On records and related processes for sequences with trends

1999 ◽  
Vol 36 (3) ◽  
pp. 668-681 ◽  
Author(s):  
K. Borovkov
Keyword(s):  
Markov Chains ◽  
Limit Theorems ◽  
Convergence Rates ◽  
Rate Function ◽  
Regular Case ◽  
Limiting Distributions ◽  
Record Times ◽  
Ergodicity Theorem ◽  
Linear Trends

We study the records and related variables for sequences with linear trends. We discuss the properties of the asymptotic rate function and relationships between the distribution of the long-term maxima in the sequence and that of a particular observation, including two characterization type results. We also consider certain Markov chains related to the process of records and prove limit theorems for them, including the ergodicity theorem in the regular case (convergence rates are given under additional assumptions), and derive the limiting distributions for the inter-record times and increments of records.

On records and related processes for sequences with trends

1999 ◽  
Vol 36 (03) ◽  
pp. 668-681 ◽  
Author(s):  
K. Borovkov
Keyword(s):  
Markov Chains ◽  
Limit Theorems ◽  
Convergence Rates ◽  
Rate Function ◽  
Regular Case ◽  
Limiting Distributions ◽  
Record Times ◽  
Ergodicity Theorem ◽  
Linear Trends

We study the records and related variables for sequences with linear trends. We discuss the properties of the asymptotic rate function and relationships between the distribution of the long-term maxima in the sequence and that of a particular observation, including two characterization type results. We also consider certain Markov chains related to the process of records and prove limit theorems for them, including the ergodicity theorem in the regular case (convergence rates are given under additional assumptions), and derive the limiting distributions for the inter-record times and increments of records.

scholarly journals Asymptotic behaviors with convergence rates of distributions of negative-binomial sums

2020 ◽  
Vol 22 (4) ◽  
pp. 415-421
Author(s):  
Tran Loc Hung ◽  
Phan Tri Kien ◽  
Nguyen Tan Nhut
Keyword(s):  
Limit Theorems ◽  
Negative Binomial ◽  
Convergence Rates ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Weak Limit ◽  
Probability Metric ◽  
Weak Limit Theorems ◽  
Distribution Theorem ◽  
Binomial Sums

The negative-binomial sum is an extension of a geometric sum. It has been arisen from the necessity to resolve practical problems in telecommunications, network analysis, stochastic finance and insurance mathematics, etc. Up to the present, the topics related to negative-binomial sums like asymptotic distributions and rates of convergence have been investigated by many mathematicians. However, in a lot of various situations, the results concerned the rates of convergence for negative-binomial sums are still restrictive. The main purpose of this paper is to establish some weak limit theorems for negative-binomial sums of independent, identically distributed (i.i.d.) random variables via Gnedenko's Transfer Theorem originated by Gnedenko and Fahim (1969). Using Zolotarev's probability metric, the rate of convergence in weak limit theorems for negativebinomial sum are established. The received results are the rates of convergence in weak limit theorem for partial sum of i.i.d random variables related to symmetric stable distribution (Theorem 1), and asymptotic distribution together with the convergence rates for negative-binomial sums of i.i.d. random variables concerning to symmetric Linnik laws and Generalized Linnik distribution (Theorem 2 and Theorem 3). Based on the results of this paper, the analogous results for geometric sums of i.i.d. random variables will be concluded as direct consequences. However, the article has just been solved for the case of 1 <a < 2; it is quite hard to estimate in the case of a 2 (0;1) via the Zolotarev's probability metric. Mathematics Subject Classification 2010: 60G50; 60F05; 60E07.

scholarly journals Global and long-term comparison of SCIAMACHY limb ozone profiles with correlative satellite data (2002–2008)

2011 ◽  
Vol 4 (4) ◽  
pp. 4867-4910
Author(s):  
S. Mieruch ◽  
M. Weber ◽  
C. von Savigny ◽  
A. Rozanov ◽  
H. Bovensmann ◽  
...  
Keyword(s):  
Satellite Data ◽  
Convective Cloud ◽  
Residual Effects ◽  
Statistical Hypothesis ◽  
Systematic Bias ◽  
Statistical Hypothesis Testing ◽  
Trend Model ◽  
The Tropics ◽  
Linear Trends

Abstract. SCIAMACHY limb scatter ozone profiles from 2002 to 2008 have been compared with MLS (2005–2008), SABER (2002–2008), SAGE II (2002–2005), HALOE (2002–2005) and ACE-FTS (2004–2008) measurements. The comparison is performed for global zonal averages and heights from 10 to 50 km in one km steps. The validation was performed by comparing monthly mean zonal means and by comparing averages over collocated profiles within a zonal band and month. Both approaches yield similar results. For most of the stratosphere SCIAMACHY agrees to within 10 % or better with other correlative data. A systematic bias of SCIAMACHY ozone of up to 100 % between 10 and 20 km in the tropics points to some remaining issues with regard to convective cloud interference. Statistical hypothesis testing reveals at which altitudes and in which region differences between SCIAMACHY and other satellite data are statistically significant. We also estimated linear trends from monthly mean data for different periods where SCIAMACHY has common observations with other satellite data using a classical trend model with QBO and seasonal terms in order to draw conclusions on potential instrumental drifts as a function of latitude and altitude. SCIAMACHY exhibits a statistically significant negative trend in the range of of about 1–3 % per year depending on latitude during the period 2002–2005 (overlapping with HALOE and SAGE II) and somewhat less during 2002–2008 (overlapping with SABER) in the altitude range of 30–40 km, while in the period 2004–2008 (overlapping with MLS and ACE-FTS) no significant trends are observed. The statistically significant negative trends only observed with SCIAMACHY data point at some residual effects from errors in the tangent height registration.

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