Algorithms for system identification, estimation, and adaptive control in stochastic systems rely mostly on different types of signal averaging to achieve uncertainty reduction, convergence, stability, and performance enhancement. The core of such algorithms is various types of laws of large numbers that reduce the effect of noises when they are averaged. Many of the noise sequences encountered are often correlated and nonwhite. In the case of state estimation using quantized information such as in networked systems, convergence must be analyzed on double-indexed and randomly weighted sums of mixing-type stochastic processes, which are correlated with the remote past and distant future being asymptotically independent. This paper presents new results on convergence analysis of such processes. Strong laws of large numbers and convergence rates for such problems are established. These results resolve some fundamental issues in state observer designs with random sampling times, quantized information processing, and other applications.

For a sequence of random elements T n n ≥ 1 in a real separable Banach space , we study the notion of T n converging completely to 0 in mean of order p where p is a positive constant. This notion is stronger than (i) T n converging completely to 0 and (ii) T n converging to 0 in mean of order p. When is of Rademacher type p 1 ≤ p ≤ 2 , for a sequence of independent mean 0 random elements V n n ≥ 1 in and a sequence of constants b n → , conditions are provided under which the normed sum n j=1 V j /b n converges completely to 0 in mean of order p. Moreover, these conditions for n j=1 V j /b n converging completely to 0 in mean of order p are shown to provide an exact characterization of Rademacher type p Banach spaces. Illustrative examples are provided.

Agarwood trade has been growing rapidly due to its active ingredients composed of volatile substances, which are widely used in the cosmetics and pharmaceuticals. However, the formation time of agarwood in nature is quite long and little is known about its formation agents and mechanisms in planted Aquilaria crassna trees. In this study, biological, chemical and mechanical treatments were applied to 5-, 8-, and 11-year-old A. crassna plantations in north central, Vietnam. Agarwood samples were collected at 1 and 2 years after treatment. Oil content (O c ) in dry wood of A. crassna and sesquiterpene content in extracted oil were analyzed using hydrodistillation method and gas chromatography-mass spectrometry. The results indicated that 1 year after treatment, control (no treatment) and mechanical treatment (nails hammered to tree stems) had highest O c (0.061-0.079 %) but lowest sesquiterpene content (2.1-7.1 %). Chemical treatment (mixture of acid sulfuric and sodium methyl bisulfate) had lowest O c (0.038-0.039 %) but highest sesquiterpene content (15.8-20.8 %). While, 1 year after treatment biological treatments had O c of 0.050-0.077 % and sesquiterpene content of 2.4-11.1 %. Two years after treatment, control still had lowest sesquiterpene content (3.2-7.0 %), while highest content (13.9-44.2 %) belonged to biological treatment (a mixture of fungi Phialophora spp. and Fusarium spp.). There was a total of 56 sesquiterpenes found in extracted oil from wood samples in biological treatments, which included eight highly commerciallyvaluable sesquiterpenes on international trade. Biological treatment with a mixture of fungi P. spp. and F. spp. should be encouraged to apply to 11-year-old A. crassna plantation, which resulted in highest sesquiterpene content (44.2 %).

Medical abortion service delivery with an MLPT to obtain a baseline (preabortion) human chorionic gonadotropin (hCG) estimate and a second follow-up MLPT 1 to 2 weeks later can establish whether there has been a drop in hCG, signifying absence of a continuing pregnancy. Used this way, MLPTs can enable women to assess their abortion status outside of a clinic setting and without serum hCG testing and/or ultrasound.

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