Large deviations for weighted random sums
In the present paper we consider weighted random sums ZN = ∑j=1 N ajXj, where 0 <= aj < ∞, N denotes a non-negative integer-valued random variable, and {X, Xj , j = 1, 2,...} is a family of independent identically distributed random variables with mean EX = µ and variance DX = σ2 > 0. Throu...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Vilnius University Press
2013-04-01
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| Series: | Nonlinear Analysis |
| Subjects: | |
| Online Access: | http://www.journals.vu.lt/nonlinear-analysis/article/view/14017 |
| _version_ | 1819018392815796224 |
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| author | Aurelija Kasparavičiūtė Leonas Saulis |
| author_facet | Aurelija Kasparavičiūtė Leonas Saulis |
| author_sort | Aurelija Kasparavičiūtė |
| collection | DOAJ |
| description | In the present paper we consider weighted random sums ZN = ∑j=1 N ajXj, where 0 <= aj < ∞, N denotes a non-negative integer-valued random variable, and {X, Xj , j = 1, 2,...} is a family of independent identically distributed random variables with mean EX = µ and variance DX = σ2 > 0. Throughout this paper N is independent of {X, Xj , j = 1, 2,...} and, for definiteness, it is assumed Z0 = 0. The main idea of the paper is to present results on theorems of large deviations both in the Cramér and power Linnik zones for a sum &Ztilde;N = (ZN − EZN )(DZN )−1/2 , exponential inequalities for a tail probability P(ZN > x) in two cases: µ = 0 and µ ≠ 0 pointing out the difference between them. Only normal approximation is considered. It should be noted that large deviations when µ ≠ 0 have been already considered in our papers. |
| first_indexed | 2024-12-21T03:18:42Z |
| format | Article |
| id | doaj.art-febe2967b5504531b5867504eddf14e3 |
| institution | Directory Open Access Journal |
| issn | 1392-5113 2335-8963 |
| language | English |
| last_indexed | 2024-12-21T03:18:42Z |
| publishDate | 2013-04-01 |
| publisher | Vilnius University Press |
| record_format | Article |
| series | Nonlinear Analysis |
| spelling | doaj.art-febe2967b5504531b5867504eddf14e32022-12-21T19:17:46ZengVilnius University PressNonlinear Analysis1392-51132335-89632013-04-01182Large deviations for weighted random sumsAurelija Kasparavičiūtė0Leonas Saulis1Vilnius Gediminas Technical University, LithuaniaVilnius Gediminas Technical University, LithuaniaIn the present paper we consider weighted random sums ZN = ∑j=1 N ajXj, where 0 <= aj < ∞, N denotes a non-negative integer-valued random variable, and {X, Xj , j = 1, 2,...} is a family of independent identically distributed random variables with mean EX = µ and variance DX = σ2 > 0. Throughout this paper N is independent of {X, Xj , j = 1, 2,...} and, for definiteness, it is assumed Z0 = 0. The main idea of the paper is to present results on theorems of large deviations both in the Cramér and power Linnik zones for a sum &Ztilde;N = (ZN − EZN )(DZN )−1/2 , exponential inequalities for a tail probability P(ZN > x) in two cases: µ = 0 and µ ≠ 0 pointing out the difference between them. Only normal approximation is considered. It should be noted that large deviations when µ ≠ 0 have been already considered in our papers.http://www.journals.vu.lt/nonlinear-analysis/article/view/14017cumulantrandom sumslarge deviation theoremsnormal approximation |
| spellingShingle | Aurelija Kasparavičiūtė Leonas Saulis Large deviations for weighted random sums Nonlinear Analysis cumulant random sums large deviation theorems normal approximation |
| title | Large deviations for weighted random sums |
| title_full | Large deviations for weighted random sums |
| title_fullStr | Large deviations for weighted random sums |
| title_full_unstemmed | Large deviations for weighted random sums |
| title_short | Large deviations for weighted random sums |
| title_sort | large deviations for weighted random sums |
| topic | cumulant random sums large deviation theorems normal approximation |
| url | http://www.journals.vu.lt/nonlinear-analysis/article/view/14017 |
| work_keys_str_mv | AT aurelijakasparaviciute largedeviationsforweightedrandomsums AT leonassaulis largedeviationsforweightedrandomsums |