A unified formulation of the central limit theorem for small and large deviations from the mean

1979 ◽  
Vol 49 (1) ◽  
pp. 105-117 ◽  
Author(s):  
Thomas H�glund
1989 ◽  
Vol 3 (4) ◽  
pp. 477-491
Author(s):  
Kyle T. Siegrist ◽  
Ashok T. Amin ◽  
Peter J. Slater

Consider the standard network reliability model in which each edge of a given (n, m)-graph G is deleted, independently of all others, with probability q = 1– p (0 <p < 1). The pair-connectivity random variable X is defined to be the number of connected pairs of vertices that remain in G. The mean of X has been proposed as a measure of reliability for failure-prone communications networks in which the edge deletions correspond to failures of the communications links. We consider deviations from the mean, the law of large numbers, and the central limit theorem for X as n → ∞. Some explicit results are obtained when G is a tree using martingale difference sequences. Stars and paths are treated in detail.


2010 ◽  
Vol 27 (11) ◽  
pp. 1822-1838 ◽  
Author(s):  
W. J. Koshak

Abstract Flashes detected by the Optical Transient Detector (OTD) that occur over the continental United States (CONUS) are intercompared with data from the National Lightning Detection Network (NLDN) in order to partition the OTD flashes into ground and cloud flashes. The entire 5-yr OTD dataset for CONUS is analyzed. The statistical distributions of a variety of optical characteristics are examined, including five flash-level attributes (radiance, area, duration, number of optical groups, and number of optical events), and two group-level attributes [the maximum number of events in a group (MNEG), and a closely related parameter, the maximum group area (MGA)]. On average, there were 5.6 optical groups per return stroke in a ground flash, which is in part due to the likelihood that OTD detects interstroke K changes. It was found that return strokes within ground flashes typically produce large optical groups; hence, the MNEG and MGA parameters serve as useful “return-stroke detectors.” The results of this study provide insight on how to construct an algorithm for retrieving the fraction of ground flashes in a set of flashes observed from a satellite lightning imager. Specifically, even though it is shown that the statistical distributions of the optical characteristics for ground and cloud flashes overlap substantially, the mean values of these distributions differ. Hence, a retrieval method that is based on an analysis of the distribution of the means, and that employs the central limit theorem of statistics, is recommended. As the sample size used to compute the means is increased, the overlap in the distributions of the means for ground and cloud flashes is diminished, making ground flash fraction retrieval feasible. Of the seven optical characteristics examined here, the mean MNEG and mean MGA parameters are suggested as being the most useful for discriminating between ground and cloud flashes in the context of this “central limit theorem” approach.


2021 ◽  
Vol 36 (2) ◽  
pp. 243-255
Author(s):  
Wei Liu ◽  
Yong Zhang

AbstractIn this paper, we investigate the central limit theorem and the invariance principle for linear processes generated by a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng [19]. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov’s central limit theorem and invariance principle to the case where probability measures are no longer additive.


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