| Summary: | Let {X(,n), n (GREATERTHEQ) 1} be a sequence of i.i.d. random variables withE(X(,1)) = 0, Var(X(,1)) = 1. Let (psi)(s) be the cumulant generating function (c.g.f.) and === (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) === be the large deviation rate of X(,1). Let S(,n) = X(,1) + ... + X(,n). Under some mild conditions on (psi), Richter (Theory Prob. Appl. (1957) 2, 206-219) showed that the probability density function f(,n) of(' )S(,n)/SQRT.(n has the asymptotic expression === (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) === whenever x(,n) = o(SQRT.(n) and SQRT.(n x(,n) > 1. In this dissertation we obtain similar large deviation local limit theorems for arbitrary sequences of random variables, not necessarily sums of i.i.d. random variables, thereby increasing the applicability of Richter's theorem. Let {T(,n), n (GREATERTHEQ) 1} be an arbitrary sequence of non-lattice random variables with characteristic function (c.f.) (phi)(,n). Let (psi)(,n), (gamma)(,n) be the c.g.f. and the large deviation rate of T(,n)/n. The main theorem in Chapter II shows that under some standard conditions on (psi)(,n), which imply that T(,n)/n converges to a constant in probability, the density function K(,n) of T(,n)/n has the asymptotic expression === (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) === where m(,n) is any sequence of real numbers and (tau)(,n) is defined by(psi)(,n)'((tau)(,n)) = m(,n). When T(,n) is the sum of n i.i.d. random variables our result reduces to Richter's theorem. Similar theorems for lattice valued random variables are also presented which are useful in obtaining asymptotic probabilities for Wilcoxon signed-rank test statistic and Kendall's tau. === In Chapter III we use the results of Chapter II to obtain central limit theorem for sums of a triangular array of dependent random variables X(,j)('(n)), j = 1, ..., n with joint distribution given by z(,n)('-1)exp{-H(,n)(x(,1), ..., x(,n))}(PI)dP(x(,j)), where x(,i) (ELEM) R (FOR ALL) i (GREATERTHEQ) 1. The function H(,n)(x(,1), ..., x(,n)) is known as the Hamiltonian. Here P is a probability measure on R. When H(,n)(x(,1), ..., x(,n)) = -log (phi)(,n)(s(,n)/n), where s(,n) = x(,1) + ... + x(,n) and the probability measure P satisfies appropriate conditions, we show that there exists an integer r (GREATERTHEQ) 1 and a sequence (tau)(,n) such that (S(,n) - n(tau)(,n))/n('1- 1/2r) has a limiting distribution which is non-Gaussian if r (GREATERTHEQ) 2. This result generalizes the theorems of Jong-Woo Jeon (Ph.D. Thesis, Dept. of Stat., F.S.U. (1979)) and Ellis and Newman (Z. Wahrscheinlichkeitstheorie und Verw. Gebiete. (1978) 44, 117-139). Chapters IV and V extend the above to the multivariate case. === Source: Dissertation Abstracts International, Volume: 43-08, Section: B, page: 2615. === Thesis (Ph.D.)--The Florida State University, 1982.
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