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1.

Dyrssen, Hannah

et al.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.

Ekström, Erik

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

We consider the sequential testing of two simple hypotheses for the drift of a Brownian motion when each observation of the underlying process is associated with a positive cost. In this setting where continuous monitoring of the underlying process is not feasible, the question is not only whether to stop or to continue at a given observation time but also, if continuing, how to distribute the next observation time. Adopting a Bayesian methodology, we show that the value function can be characterized as the unique fixed point of an associated operator and that it can be constructed using an iterative scheme. Moreover, the optimal sequential distribution of observation times can be described in terms of the fixed point.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

Vaicenavicius, Juozas

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Systems and Control. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Automatic control.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.

Anscombe's Theorem 60 Years Later2012In: Sequential Analysis, ISSN 0747-4946, E-ISSN 1532-4176, Vol. 31, no 3, p. 368-396Article in journal (Refereed)

Abstract [en]

The point of departure of the present article is Anscombe's seminal 1952 paper on limit theorems for randomly indexed processes. We discuss the importance of this result and mention some of its impact, mainly on stopped random walks. The main aim of the article is to illustrate the beauty and efficiency of what will be called the stopped random walk (SRW) method.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.

Hsu and Robbins (1947) introduced the concept of complete conver-gence as a complement to the Kolmogorov strong law in that they proved that Sigma(infinity)(n=1) P(|Sn| > n epsilon) < infinity provided the mean of the summands is zero and that the variance is finite. Later, Erdos proved the necessity (1949, 1950). Heyde (1975) proved that, under the same conditions, lim(epsilon)SE arrow 0 epsilon(2) Sigma(infinity)(n=1) P(|Sn| > n epsilon) = EX2, thereby opening an area of research that has been called precise asymptotics. Both results above have been extended and generalized in various directions. Kao (1978) proved a pointwise version of Heyde's result, viz. for the counting process N(epsilon) = Sigma(infinity)(n=1) 1{|Sn| > n epsilon}, he showed that lim(epsilon)SE arrow 0 epsilon N-2 (epsilon) ->(d) EX2 integral(infinity)(0) 1 {|W(u)| > u} du, where W(.) is the standard Wiener process. In this article, we prove an analog for perturbed random walks and illustrate how they enter naturally within the theory of repeated significance tests in exponential families.