Distribution of Patterns of Constrained Length in Binary Sequences
On a finite sequence of binary (0-1) trials we define a random variable enumerating patterns of length subject to certain constraints. For sequences of independent and identically distributed binary trials exact probability mass functions are established in closed forms by means of combinatorial ana...
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| Published in | Methodology and computing in applied probability Vol. 25; no. 4; p. 90 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.12.2023
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1387-5841 1573-7713 |
| DOI | 10.1007/s11009-023-10068-5 |
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| Abstract | On a finite sequence of binary (0-1) trials we define a random variable enumerating patterns of length subject to certain constraints. For sequences of independent and identically distributed binary trials exact probability mass functions are established in closed forms by means of combinatorial analysis. An explicit expression of the mean value of this random variable is obtained. The results associated with the probability mass functions are extended on sequences of exchangeable binary trials. An application in Information theory concerning counting of a class of run-length-limited binary sequences is provided as a direct byproduct of our study. Illustrative numerical examples exemplify further the results. |
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| AbstractList | On a finite sequence of binary (0-1) trials we define a random variable enumerating patterns of length subject to certain constraints. For sequences of independent and identically distributed binary trials exact probability mass functions are established in closed forms by means of combinatorial analysis. An explicit expression of the mean value of this random variable is obtained. The results associated with the probability mass functions are extended on sequences of exchangeable binary trials. An application in Information theory concerning counting of a class of run-length-limited binary sequences is provided as a direct byproduct of our study. Illustrative numerical examples exemplify further the results. |
| ArticleNumber | 90 |
| Author | Makri, Frosso S. Psillakis, Zaharias M. |
| Author_xml | – sequence: 1 givenname: Frosso S. surname: Makri fullname: Makri, Frosso S. email: makri@math.upatras.gr organization: Department of Mathematics, University of Patras – sequence: 2 givenname: Zaharias M. surname: Psillakis fullname: Psillakis, Zaharias M. organization: Department of Physics, University of Patras |
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| Cites_doi | 10.1080/03610918.2019.1642486 10.1002/j.1538-7305.1948.tb00917.x 10.1007/s11009-020-09810-0 10.1109/ISIT.2006.262115 10.1016/0167-7152(88)90069-7 10.2307/3215201 10.1080/00207720903144537 10.1214/aop/1022677272 10.1016/j.camwa.2010.12.023 10.1080/00224065.2019.1571334 10.1007/s00362-012-0462-1 10.1239/jap/1261670698 10.1016/j.jspi.2011.10.015 10.1214/aoap/1019487356 10.1007/s00184-021-00842-1 10.1007/978-1-4757-3460-7 10.1007/978-1-4614-8414-1_48-1 10.2174/1876527001708010001 10.1147/rd.144.0376 10.1109/RAIT.2012.6194501 10.1007/s11009-022-09948-z 10.1007/s00184-018-0668-x 10.1016/S0019-9958(70)90369-4 10.1016/j.camwa.2009.07.057 10.1142/4669 10.1239/aap/1198177236 10.1007/s00362-010-0340-7 10.1016/j.spl.2017.11.018 |
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| Copyright | The Author(s) 2023 The Author(s) 2023. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
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| Keywords | Constrained binary patterns and codes 60C05 Exact distributions 60G09 Exchangeable trials 62E12 Independent and identical trials Combinatorial analysis Runs 60E05 |
