Longest runs in a sequence ofm-dependent random variables
Article
Received:
Revised:
- 81 Downloads
- 18 Citations
Summary
Let {X i ,i≧1} be a sequence of random 0's and 1's. We define the lengthR n of the longest head run as follows:
$$R_n = \max \left\{ {k \leqq n:\mathop {\max }\limits_{0 \leqq i \leqq n - k} 1\{ X_{i + 1} = \cdots = X_{i + k} = 1\} = 1} \right\}$$
The limit law for the distribution ofR n asn→∞ was found in 1943. The rate of convergence in that limit theorem remained unknown until recently even for Bernoulli independent trials. Supposing that {X i ,i≧1} is a stationary sequence ofm-dependent r.v.'s, we given an estimate on the rate of convergence in that limit theorem. This estimate provides the unimprovable raten−1(lnn) for the independent case. A result of lim inf type is obtained too.
Keywords
Stochastic Process Probability Theory Limit Theorem Mathematical Biology Stationary Sequence
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Preview
Unable to display preview. Download preview PDF.
References
- 1.Anisimov, V.V., Chernyak, A.I.: Limit theorems for certain rare functionals on Markov chains and semimarkov processes. Theory Probab. Math. Stat.26, 1–6 (1982)Google Scholar
- 2.Arratia, R., Golgstein, L., Gordon, L.: Two moments suffice for Poisson approximation. Ann. Probab.17, 9–25 (1989)Google Scholar
- 3.Arratia, R., Gordon, L., Waterman, M.S.: The Erdös-Renyi law in distribution, for coin tossing and sequence matching. Ann. Stat.18, 539–570 (1990)Google Scholar
- 4.Erdös, P., Renyi, A.: On a new law of large numbers. J. Anal. Math.23, 103–111 (1970)Google Scholar
- 5.Erdös, P., Révész, P.: On the length of the longest head-run. Colloq. Math. Soc. Janos Bolyai16, 219–229 (1975)Google Scholar
- 6.Földes, A.: The limit distribution of the length of the longest head-run. Trans. 8th Prague Conf. Inform. Theory Stat. Decis. Funct. Random Processes, vol.C, pp. 95–104 (1979)Google Scholar
- 7.Goncharov, V.L.: On the field of combinatory analysis. Am. Math. Soc. Transl.19(2), 1–46 (1943)Google Scholar
- 8.Gordon, L., Schilling, M.F., Waterman, M.S.: An extreme value theory for long head-run. Probab. Theory Relat. Field72(2), 279–287 (1986)Google Scholar
- 9.Grill, K.: Erdös-Révész type bounds for the length of the longest run from a stationary mixing sequence. Probab. Theory Relat. Fields75(1), 169–179 (1987)Google Scholar
- 10.Guibas, L.J., Odlyzko, A.M.: Long repetitive patterns in random sequence. Z. Wahrscheinlichkeitstheor. Verw. Geb.53(3), 241–262 (1980)Google Scholar
- 11.Kusolitsch, N.: Longest runs in Markov chains. In: Grossman, W. et al. (eds.) Probability and statistical inference, pp. 223–230. Dordrecht: Reidel 1982Google Scholar
- 12.De Moivre, A.: Doctrine of chances. London: 1738Google Scholar
- 13.Novak, S.Yu.: Time intervals of constant sojourne of a homogeneous Markov chain in a fixed subset of states. Sib. Math. J.29(1), 100–109 (1988)Google Scholar
- 14.Novak, S.Yu.: Asymptotic expansions in the problem of the longest head-run for Markov chain with two states. Tr. Inst. Mat.13, 136–147 (1989)Google Scholar
- 15.Novak, S.Yu.: On the length of the longest increasing run. In: 1st Intern. Congr. Bernoulli Soc. Math. Stat. Probab. Theory Abstracts Commun., Vol. 2, p. 767. Tashkent, 1986Google Scholar
- 16.Pittel, B.G.: Limiting behavior of a process of runs. Ann. Probab.9, 119–129 (1981)Google Scholar
- 17.Rajarshi, M.B.: Success runs in a two-state Markov chain. J. Appl. Probab.11, 190–192 (1974)Google Scholar
- 18.Révész, P.: Three problems on the lengths of increasing runs. Stochastic Processes Appl.15, 169–179 (1983)Google Scholar
- 19.Samarova, S.S.: On the length of the longest head-run for Markov chain with two states. Theory Probab. Appl.26(3), 499–509 (1981)Google Scholar
- 20.Variakoyis, L. On the maximal length of “success” run in a countable Markov chain. Lit. Math. J.26(4), 605–615 (1986)Google Scholar
Copyright information
© Springer-Verlag 1992