Abstract
Let {X k ,J k } be a bivariate sequence of random variables, where J k is a finite ergodic Markov chain. Assume the random variables X k are conditionally independent given {Jk}. By decomposing \( {{S}_{n}} = \sum\nolimits_{{k = 1}}^{n} {{{X}_{k}}} \) the sum of i.i.d. random variables plus two ‘remainder’ terms, it is proved that Sn satisfies both the Strong Law of Large Numbers and the Law of the Iterated Logarithm under the conditions of finite first and second moments, respectively, of [Xk|Jk-1, Jk]
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© 1987 D. Reidel Publishing Company, Dordrecht, Holland
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Ramsay, C.M. (1987). Strong Limit Theorems for Sums of Random Variables Defined on a Finite Markov Chain. In: MacNeill, I.B., Umphrey, G.J., Bellhouse, D.R., Kulperger, R.J. (eds) Advances in the Statistical Sciences: Applied Probability, Stochastic Processes, and Sampling Theory. The University of Western Ontario Series in Philosophy of Science, vol 34. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-4786-3_14
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DOI: https://doi.org/10.1007/978-94-009-4786-3_14
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