Exact bounds for tail probabilities of martingales with bounded differences
We consider random walks, say Wn = {0, M1, . . ., Mn} of length n starting at 0 and based on a martingale sequence Mk = X1 + ··· + Xk with differences Xm. Assuming |Xk| \leq 1 we solve the isoperimetric problem Bn(x) = supP\{Wn visits an interval [x,∞)\}, (1) where sup is taken over all poss...
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| Format: | Article |
| Language: | English |
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Vilnius University Press
2009-12-01
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| Series: | Lietuvos Matematikos Rinkinys |
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| Online Access: | https://www.journals.vu.lt/LMR/article/view/18001 |
| _version_ | 1818382770932547584 |
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| author | Dainius Dzindzalieta |
| author_facet | Dainius Dzindzalieta |
| author_sort | Dainius Dzindzalieta |
| collection | DOAJ |
| description | We consider random walks, say Wn = {0, M1, . . ., Mn} of length n starting at 0 and based on a martingale sequence Mk = X1 + ··· + Xk with differences Xm. Assuming |Xk| \leq 1 we solve the isoperimetric problem
Bn(x) = supP\{Wn visits an interval [x,∞)\}, (1)
where sup is taken over all possible Wn. We describe random walks which maximize the probability in (1). We also extend the results to super-martingales.For martingales our results can be interpreted as a maximal
inequalities
P\{max 1\leq k\leq n Mk \geq x\} \leq Bn(x).
The maximal inequality is optimal since the equality is achieved by martingales related to the maximizing random walks. To prove the result we introduce a general principle – maximal inequalities for (natural classes of) martingales are equivalent to (seemingly weaker) inequalities for tail probabilities, in our case
Bn(x) = supP{Mn \geq x}.
Our methods are similar in spirit to a method used in [1], where a solution of an isoperimetric problem (1), for integer x is provided and to the method used in [4], where the isoperimetric problem of type (1) for conditionally symmetric bounded martingales was solved for all x ∈ R. |
| first_indexed | 2024-12-14T02:55:46Z |
| format | Article |
| id | doaj.art-553090f0d8fc46d8ac1161c7552a7d0d |
| institution | Directory Open Access Journal |
| issn | 0132-2818 2335-898X |
| language | English |
| last_indexed | 2024-12-14T02:55:46Z |
| publishDate | 2009-12-01 |
| publisher | Vilnius University Press |
| record_format | Article |
| series | Lietuvos Matematikos Rinkinys |
| spelling | doaj.art-553090f0d8fc46d8ac1161c7552a7d0d2022-12-21T23:19:38ZengVilnius University PressLietuvos Matematikos Rinkinys0132-28182335-898X2009-12-0150proc. LMS10.15388/LMR.2009.73Exact bounds for tail probabilities of martingales with bounded differencesDainius Dzindzalieta0Institute of Mathematics and InformaticsWe consider random walks, say Wn = {0, M1, . . ., Mn} of length n starting at 0 and based on a martingale sequence Mk = X1 + ··· + Xk with differences Xm. Assuming |Xk| \leq 1 we solve the isoperimetric problem Bn(x) = supP\{Wn visits an interval [x,∞)\}, (1) where sup is taken over all possible Wn. We describe random walks which maximize the probability in (1). We also extend the results to super-martingales.For martingales our results can be interpreted as a maximal inequalities P\{max 1\leq k\leq n Mk \geq x\} \leq Bn(x). The maximal inequality is optimal since the equality is achieved by martingales related to the maximizing random walks. To prove the result we introduce a general principle – maximal inequalities for (natural classes of) martingales are equivalent to (seemingly weaker) inequalities for tail probabilities, in our case Bn(x) = supP{Mn \geq x}. Our methods are similar in spirit to a method used in [1], where a solution of an isoperimetric problem (1), for integer x is provided and to the method used in [4], where the isoperimetric problem of type (1) for conditionally symmetric bounded martingales was solved for all x ∈ R.https://www.journals.vu.lt/LMR/article/view/18001random walksmaximal inequalitiesprobability to visit an intervallarge deviationsmartingalesuper-martingale |
| spellingShingle | Dainius Dzindzalieta Exact bounds for tail probabilities of martingales with bounded differences Lietuvos Matematikos Rinkinys random walks maximal inequalities probability to visit an interval large deviations martingale super-martingale |
| title | Exact bounds for tail probabilities of martingales with bounded differences |
| title_full | Exact bounds for tail probabilities of martingales with bounded differences |
| title_fullStr | Exact bounds for tail probabilities of martingales with bounded differences |
| title_full_unstemmed | Exact bounds for tail probabilities of martingales with bounded differences |
| title_short | Exact bounds for tail probabilities of martingales with bounded differences |
| title_sort | exact bounds for tail probabilities of martingales with bounded differences |
| topic | random walks maximal inequalities probability to visit an interval large deviations martingale super-martingale |
| url | https://www.journals.vu.lt/LMR/article/view/18001 |
| work_keys_str_mv | AT dainiusdzindzalieta exactboundsfortailprobabilitiesofmartingaleswithboundeddifferences |