We first establish a local limit theorem for a linear random field, which is constructed using a set of real coefficients and a set (referred to as the innovations) of i.id. random variables in L2. Each set is indexed by Zd, where d is the dimension. For each j ∈ Zd, the random variable Xj of the linear random field is constructed as a combination of the innovations using the real coefficients mentioned earlier. We study a sequence Sn of partial sums of the Xj. The manner in which we construct these sums varies and is wholly determined by a sequence of index sets of summation, which we generally refer to as the regions of summation. Under reasonable conditions, we establish the local limit theorem for these sums. That is, we show that the sequence of measures √2π Bn P(Sn ∈ [a, b]) of [a, b] converges to b − a, where Bn2 = Var Sn). We also estimate the Shannon entropy S(f) = −E[log f(x)] of a one-sided linear process with probability density function f(x) employing the integral estimator Sn (f), which utilizes the standard kernel density estimator fn(x) of f(x). We show that Sn (f) converges to S(f) almost surely and in L2, under reasonable conditions.
|Commitee:||Dang, Xin, Díaz, Erwin Miña, Bu, Qingying|
|School:||The University of Mississippi|
|School Location:||United States -- Mississippi|
|Source:||DAI-B 82/4(E), Dissertation Abstracts International|
|Keywords:||Entropy, Field, Limit, Local, Random, Shannon|
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