402 KB31 KB200820252016Gajser, Davidletnik:10številka:2str. 393-410COBISSID:17736025ISSN:1855-3966URN:URN:NBN:SI:doc-FXRL3RVNenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneabinomial meanbinomska sredinaCesaro meanCesarova sredinaconvergencefinite Markov chainkončna Markovska verigakonvergencasequencezaporedjeOn convergence of binomial means, and an application to finite Markov chains|For a sequence ?$\{a_n\}_{n \ge 0}$? of real numbers, we define the sequence of its arithmetic means ?$\{a_n^\ast\}_{n \ge 0}$? as the sequence of averages of the first ?$n$? elements of ?$\{a_n\}_{n \ge 0}$?. For a parameter ?$0 < p < 1$?, we define the sequence of ?$p$?-binomial means ?$\{a_n^p\}_{n \ge 0}$? of the sequence ?$\{a_n\}_{n \ge 0}$? as the sequence of ?$p$?-binomially weighted averages of the first ?$n$? elements of ?$\{a_n\}_{n \ge 0}$?. We compare the convergence of sequences ?$\{a_n\}_{n \ge 0}$?, ?$\{a_n^\ast\}_{n \ge 0}$? and ?$\{a_n^p\}_{n \ge 0}$? for various ?$0 < p < 1$?, i.e., we analyze when the convergence of one sequence implies the convergence of the other. While the sequence ?$\{a_n^\ast\}_{n \ge 0}$?, known also as the sequence of Cesaro means of a sequence, is well studied in the literature, the results about ?$\{a_n^p\}_{n \ge 0}$? are hard to find. Our main result shows that, if ?$\{a_n\}_{n \ge 0}$? is a sequence of non-negative real numbers such that ?$\{a_n^p\}_{n \ge 0}$? converges to ?$a \in \mathbb{R} \cup \{\infty\}$? for some ?$0 < p < 1$, then $\{a_n^p\}_{n \ge 0}$? also converges to ?$a$?. We give an application of this result to finite Markov chainsZa zaporedje ?$\{a_n\}_{n \ge 0}$? realnih števil definiramo zaporedje njegovih aritmetičnih sredin ?$\{a_n^\ast\}_{n \ge 0}$? kot zaporedje povprečij prvih ?$n$? členov zaporedja ?$\{a_n\}_{n \ge 0}$?. Za parameter ?$0 < p < 1$? definiramo zaporedje ?$p$?-binomskih sredin ?$\{a_n^p\}_{n \ge 0}$? zaporedja ?$\{a_n\}_{n \ge 0}$? kot zaporedje ?$p$?-binomsko uteženih povprečij prvih ?$n$? členov zaporedja ?$\{a_n\}_{n \ge 0}$?. Primerjamo konvergenco zaporedij ?$\{a_n\}_{n \ge 0}$?, ?$\{a_n^\ast\}_{n \ge 0}$? in ?$\{a_n^p\}_{n \ge 0}$? za različne ?$0 < p < 1$? oziroma analiziramo, kdaj konvergenca enega zaporedja implicira konvergenco drugega. Medtem ko je zaporedje ?$\{a_n^\ast\}_{n \ge 0}$?, znano tudi kot zaporedje Cesarovih sredin zaporedja, dobro raziskano v literaturi, pa je rezultate v zvezi z ?$\{a_n^p\}_{n \ge 0}$? težko najti. Naš glavni rezultat pokaže, da če je ?$\{a_n\}_{n \ge 0}$? zaporedje nenegativnih realnih števil in ?$\{a_n^p\}_{n \ge 0}$? konvergira k ?$a \in \mathbb{R} \cup \{\infty\}$? za neki ?$0 < p < 1$?, potem ?$\{a_n^\ast\}_{n \ge 0}$? prav tako konvergira k ?$a$?. Podamo aplikacijo tega rezultata v končnih Markovskih verigahTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije