An exact bound for tail probabilities for a class of conditionally symmetric bounded martingales
We consider the class, say ℳn,sym, of martingales Mn = X1 + ⋯ + Xn with conditionally symmetric bounded differences Xk such that |Xk | ≤ 1. We find explicitly a solution, say Dn(x), of the variational problem Dn(x) ≝ sup Mn ∈ℳn,sym ℙ {Mn ≥ x}. We show that this problem is equivalent to one when yo...
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| Format: | Article |
| Language: | English |
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Vilnius University Press
2023-09-01
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| Series: | Lietuvos Matematikos Rinkinys |
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| Online Access: | https://www.zurnalai.vu.lt/LMR/article/view/30792 |
| _version_ | 1797198470545670144 |
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| author | Dainius Dzindzalieta |
| author_facet | Dainius Dzindzalieta |
| author_sort | Dainius Dzindzalieta |
| collection | DOAJ |
| description |
We consider the class, say ℳn,sym, of martingales Mn = X1 + ⋯ + Xn with conditionally symmetric bounded differences Xk such that |Xk | ≤ 1. We find explicitly a solution, say Dn(x), of the variational problem Dn(x) ≝ sup Mn ∈ℳn,sym ℙ {Mn ≥ x}. We show that this problem is equivalent to one when you want to find out the symmetric random walk with bounded length of steps which maximizes the probability to visit an interval [x;∞]. The function x \mapsto Dn(x) allows a simple description and is closely related to the binomial tail probabilities. We can interpret the result as a final and optimal upper bound ℙ{Mn ≥ x} ≤ Dn(x), x ∈ ℝ, for the tail probability ℙ {Mn ≥ x}.
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| first_indexed | 2024-03-07T14:22:48Z |
| format | Article |
| id | doaj.art-ead85abd36a6433288745d330cdc1b18 |
| institution | Directory Open Access Journal |
| issn | 0132-2818 2335-898X |
| language | English |
| last_indexed | 2024-04-24T07:00:22Z |
| publishDate | 2023-09-01 |
| publisher | Vilnius University Press |
| record_format | Article |
| series | Lietuvos Matematikos Rinkinys |
| spelling | doaj.art-ead85abd36a6433288745d330cdc1b182024-04-22T09:00:38ZengVilnius University PressLietuvos Matematikos Rinkinys0132-28182335-898X2023-09-0146spec.10.15388/LMR.2006.30792An exact bound for tail probabilities for a class of conditionally symmetric bounded martingalesDainius Dzindzalieta0Institute of Mathematics and Informatics We consider the class, say ℳn,sym, of martingales Mn = X1 + ⋯ + Xn with conditionally symmetric bounded differences Xk such that |Xk | ≤ 1. We find explicitly a solution, say Dn(x), of the variational problem Dn(x) ≝ sup Mn ∈ℳn,sym ℙ {Mn ≥ x}. We show that this problem is equivalent to one when you want to find out the symmetric random walk with bounded length of steps which maximizes the probability to visit an interval [x;∞]. The function x \mapsto Dn(x) allows a simple description and is closely related to the binomial tail probabilities. We can interpret the result as a final and optimal upper bound ℙ{Mn ≥ x} ≤ Dn(x), x ∈ ℝ, for the tail probability ℙ {Mn ≥ x}. https://www.zurnalai.vu.lt/LMR/article/view/30792tail probabilitiesmartingalesrandomwalkisoperimetric |
| spellingShingle | Dainius Dzindzalieta An exact bound for tail probabilities for a class of conditionally symmetric bounded martingales Lietuvos Matematikos Rinkinys tail probabilities martingales randomwalk isoperimetric |
| title | An exact bound for tail probabilities for a class of conditionally symmetric bounded martingales |
| title_full | An exact bound for tail probabilities for a class of conditionally symmetric bounded martingales |
| title_fullStr | An exact bound for tail probabilities for a class of conditionally symmetric bounded martingales |
| title_full_unstemmed | An exact bound for tail probabilities for a class of conditionally symmetric bounded martingales |
| title_short | An exact bound for tail probabilities for a class of conditionally symmetric bounded martingales |
| title_sort | exact bound for tail probabilities for a class of conditionally symmetric bounded martingales |
| topic | tail probabilities martingales randomwalk isoperimetric |
| url | https://www.zurnalai.vu.lt/LMR/article/view/30792 |
| work_keys_str_mv | AT dainiusdzindzalieta anexactboundfortailprobabilitiesforaclassofconditionallysymmetricboundedmartingales AT dainiusdzindzalieta exactboundfortailprobabilitiesforaclassofconditionallysymmetricboundedmartingales |