequality, but it is also more applicable because we only require the random variables to be pairwise independent. An extension to separable Banach space-valuedr-dimensional arrays of random vectors is also discussed. For the weak law of large numbers concerning pairwise independent random variables, which follows from our result, see Theorem 5.2.2 in Chung [1].

On the laws of large numbers for nonnegative random variables

Strong laws of large numbers concerning nonnegative random variables are obtained and then they are utilized to establish stability results, among other things, for sums of pairwise independent random variables and the range of random walks.

Convergence of weighted averages of random variables revisited

We show that for a large class of positive weights including the ones that are eventually monotone decreasing and those that are eventually monotone increasing but vary regularly, if the averages of random variables converge in some sense, then their corresponding weighted averages also converge in the same sense. We will also replace the sufficient conditions in the fundamental result of Jamison, Pruitt, and Orey for i.i.d. random variables that make their work more transparent.

Stability of sums of weighted nonnegative random variables

A stability result for sums of weighted nonnegative random variables is established and then it is utilized to obtain, among other things, a slight generalization of the Borel-Cantelli lemma and to show that the work of Jamison, Orey, and Pruitt (Z. Wahrsch. Verw. Gebiete 4 (1965), 40-44) on almost sure convergence of weighted averages of independent random variables remains valid if the assumption of independence on the random variables is replaced by pairwise independence.

Strong law of large numbers for 2-exchangeable random variables

The investigation of the role of independence in the classical SLLN leads to a natural generalization of the SLLN to the case where the random variables are 2-exchangeable; namely, let {Xi: i [greater-or-equal, slanted] 1} be a sequence of random variables such that all ordered pairs (Xi, Xj), i [not equal to] j, are identically distributed. Then we show, among other things, that where X is in general a non-degenerate random variable. This provids a unified treatment of the SLLN for both exchangeable and pairwise independent random variables. We also show that, under 2-exchangeability, to preserve the Glivenko-Cantelli Theorem - sometimes refered to as the fundamental theorem of statistics - it is necessary that the random variables be pairwise independent.

Nonanticipative transformations of the two-parameter Wiener process and a Girsanov theorem

Nonanticipative linear transformations of the two-parameter Wiener process W are studied. It is shown that they induce measures equivalent to two-parameter Wiener measure and the corresponding Radon-Nikodym derivatives are calculated. A two-parameter extension of Girsanovs theorem is established for a class of nonanticipative, possibly nonlinear transformations of W.

On the maximal inequalities for the average of pairwise i.i.d. random variables

We obtain two sided inequalities for the tail of the maximal function of the averages of a multiple sequence of pairwise i.i.d. random variables taking values in a separable Banach space. We then use the results to establish a necessary and sufficient con¬dition, in terms of the common distribution of the norm of the random variables, for the maximal function to be in L , 1< p << z

Convergence of sequences of pairwise independent random variables

In spite of the fact that the tail σ-algebra of a sequence of pairwise independent random variables may not be trivial, we have discovered that if such a sequence converges in probability or almost everywhere, then the limit has to be a constant. This enables us to provide necessary and sufficient conditions for its convergence, in terms of its marginal distribution functions.

Maximal inequalities for partial sums of independent random vectors with multi dimensional time parameters

We obtain upper and lower bounds on the distribution of the partial sums constructed from a multi-dimensional array of independent random vectors. These bounds include, among others, generalizations of some of the well known classical inequalities such as the converse Kolmogorov and the Skorokhod-Ottaviani maximal inequalities.

On curve and surface stretching in isotropic turbulent flow

Cocke (1969) showed that, on average, infinitesimal material lines and surfaces are stretched in incompressible isotropic turbulence. We have extended those results to obtain upper and lower bounds for the stretching of such infinitesimal elements in terms of the eigenvalues of the Green deformation tensor. These bounds are in turn used to find bounds for the stretching of finite material lines and surfaces.