The exact order of subsets of additive bases

1984 ◽  
pp. 273-277 ◽  
Author(s):  
Melvyn B. Nathanson
Keyword(s):  
Exact Order

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

On exact order of convergence of random polynomials

2000 ◽  
Vol 99 (3) ◽  
pp. 1234-1243
Author(s):  
A. Basalykas ◽  
O. Yanushkevichiene
Keyword(s):  
Order Of Convergence ◽  
Exact Order ◽  
Random Polynomials

Markov chains in small time intervals

1981 ◽  
Vol 18 (3) ◽  
pp. 747-751
Author(s):  
Stig I. Rosenlund
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Transition Probability ◽  
Small Time ◽  
Exact Order ◽  
Time Intervals ◽  
Continuous Parameter ◽  
Minimum Number

For a time-homogeneous continuous-parameter Markov chain we show that as t → 0 the transition probability pn,j (t) is at least of order where r(n, j) is the minimum number of jumps needed for the chain to pass from n to j. If the intensities of passage are bounded over the set of states which can be reached from n via fewer than r(n, j) jumps, this is the exact order.

scholarly journals A4Mb High Resolution BAC Contig on Bovine Chromosome1q12and Comparative Analysis With Human Chromosome21q22

2005 ◽  
Vol 6 (4) ◽  
pp. 194-203 ◽  
Author(s):  
Cord Drögemüller ◽  
Anne Wöhlke ◽  
Tosso Leeb ◽  
Ottmar Distl
Keyword(s):  
Positional Cloning ◽  
Physical Map ◽  
Dominant Gene ◽  
Bac Library ◽  
Exact Order ◽  
Bac Contig ◽  
Nucleotide Sequence Data ◽  
New Genes ◽  
Human Genes ◽  
First Time

The bovine RPCI-42 BAC library was screened to construct a sequence-ready ~4 Mb single contig of 92 BAC clones on BTA 1q12. The contig covers the region between the genesKRTAP8P1andCLIC6. This genomic segment in cattle is of special interest as it contains the dominant gene responsible for the hornless or polled phenotype in cattle. The construction of the BAC contig was initiated by screening the bovine BAC library with heterologous cDNA probes derived from 12 human genes of the syntenic region on HSA 21q22. Contig building was facilitated by BAC end sequencing and chromosome walking. During the construction of the contig, 165 BAC end sequences and 109 single-copy STS markers were generated. For comparative mapping of 25 HSA 21q22 genes, genomic PCR primers were designed from bovine EST sequences and the gene-associated STSs mapped on the contig. Furthermore, bovine BAC end sequence comparisons against the human genome sequence revealed significant matches to HSA 21q22 and allowed thein silicomapping of two new genes in cattle. In total, 31 orthologues of human genes located on HSA 21q22 were directly mapped within the bovine BAC contig, of which 16 genes have been cloned and mapped for the first time in cattle. In contrast to the existing comparative bovine–human RH maps of this region, these results provide a better alignment and reveal a completely conserved gene order in this 4 Mb segment between cattle, human and mouse. The mapping of known polled linked BTA 1q12 microsatellite markers allowed the integration of the physical contig map with existing linkage maps of this region and also determined the exact order of these markers for the first time. Our physical map and transcript map may be useful for positional cloning of the putative polled gene in cattle. The nucleotide sequence data reported in this paper have been submitted to EMBL and have been assigned Accession Numbers AJ698510–AJ698674.

scholarly journals On the 2-adic order of Stirling numbers of the second kind and their differences

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Binary Representation ◽  
Exact Order ◽  
Special Cases ◽  
International Audience ◽  
The Difference ◽  
Divisibility Properties ◽  
Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

scholarly journals RESIDUAL-BASED GARCH BOOTSTRAP AND SECOND ORDER ASYMPTOTIC REFINEMENT

2016 ◽  
Vol 33 (3) ◽  
pp. 779-790 ◽  
Author(s):  
Minsoo Jeong
Keyword(s):  
Second Order ◽  
Moment Condition ◽  
Garch Models ◽  
Exact Order ◽  
Block Bootstrap ◽  
Edgeworth Expansions ◽  
Coverage Probabilities ◽  
The Moment ◽  
Autoregressive Conditional Heteroscedasticity ◽  
Reliable Methods

The residual-based bootstrap is considered one of the most reliable methods for bootstrapping generalized autoregressive conditional heteroscedasticity (GARCH) models. However, in terms of theoretical aspects, only the consistency of the bootstrap has been established, while the higher order asymptotic refinement remains unproven. For example, Corradi and Iglesias (2008) demonstrate the asymptotic refinement of the block bootstrap for GARCH models but leave the results of the residual-based bootstrap as a conjecture. To derive the second order asymptotic refinement of the residual-based GARCH bootstrap, we utilize the analysis in Andrews (2001, 2002) and establish the Edgeworth expansions of the t-statistics, as well as the convergence of their moments. As expected, we show that the bootstrap error in the coverage probabilities of the equal-tailed t-statistic and the corresponding test-inversion confidence intervals are at most of the order of O(n−1), where the exact order depends on the moment condition of the process. This convergence rate is faster than that of the block bootstrap, as well as that of the first order asymptotic test.

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