Quadratic Approximation for Log-Likelihood Ratio Processes
For the stochastic differential equationdX(t)=a X ( t ) + b X ( t - 1 )dt+dW(t),t≥0,
the local asymptotic properties of the likelihood function are studied. They depend strongly on the true value of the parameter ϑ=(a,b) * . Eleven different cases are possible if ϑ runs through ℝ 2 . Let ϑ ^ T be the maximum likelihood estimator of ϑ based on (X(t), t≥T). Applications to the asymptotic behaviour of ϑ ^ T as T→∞ are given.
The role of Yuri Vasilyevich Prokhorov as a prominent mathematician and leading expert in the theory of probability is well known. Even early in his career he obtained substantial results on the validity of the strong law of large numbers and on the estimates (bounds) of the rates of convergence, some of which are the best possible. His findings on limit theorems in metric spaces and particularly functional limit theorems are of exceptional importance. Y.V. Prokhorov developed an original approach to the proof of functional limit theorems, based on the weak convergence of finite dimensional distributions and the condition of tightness of probability measures.
The present volume commemorates the 80th birthday of Yuri Vasilyevich Prokhorov. It includes scientific contributions written by his colleagues, friends and pupils, who would like to express their deep respect and sincerest admiration for him and his scientific work.
Let X be a semimartingale which is locally square integrable and admitting the canonical decompositions X=M+A and X=M ' +A ' with respect to measures P and P ' . Let γ be the density of A-A ' with respect to C=(〈M〉+〈M ' 〉) in the Lebesgue decomposition. Then there is a version h of the Hellinger process h(1/2;P,P ' ) such that (1-Δh) -2 ·h⪰(1/8)γ 2 ·C P- and P ' -a.s. This inequality is related with a generalization of the Cramér-Rao inequality to the case of filtered space. The author gives some applications to a continuous-time linear regression model as well as to a discrete-time autoregression model with martingale errors.
Предлагается общая вероятностная модель для булевых функций от n переменных, задаваемая произвольной вероятностной мерой на множестве всех таких функций. Выводится характеристическая функция спектра Уолша случайной функции и находятся точные и асимптотические (при n→∞) распределения некоторых его характеристик для случая параметрической меры.