| Let {X(,n), n (GREATERTHEQ) 1} be a sequence of i.i.d. random variables with.;(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).;E(X(,1)) = 0, Var(X(,1)) = 1. Let (psi)(s) be the cumulant generating function (c.g.f.) and.;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. |
