Some results on convergence rates for probabilities of moderate deviations for sums of random variables
Let X, Xn, n≥1 be a sequence of iid real random variables, and Sn=∑k=1nXk, n≥1. Convergence rates of moderate deviations are derived, i.e., the rate of convergence to zero of certain tail probabilities of the partial sums are determined. For example, we obtain equivalent conditions for the convergen...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
1992-01-01
|
| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171292000644 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832565572286545920 |
|---|---|
| author | Deli Li Xiangchen Wang M. Bhaskara Rao |
| author_facet | Deli Li Xiangchen Wang M. Bhaskara Rao |
| author_sort | Deli Li |
| collection | DOAJ |
| description | Let X, Xn, n≥1 be a sequence of iid real random variables, and Sn=∑k=1nXk, n≥1. Convergence rates of moderate deviations are derived, i.e., the rate of convergence to zero of certain tail probabilities of the partial sums are determined. For example, we obtain equivalent conditions for the convergence of series ∑n≥1(ψ2(n)/n)P(|Sn|≥nφ(n)) only under the assumptions convergence that EX=0 and EX2=1, where φ and ψ are taken from a broad class of functions. These results generalize and improve some recent results of Li (1991) and Gafurov (1982) and some previous work of Davis (1968). For b∈[0,1] and ϵ>0, letλϵ,b=∑n≥3((loglogn)b/n)I(|Sn|≥(2+ϵ)nloglogn).The behaviour of Eλϵ,b as ϵ↓0 is also studied. |
| format | Article |
| id | doaj-art-352c1414b1144c6eb62da4d8731bdc71 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1992-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-352c1414b1144c6eb62da4d8731bdc712025-02-03T01:07:20ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251992-01-0115348149710.1155/S0161171292000644Some results on convergence rates for probabilities of moderate deviations for sums of random variablesDeli Li0Xiangchen Wang1M. Bhaskara Rao2Institute of Mathematics, Jilin University, Changchun 130023, ChinaDepartment of Mathematics, Jilin University, Changchun 130023, ChinaDepartment of Statistics, North Dakota State University, Fargo, ND 58105, USALet X, Xn, n≥1 be a sequence of iid real random variables, and Sn=∑k=1nXk, n≥1. Convergence rates of moderate deviations are derived, i.e., the rate of convergence to zero of certain tail probabilities of the partial sums are determined. For example, we obtain equivalent conditions for the convergence of series ∑n≥1(ψ2(n)/n)P(|Sn|≥nφ(n)) only under the assumptions convergence that EX=0 and EX2=1, where φ and ψ are taken from a broad class of functions. These results generalize and improve some recent results of Li (1991) and Gafurov (1982) and some previous work of Davis (1968). For b∈[0,1] and ϵ>0, letλϵ,b=∑n≥3((loglogn)b/n)I(|Sn|≥(2+ϵ)nloglogn).The behaviour of Eλϵ,b as ϵ↓0 is also studied.http://dx.doi.org/10.1155/S0161171292000644moderate deviationsrates of convergencetail probabilities. |
| spellingShingle | Deli Li Xiangchen Wang M. Bhaskara Rao Some results on convergence rates for probabilities of moderate deviations for sums of random variables International Journal of Mathematics and Mathematical Sciences moderate deviations rates of convergence tail probabilities. |
| title | Some results on convergence rates for probabilities of moderate deviations for sums of random variables |
| title_full | Some results on convergence rates for probabilities of moderate deviations for sums of random variables |
| title_fullStr | Some results on convergence rates for probabilities of moderate deviations for sums of random variables |
| title_full_unstemmed | Some results on convergence rates for probabilities of moderate deviations for sums of random variables |
| title_short | Some results on convergence rates for probabilities of moderate deviations for sums of random variables |
| title_sort | some results on convergence rates for probabilities of moderate deviations for sums of random variables |
| topic | moderate deviations rates of convergence tail probabilities. |
| url | http://dx.doi.org/10.1155/S0161171292000644 |
| work_keys_str_mv | AT delili someresultsonconvergenceratesforprobabilitiesofmoderatedeviationsforsumsofrandomvariables AT xiangchenwang someresultsonconvergenceratesforprobabilitiesofmoderatedeviationsforsumsofrandomvariables AT mbhaskararao someresultsonconvergenceratesforprobabilitiesofmoderatedeviationsforsumsofrandomvariables |