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.

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

We introduce real second-order freeness in second-order noncommutative probability spaces. We demonstrate that under this definition, independent ensembles of the three real models of random matrices which we consider, namely real Ginibre matrices, Gaussian orthogonal matrices, and real Wishart matrices, are asymptotically second-order free. These ensembles do not satisfy the complex definition of second-order freeness satisfied by their complex analogues. This definition may be used to calculate the asymptotic fluctuations of products of matrices in terms of the fluctuations of each ensemble. We use a combinatorial approach to the matrix calculations similar to genus expansion, but in which nonorientable surfaces appear, demonstrating the commonality between the real ensembles and the distinction from their complex analogues, motivating this distinct definition. We generalize the description of graphs on surfaces in terms of the symmetric group to the nonorientable case. In the real case we find, in addition to the terms appearing in the complex case corresponding to annular spoke diagrams, an extra set of terms corresponding to annular spoke diagrams in which the two circles of the annulus are oppositely oriented, and in which the matrix transpose appears. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2011-09-09 11:07:37.414

When high-frequency data is available, realised variance and realised absolute variation can be calculated from intra-day prices. In the context of a stochastic volatility model, realised variance and realised absolute variation can estimate the integrated variance and the integrated spot volatility respectively. A central limit theory enables us to do filtering and smoothing using model-based and model-free approaches in order to improve the precision of these estimators. When the log-price process involves a finite activity jump process, realised variance estimates the quadratic variation of both continuous and jump components. Other consistent estimators of integrated variance can be constructed on the basis of realised multipower variation, i.e., realised bipower, tripower and quadpower variation. These objects are robust to jumps in the log-price process. Therefore, given adequate asymptotic assumptions, the difference between realised multipower variation and realised variance can provide a tool to test for jumps in the process. Realised variance becomes biased in the presence of market microstructure effect, meanwhile realised bipower, tripower and quadpower variation are more robust in such a situation. Nevertheless there is always a trade-off between bias and variance; bias is due to market microstructure noise when sampling at high frequencies and variance is due to the asymptotic assumptions when sampling at low frequencies. By subsampling and averaging realised multipower variation this effect can be reduced, thereby allowing for calculations with higher frequencies.

Thesis (Ph.D.)--Tufts University, 2001. / Adviser: Marjorie G. Hahn. Submitted to the Dept. of Mathematics. Includes bibliographical references (leaves44-46). Access restricted to members of the Tufts University community. Also available via the World Wide Web;