Abstract
Let \(\{ X_{\bf n}, {\bf n}\in \mathbb{N}^d \}\) be a random field of negatively dependent random variables. The complete convergence results for negatively dependent random fields are refined. To obtain the main theorem several lemmas for convergence of families indexed by \(\mathbb{N}^d\) have been proved. Auxiliary lemmas have wider application to study the random walks on the lattice.
Highlights
The concept of complete convergence was introduced in [6] by Hsu and Robbins. They proved that the sequence of arithmetic means of independent, identically distributed (i.i.d.) random variables converges completely to the expected value of the variables, provided the random variables are square-integrable
We proved several results for the convergence of families {an, n ∈ Nd} and their partial sums
On Nd we introduce a relation and functions restricted to subsets of D, i.e. for a given J ⊆ D and m, n ∈ Nd we define mRJ n if and only if miRni, for i ∈ J
Summary
The concept of complete convergence was introduced in [6] by Hsu and Robbins. They proved that the sequence of arithmetic means of independent, identically distributed (i.i.d.) random variables converges completely to the expected value of the variables, provided the random variables are square-integrable. We proved several results for the convergence of families {an, n ∈ Nd} and their partial sums These auxiliary lemmas can be very useful to study the complete convergence in more general context. Lemma 2.2 means that we can straightforwardly transfer metrizable convergence results for random variables to convergence of random fields as max n → ∞. This tool is very helpful to study the convergence of partial sums, as we will see in the proof of Lemma 2.4. Let {an, n ∈ Nd} be a family of non-negative real numbers and Sn = ak, the following conditions are equivalent: k≤n (i) lim Sn = S, min n→∞.
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