VALIDITY OF THE SAMPLING WINDOW METHOD FOR LONG-RANGE DEPENDENT LINEAR PROCESSES

Abstract The Marcinkiewicz strong law, limn→∞(1 / n1/p)∑k=1n(Dk - D) = 0 almost surely with p ∈ (1, 2), is studied for outer products Dk = {XkX̅kT}, where {Xk} and {X̅k} are both two-sided (multivariate) linear processes (with coefficient matrices (Cl), (C̅l) and independent and identically distributed zero-mean innovations {Ξ} and {Ξ̅}). Matrix sequences Cl and C ̅l can decay slowly enough (as |l| → ∞) that {Xk,X ̅k} have long-range dependence, while {Dk} can have heavy tails. In particular, the heavy-tail and long-range-dependence phenomena for {Dk} are handled simultaneously and a new decoupling property is proved that shows the convergence rate is determined by the worst of the heavy tails or the long-range dependence, but not the combination. The main result is applied to obtain a Marcinkiewicz strong law of large numbers for stochastic approximation, nonlinear function forms, and autocovariances.

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Convergence rates in the law of large numbers for long-range dependent linear processes

Journal of Inequalities and Applications ◽

10.1186/s13660-017-1517-6 ◽

2017 ◽

Vol 2017 (1) ◽

Cited By ~ 1

Author(s):

Tao Zhang ◽

Pingyan Chen ◽

Soo Hak Sung

Keyword(s):

Long Range ◽

Law Of Large Numbers ◽

Convergence Rates ◽

Linear Processes ◽

Large Numbers ◽

The Law

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Central limit theorems for nearly long range dependent subordinated linear processes

Journal of Applied Probability ◽

10.1017/jpr.2020.10 ◽

2020 ◽

Vol 57 (2) ◽

pp. 637-656

Author(s):

Martin Wendler ◽

Wei Biao Wu

Keyword(s):

Long Range ◽

Central Limit ◽

Short Range ◽

Limit Theorems ◽

Central Limit Theorems ◽

Partial Sums ◽

Long Range Dependence ◽

Stationary Sequences ◽

Linear Processes ◽

Arbitrary Power

AbstractThe limit behavior of partial sums for short range dependent stationary sequences (with summable autocovariances) and for long range dependent sequences (with autocovariances summing up to infinity) differs in various aspects. We prove central limit theorems for partial sums of subordinated linear processes of arbitrary power rank which are at the border of short and long range dependence.

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High Voltage Transmission Microscopy of Surveyor III Camera Shrouds

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100064748 ◽

1971 ◽

Vol 29 ◽

pp. 138-139

Author(s):

W. R. Duff ◽

L. E. Thomas ◽

R. M. Fisher ◽

S. V. Radcliffe

Keyword(s):

Alloy Sheet ◽

Transmission Microscopy ◽

Television Camera ◽

Window Method ◽

Transmission Electron ◽

Microstructural Effects ◽

Lunar Environment ◽

Successful Retrieval ◽

Materials Used ◽

Abrasive Paper

Successful retrieval of the television camera and other components from the Surveyor III spacecraft by the Apollo 12 astronauts has provided a unique opportunity to study the effects of a known and relatively extensive exposure to the lunar environment. Microstructural effects including those produced by micro-meteorite impact, radiation damage (by both the solar wind and cosmic rays) and solar heating might be expected in the materials used to fabricate the spacecraft. Samples received were in the form of 1 cm2 of painted unpainted aluminum alloy sheet from the top of the camera visor (JPL Code 933) and the sides (935,936) and bottom (934) of the lower camera shroud. They were prepared for transmission electron microscopy by first hand-grinding with abrasive paper to a thickness of 0.006". The edges were lacquered and the sample electropolished in 10% perchloric methanol using the “window” method, to a thickness of ~0.001". Final thinning was accomplished by polishing 3 mm punched disks in an acetic-phosphoric-nitric acid solution.

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Peak-to-background measurements on a 300-kV TEM/STEM

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s042482010013081x ◽

1992 ◽

Vol 50 (2) ◽

pp. 1236-1237 ◽

Cited By ~ 1

Author(s):

S. M. Zemyan ◽

D. B. Williams

Keyword(s):

Electron Microscope ◽

X Ray ◽

Background Ratio ◽

Window Method ◽

Analytical Electron ◽

Analytical Electron Microscope ◽

Cr Film

As has been reported elsewhere, a thin evaporated Cr film can be used to monitor the x-ray peak to background ratio (P/B) in an analytical electron microscope. Presented here are the results of P/B measurements for the Cr Ka line on a Philips EM430 TEM/STEM, with Link Si(Li) and intrinsic Ge (IG) x-ray detectors. The goal of the study was to determine the best conditions for x-ray microanalysis.We used the Fiori P/B definition, in which P/B is the ratio of the total peak integral to the average background in a 10 eV channel beneath the peak. Peak and background integrals were determined by the window method, using a peak window from 5.0 to 5.7 keV about Cr Kα, and background windows from 4.1 to 4.8 keV and 6.3 to 7.0 keV.

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