A functional central limit theorem for a strictly stationary associated random field in the general d-dimension case with an added moment condition is proven. Functional central limit theorems for associated random measures are also proven. More specifically, conditions are given that imply weak convergence in the Skorohod topology of a renormalized random measure to the d-dimensional Wiener process. These results are applied to show new functional central limit theorems for doubly stochastic point random fields and Poisson cluster random measures. / Graduation date: 1986

Thesis (Ph. D.)--Oregon State University, 1986. / Typescript (photocopy). Includes bibliographical references (leaves 72-74). Also available on the World Wide Web.

Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2003. / Includes bibliographical references (leaves 44). Also available in electronic version. Access restricted to campus users.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE / Vita. / Bibliography: leaves 135-137. / by Kenneth Sidney Alexander. / Ph.D.

In this thesis, we study time-dependent empirical processes, which extend the classical empirical processes to have a time parameter; for example the empirical process for a sequence of independent stochastic processes {Yi : i ∈ N}: (1) ν_n(t, y) = n^(−1/2 )Sigma[1_(Y i(t)¬<=y) – P(Yi(t) <= y)] from i=1 to n, t ∈ E, y ∈ R. In the case of independent identically distributed samples (that is {Yi(t) : i ∈ N} are iid), Kuelbs et al. (2013) proved a Central Limit Theorem for ν_n(t, y) for a large class of stochastic processes. In Chapter 3, we give a sufficient condition for the weak convergence of the weighted empirical process for iid samples from a uniform process: (2) α_n(t, y) := n^(−1/2 )Sigma[w(y)(1_(X (t)<=y) – y)] from i=1 to n, t ∈ E, y ∈ [0, 1] where {X (t), X1(t), X2(t), • • • } are independent and identically distributed uniform processes (for each t ∈ E, X (t) is uniform on (0, 1)) and w(x) is a “weight” function satisfying some regularity properties. Then we give an example when X (t) := Ft(Bt) : t ∈ E = [1, 2], where Bt is a Brownian motion and Ft is the distribution function of Bt. In Chapter 4, we investigate the weak convergence of the empirical processes for non-iid samples. We consider the weak convergence of the empirical process: (3) β_n(t, y) := n^(−1/2 )Sigma[(1_(Y (t)<=y) – Fi(t,y))] from i=1 to n, t ∈ E ⊂ R, y ∈ R where {Yi(t) : i ∈ N} are independent processes and Fi(t, y) is the distribution function of Yi(t). We also prove that the covariance function of the empirical process for non-iid samples indexed by a uniformly bounded class of functions necessarily uniformly converges to the covariance function of the limiting Gaussian process for a CLT.

In this paper limit theorems for the conditional distributions of linear test statistics are proved. The assertions are conditioned by the sigma-field of permutation symmetric sets. Limit theorems are proved both for the conditional distributions under the hypothesis of randomness and under general contiguous alternatives with independent but not identically distributed observations. The proofs are based on results on limit theorems for exchangeable random variables by Strasser and Weber. The limit theorems under contiguous alternatives are consequences of an LAN-result for likelihood ratios of symmetrized product measures. The results of the paper have implications for statistical applications. By example it is shown that minimum variance partitions which are defined by observed data (e.g. by LVQ) lead to asymptotically optimal adaptive tests for the k-sample problem. As another application it is shown that conditional k-sample tests which are based on data-driven partitions lead to simple confidence sets which can be used for the simultaneous analysis of linear contrasts. (author's abstract) / Series: Report Series SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

We consider random simplicial complexes constructed on a Poisson point process within a convex set in a Euclidean space, especially the Vietoris-Rips complex and the Cech complex both of whose 1-skeleton is the Gilbert graph. We investigate at first the Vietoris-Rips complex by considering the volume-power functionals defined by summing powers of the volume of all k-dimensional faces in the complex. The asymptotic behaviour of these functionals is investigated as the intensity of the underlying Poisson point process tends to infinity and the distance parameter goes to zero. This behaviour is observed in different regimes. Univariate and multivariate central limit theorems are proven, and analogous results for the Cech complex are then given. Finally we provide a Poisson limit theorem for the components of the f-vector in the sparse regime.

No description available.

Under an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-Gaussian stable random variable. A functional version of this is known to be true as well, the limit process being a stable L´evy process. In this thesis, we developed an explicit construction for the α-stable L´evy process motion with values in D([0, 1]), by considering the cases α < 1 and α > 1. The case α < 1 is the simplest since we can work with the uniform topology of the sup-norm on D([0, 1]) and the construction follows more or less by classical techniques. The case α > 1 required more work. In particular, we encountered two problems : one was related to the construction of a modification of this process (for all time), which is right-continuous and has left-limit with respect to the J1 topology. This problem was solved by using the Itob-Nisio theorem. The other problem was more difficult and we only managed to solve it by developing a criterion for tightness of probability measures on the space of cadlag fonction on [0, T] with values in D([0, 1]), equipped with a generalization of Skorohod’s J1 topology. In parallel with the construction of the infinite-dimensional process Z, we focus on the functional extension of Roueff and Soulier [29]. This part of the thesis was completed using the method of point process, which gave the convergence of the truncated sum. The case α > 1 required more work due to the presence of centering. For this case, we developed an ad-hoc result regarding the continuity of the addition for functions on [0, T] with values in D([0, 1]), which was tailored for our problem.

Thesis. 1975. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / Vita. / Bibliography: leaves 111-114. / by Marjorie G. Hahn. / Ph.D.