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| References | GeraASimultaneous demonstration tests involving sparse failuresStatist Probab Lett20181352631375825710.1016/j.spl.2017.11.018 MakriFSPsillakisZMCounting certain binary stringsJ Stat Plan Inference2012142908924286387910.1016/j.jspi.2011.10.015 DafnisSDPhilippouANAntzoulakosDLDistributions of patterne of two successes separated by a string of k-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k-2$$\end{document} failuresStat Pap20125332334410.1007/s00362-010-0340-7 SenKGoyalBDistributions of patterns of two failures separated by success runs of length k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}J Korean Stat Soc20043335582059206 GeskeMXGodboleAPSchaffnerAAScolnickAMWallstromGLCompound Poisson approximations for word patterns under Markovian hypothesesJ Appl Probab199532877892136333010.2307/3215201 Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423, 623–656 MakriFSPhilippouANPsillakisZMSuccess run statistics defined on an urn modelAdv Appl Probab2007399911019238158510.1239/aap/1198177236 FranaszekPASequence-state methods for run-length-limited codingIBM J Res Dev19701437638326941710.1147/rd.144.0376 SinhaKSinhaBPOn the distribution of ones in binary stringsComput Math with Appl20095818161829255756010.1016/j.camwa.2009.07.057 DafnisSDMakriFSKoutrasMVGeneralizations of runs and patterns distributions for sequences of binary trialsMethodol Comput Appl Probab202123165185422491010.1007/s11009-020-09810-0 EryilmazSStatistical inference for a class of start-up demonstration testsJ Qual Technol20195131432410.1080/00224065.2019.1571334 MakriFSPsillakisZMOn limited length binary strings with an application in statistical controlThe Open Statistics & Probability Journal201781610.2174/1876527001708010001 GeraAFrom runs to patternsCommun Stat Simul Comput20215043004314434332310.1080/03610918.2019.1642486 EryilmazSZuoMConstrained (k,d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k, d)$$\end{document}-out-of-n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} systemsInt J Syst Sci201041679685266273710.1080/00207720903144537 ZhaoXSongYLvZDistributions of (k1,k2,…,km)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k_{1}, k_{2},\ldots, k_{m})$$\end{document}-runs with multi-state trialsMethodol Comput Appl Probab20222426892702452839810.1007/s11009-022-09948-z BalakrishnanNKoutrasMVRuns and scans with applications2002New YorkWiley HolstLOn consecutive records in certain binary sequencesJ Appl Probab20094612011208258271510.1239/jap/1261670698 GlazJNaussJWallensteinSScan statistics2001New YorkSpringer10.1007/978-1-4757-3460-7 StefanovVTSzpankowskiWWaiting time distributions for patterns occurrence in a constrained sequenceDiscret Math Theor Comput Sci200793053202369148 ErhardssonTCompound Poisson approximation for Markov chains using Stein’s methodAnn Probab199927565596168114910.1214/aop/1022677272 ImminkKASCodes for mass data storage systems20042Eindhoven, The NetherlandsShannon Foundation Publishers LingKDOn binomial distributions of order k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}Statist Probab Lett1988624725092053410.1016/0167-7152(88)90069-7 FuJCLouWYWDistribution theory of runs and patterns and its applications: a finite Markov chain imbedding approach2003River EdgeWorld Scientific10.1142/4669 JohnsonNKotzSUrn models and their applications1977New YorkJohn Wiley MakriFSPsillakisZMOn success runs of a fixed length in Bernoulli sequences: exact and asymptotic resultsComput Math with Appl201161761772277048010.1016/j.camwa.2010.12.023 ErhardssonTCompound Poisson approximation for counts of rare patterns in Markov Chains and extreme sojourns in birth-death chainsAnn Appl Probab200010573591176822210.1214/aoap/1019487356 Makri FS, Psillakis ZM (2019) On the exact distributions of pattern statistics for a sequence of binary trials: a combinatorial approach. In: Glaz J, Koutras MV (eds) Handbook of Scan Statistics. pp 1-20. https://doi.org/10.1007/978-1-4614-8414-1_48-1 Sinha K, Sinha BP (2012) Energy efficient communication: understanding the distribution of runs in binary strings. In: 1st International Conference on Recent Advances in Information Technology (Rait-2012), pp 177–181 Jacquet P, Szpankowski W (2006) On (d,k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(d,k)$$\end{document} sequences not containing a given word. IEEE International Symposium on Information Theory (ISIT) Seatle Jul 2006, pp 1486–1489 DafnisSDMakriFSWeak runs in sequences of binary trialsMetrika202285573603442898810.1007/s00184-021-00842-1 Feller W (1968) An introduction to probability theory and its applications, vol I, 3rd edn. Wiley, New York KumarANUpadhyeNSGeneralizations of distributions related to (k1,k2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k_{1}, k_{2})$$\end{document}-runsMetrika201982249268392252910.1007/s00184-018-0668-x