Stein's method is applied to study the rate of convergence of maximum likelihood estimators to the normal law. Proximity of distributions of random variables is studied via Zolotarev ideal metrics. The previously available results are strengthened. As an example, the family of exponential distributions is considered.
We prove weak convergence of a net of Baire probability measures on a locally convex space with the weak topology under the condition of weak convergence of finite-dimensional projections and in the case of a sequence under the condition of convergence of their Fourier transforms.
We find necessary and sufficient conditions for the Marchenko-Pastur law to be the limiting spectral distribution of Gram matrices n-1XX & lowast; associated with random matrices X in Cp & times;n consisting of zero-mean independent entries with finite variances whose arithmetic mean is equal to one.
Let Z = {Z(t) : t is an element of R} be a stochastic process with trajectories in the space D(R). It is assumed that there exists an essentially smooth function A: R ->(-infinity, infinity] such that (1) (T) ln E(e(& micro;(Z(T )-alpha T) ) Z(s), s <= 0) = A(& micro;) + o(1) for all alpha is an element of R, & micro; is an element of dom A, uniformly on the event C(T) := {|Z(0)/T-alpha| < eta T }, where eta T -> 0 as T -> infinity. Under this condition, the following uniform conditional local large deviation principle (l.l.d.p.) is established: lim (1)(T ->infinity)(T) ln P(Z(T)/T-alpha is an element of beta-epsilon(T), 3 + epsilon T) Z(s), s <= 0) = -D ( beta) for all fixed alpha, beta is an element of R and a positive function eta T = o(1), as epsilon(T)-> 0 sufficiently slowly as T -> infinity, uniformly on C(T), where D is the Legendre transform of the function A. This result is used to establish a conditional l.l.d.p. for the finite-dimensional distributions of the process {z(T)(s) = Z(sT)/T: s is an element of [0, 1]}. Under additional conditions on the magnitude of oscillations of the trajectories z(T), a functional l.l.d.p. is obtained for the asymptotics of lnP(z(T) is an element of (f)epsilon T) as T ->infinity, where f is an element of D(0, 1), (f)(epsilon) is the epsilon-neighborhood off in the space D(0, 1) with respect to the uniform metric, and epsilon(T)-> 0 sufficiently slowly. The obtained results can be extended to a more general triangular array scheme where the process Z = Z((T)) itself also depends on the parameter T.
An analogue of Tanaka's formula is derived for the two-dimensional Brownian motion process.
