Strong laws of large numbers for certain sequences and arrays of dependent random variables
Date of Award
Doctor of Philosophy (PhD)
Terry R. McConnell
identically distributed random variables, dependence structure, maximal function
In this work, we study the almost sure convergence of the averages of certain classes of sequences and arrays of dependent random variables, and the behaviour of the associated maximal functions. We first consider sequences of identically distributed random variables that are strongly mixing. Our main interest here is to obtain some insight into how the dependence structure of such sequences affects the relationship between the moments of the random variables and the behaviour of the maximal function. We find that for strictly stationary sequences, there is a natural division between those sequences for which the existence of the first moment is necessary for the almost sure finiteness of the ergodic maximal function, and those for which this is not the case. When the condition of stationarity is relaxed, we can still obtain some partial results in this direction.
In the two parameter setting, we study arrays whose joint distributions are invariant under interchange of rows and columns. We have formulated an analog of the Marcinkiewicz strong law of large numbers for such arrays. The results extend earlier work by Smythe (1974), Gut (1978) and McConnell (1987). We have also obtained a dominated ergodic theorem for a particular subclass of this type of array. This result facilitates the proof of a two parameter strong law for degenerate U-statistics of degree two which complements some results due to McConnell (1987).
The key role that Kronecker's lemma plays in many proofs of the classical one parameter results has led us to formulate and prove an analogous two parameter result appropriate for our purposes. Although the general approach we use is classical, relatively modern methods concerning multiparameter martingales and Banach space valued random variables are heavily relied upon. We also make substantial use of symmetrization techniques. Fairly recent results concerning Rademacher quadratic forms play an important role in the successful application of these techniques.
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Rieders, Eric Forrest, "Strong laws of large numbers for certain sequences and arrays of dependent random variables" (1988). Mathematics - Dissertations. 53.