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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 L^{2}. Each set is indexed by Z^{d}, where d is the dimension. For each j ∈ Z^{d}, the random variable X_{j} of the linear random field is constructed as a combination of the innovations using the real coefficients mentioned earlier. We study a sequence S_{n} of partial sums of the X_{j}. 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 B_{n} P(S_{n} ∈ [a, b]) of [a, b] converges to b − a, where B_{n}^{2} = Var S_{n}). 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 S_{n} (f), which utilizes the standard kernel density estimator f_{n}(x) of f(x). We show that S_{n} (f) converges to S(f) almost surely and in L^{2}, under reasonable conditions.
Advisor: | Sang, Hailin |
Commitee: | Dang, Xin, Díaz, Erwin Miña, Bu, Qingying |
School: | The University of Mississippi |
Department: | Mathematics |
School Location: | United States -- Mississippi |
Source: | DAI-B 82/4(E), Dissertation Abstracts International |
Source Type: | DISSERTATION |
Subjects: | Statistics |
Keywords: | Entropy, Field, Limit, Local, Random, Shannon |
Publication Number: | 28087868 |
ISBN: | 9798678140555 |