Stochastic approximation and the final value theorem

The aim here is to show how to obtain many of the well-known limit results (i.e., central limit theorem, law of the iterated logarithm, invariance principle) of stochastic approximation (SA) by a shorter argument and under weaker conditions. The idea is to introduce an artificial sequence, related to the SA scheme, and which clearly obeys the limit law. This sequence is subtracted from the SA scheme and then simple deterministic limit theory is used to show the remainder is negligible. As a consequence of this approach proofs are shorter and the meaning of conditions becomes clearer. Because the difference equations are not summed up it is simple to state results for general a_n, c_n sequences.     

Stochastic approximation with dependent noise

In this work we derive the usual limit laws (weak and strong convergence, central limit theorem, invariance principle) for stochastic approximation with stationary noise. The idea is to introduce an artificial sequence, related to the SA scheme, but which clearly obeys the desired limit law. This sequence is subtracted from the SA scheme and the remainder, which behaves more or less deterministically, is shown to vanish using simple limit arguments.     

The exact likelihood for a multivariate ARMA model

A number of algorithms are presented for calculating the exact likelihood of a multivariate ARMA model. There are two aspects to the algorithms. Firstly, the parameterization is in terms of AR parameters and autocovariances. This obviates difficulties with initial MA estimates. Secondly, the algorithms explicitly account for specification of the lag structure of the multivariate time series. Additionally, an algorithm is presented to deal with missing data. The algorithms are, of themselves, not new but they have not been applied to likelihood construction in the manner discussed here.     

Productive Syntheses of Privileged Scaffolds Inspired by the Recognition of a Diels–Alder Pattern Common to Three Classes of Natural Products

Identification of a common Diels–Alder pattern in three classes of bioactive natural products led us to study the synthesis and cycloaddition of a new class of cyclic dienes readily available from β,γ-unsaturated lactams. A practical and readily scalable route to the parent p-methoxybenzyl-protected 6- and 7-membered β,γ-unsaturated lactams was developed. These were readily transformed into the corresponding O-silylated dienes, which were reacted with dimethyl and diethyl fumarate to yield stereoselectively highly functionalized bicyclic adducts. These exhibited unexpected and versatile transformations upon acid hydrolysis depending on the nature of the dienophile substituents and the acid catalyst. All reactions have been performed on multigram quantities. These transformations provide a convenient, economical, and easily scalable pathway for the rapid construction of functionally and stereochemically dense privileged scaffolds for the construction of libraries of natural products-inspired molecules of pharmacological relevance.