scholarly journals Asymptotic normality for the counting process of weak records and δ -records in discrete models

Bernoulli ◽  
2007 ◽  
Vol 13 (3) ◽  
pp. 754-781 ◽  
Author(s):  
Raúl Gouet ◽  
F. Javier López ◽  
Gerardo Sanz
2005 ◽  
Vol 37 (3) ◽  
pp. 781-800 ◽  
Author(s):  
Raul Gouet ◽  
F. Javier López ◽  
Gerardo Sanz

Consider a sequence (Xn) of independent and identically distributed random variables taking nonnegative integer values, and call Xn a record if Xn> max{X1,…,Xn−1}. By means of martingale arguments it is shown that the counting process of records among the first n observations, suitably centered and scaled, is asymptotically normally distributed.


2005 ◽  
Vol 37 (03) ◽  
pp. 781-800 ◽  
Author(s):  
Raul Gouet ◽  
F. Javier López ◽  
Gerardo Sanz

Consider a sequence (X n ) of independent and identically distributed random variables taking nonnegative integer values, and call X n a record if X n > max{X 1,…,X n−1}. By means of martingale arguments it is shown that the counting process of records among the first n observations, suitably centered and scaled, is asymptotically normally distributed.


Author(s):  
Marek Capinski ◽  
Ekkehard Kopp

1998 ◽  
Vol 14 (4) ◽  
pp. 833-848
Author(s):  
Malcolm P. Quine ◽  
Władysław Szczotka
Keyword(s):  

Author(s):  
Shiwei Wang ◽  
Anton Chavez ◽  
Simil Thomas ◽  
Hong Li ◽  
Nathan C. Flanders ◽  
...  

This work reports on the assembly of imine-linked macrocycles that serve as models of two-dimensional covalent organic frameworks (2D COFs). Interlayer interactions play an important role in the formation of 2D COFs, yet the effect of monomer structure on COF formation, crystallinity, and susceptibility to exfoliation are not well understood. For example, monomers with both electron-rich and electron-poor π-electron systems have been proposed to strengthen interlayer inter-actions and improve crystallinity. Here we probe these effects by studying the stacking behavior of imine-linked macrocycles that represent discrete models of 2D COFs. <div><br></div><div>Specifically, macrocycles based on terephthaldehyde (PDA) or 2,5-dimethoxyterephthaldehyde (DMPDA) stack upon cooling molecularly dissolved solutions. Both macrocycles assemble cooperatively with similar ΔHe values of -97 kJ/mol and -101 kJ/mol, respectively, although the DMPDA macrocycle assembly process showed a more straightforward temperature dependence. Circular dichroism spectroscopy performed on macrocycles bearing chiral side chains revealed a helix reversion process for the PDA macrocycles that was not observed for the DMPDA macrocycles. <br></div><div><br></div><div>Given the structural similarity of these monomers, these findings demonstrate that the stacking processes associated with nanotubes derived from these macrocycles, as well as for the corresponding COFs, are complex and susceptible to kinetic traps, casting doubt on the relevance of thermodynamic arguments for improving materials quality. <br></div>


1998 ◽  
Vol 38 (6) ◽  
pp. 209-217 ◽  
Author(s):  
Jianhua Lei ◽  
Sveinung Sægrov

This paper demonstrates the statistical approach for describing failures and lifetimes of water mains. The statistical approach is based on pipe inventory data and the maintenance data registered in the data base. The approach consists of data pre-processing and statistical analysis. Two classes of statistical models are applied, namely counting process models and lifetime models. With lifetime models, one can estimate the probability which a pipe will fail within a time horizon. With counting process models one can see the deteriorating (or improving) trend in time of a group of “identical” pipes and their rates of occurrence of failure (ROCOF). The case study with the data base from Trondheim municipality (Norway) demonstrates the applicability of the statistical approach and leads to the following results: 1). In the past 20 years, Trondheim municipality has experienced approximately 250 to 300 failures per year. However, the number of failures per year will significantly increase in the near future unless better maintenance practice is implemented now. 2). Unprotected ductile iron pipes have a higher probability of failures than other materials. The average lifetime of unprotected ductile iron pipes is approximately 30 to 40 years shorter than the lifetime of a cast iron pipe. 3). Pipes installed 1963 and 1975 are most likely to fail in the future; 4) The age of a pipe does not play a significant role for the remaining lifetime of the pipe; 5). After 2 to 3 failures, a pipe enters a fast-failure stage (i.e., frequent multiple between failures).


Author(s):  
Junshan Qiu ◽  
Dali Zhou ◽  
H.M. Jim Hung ◽  
John Lawrence ◽  
Steven Bai

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