MakriFSPsillakisZMExact distributions of constrained (k,l)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k, l)$$\end{document} strings of failures between subsequent successesStat Pap201354783806307290110.1007/s00362-012-0462-1 TangDTBahlLRBlock codes for a class of constrained noiseless channelsInf Control19701743646127252210.1016/S0019-9958(70)90369-4 FS Makri (10068_CR22) 2007; 39 SD Dafnis (10068_CR3) 2021; 23 SD Dafnis (10068_CR4) 2012; 53 AN Kumar (10068_CR20) 2019; 82 DT Tang (10068_CR33) 1970; 17 S Eryilmaz (10068_CR7) 2019; 51 KD Ling (10068_CR21) 1988; 6 K Sen (10068_CR28) 2004; 33 S Eryilmaz (10068_CR8) 2010; 41 JC Fu (10068_CR11) 2003 10068_CR31 FS Makri (10068_CR24) 2012; 142 10068_CR29 MX Geske (10068_CR14) 1995; 32 10068_CR27 N Balakrishnan (10068_CR1) 2002 K Sinha (10068_CR30) 2009; 58 J Glaz (10068_CR15) 2001 L Holst (10068_CR16) 2009; 46 T Erhardsson (10068_CR5) 1999; 27 PA Franaszek (10068_CR10) 1970; 14 VT Stefanov (10068_CR32) 2007; 9 A Gera (10068_CR12) 2018; 135 N Johnson (10068_CR19) 1977 X Zhao (10068_CR34) 2022; 24 FS Makri (10068_CR25) 2013; 54 10068_CR18 T Erhardsson (10068_CR6) 2000; 10 FS Makri (10068_CR26) 2017; 8 A Gera (10068_CR13) 2021; 50 10068_CR9 KAS Immink (10068_CR17) 2004 FS Makri (10068_CR23) 2011; 61 SD Dafnis (10068_CR2) 2022; 85 |
| References_xml | – reference: GeskeMXGodboleAPSchaffnerAAScolnickAMWallstromGLCompound Poisson approximations for word patterns under Markovian hypothesesJ Appl Probab199532877892136333010.2307/3215201 – reference: BalakrishnanNKoutrasMVRuns and scans with applications2002New YorkWiley – reference: DafnisSDMakriFSWeak runs in sequences of binary trialsMetrika202285573603442898810.1007/s00184-021-00842-1 – reference: MakriFSPhilippouANPsillakisZMSuccess run statistics defined on an urn modelAdv Appl Probab2007399911019238158510.1239/aap/1198177236 – reference: LingKDOn binomial distributions of order k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}Statist Probab Lett1988624725092053410.1016/0167-7152(88)90069-7 – reference: EryilmazSZuoMConstrained (k,d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k, d)$$\end{document}-out-of-n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} systemsInt J Syst Sci201041679685266273710.1080/00207720903144537 – reference: MakriFSPsillakisZMExact distributions of constrained (k,l)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k, l)$$\end{document} strings of failures between subsequent successesStat Pap201354783806307290110.1007/s00362-012-0462-1 – reference: FranaszekPASequence-state methods for run-length-limited codingIBM J Res Dev19701437638326941710.1147/rd.144.0376 – reference: KumarANUpadhyeNSGeneralizations of distributions related to (k1,k2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k_{1}, k_{2})$$\end{document}-runsMetrika201982249268392252910.1007/s00184-018-0668-x – reference: StefanovVTSzpankowskiWWaiting time distributions for patterns occurrence in a constrained sequenceDiscret Math Theor Comput Sci200793053202369148 – reference: Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423, 623–656 – reference: DafnisSDMakriFSKoutrasMVGeneralizations of runs and patterns distributions for sequences of binary trialsMethodol Comput Appl Probab202123165185422491010.1007/s11009-020-09810-0 – reference: SinhaKSinhaBPOn the distribution of ones in binary stringsComput Math with Appl20095818161829255756010.1016/j.camwa.2009.07.057 – reference: JohnsonNKotzSUrn models and their applications1977New YorkJohn Wiley – reference: FuJCLouWYWDistribution theory of runs and patterns and its applications: a finite Markov chain imbedding approach2003River EdgeWorld Scientific10.1142/4669 – reference: HolstLOn consecutive records in certain binary sequencesJ Appl Probab20094612011208258271510.1239/jap/1261670698 – reference: GeraASimultaneous demonstration tests involving sparse failuresStatist Probab Lett20181352631375825710.1016/j.spl.2017.11.018 – reference: Makri FS, Psillakis ZM (2019) On the exact distributions of pattern statistics for a sequence of binary trials: a combinatorial approach. 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| Title | Distribution of Patterns of Constrained Length in Binary Sequences |
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