The purpose of this application paper is to apply the Stein-Chen (SC) method to provide a Poisson-based approximation and corresponding total variation distance bounds in a time series context. The SC method that is u...The purpose of this application paper is to apply the Stein-Chen (SC) method to provide a Poisson-based approximation and corresponding total variation distance bounds in a time series context. The SC method that is used approximates the probability density function (PDF) defined on how many times a pattern such as I<sub>t</sub>,I<sub>t</sub><sub>+1</sub>,I<sub>t</sub><sub>+2</sub> = {1 0 1} occurs starting at position t in a time series of length N that has been converted to binary values using a threshold. The original time series that is converted to binary is assumed to consist of a sequence of independent random variables, and could, for example, be a series of residuals that result from fitting any type of time series model. Note that if {1 0 1} is known to not occur, for example, starting at position t = 1, then this information impacts the probability that {1 0 1} occurs starting at position t = 2 or t = 3, because the trials to obtain {1 0 1} are overlapping and thus not independent, so the Poisson distribution assumptions are not met. Nevertheless, the results shown in four examples demonstrate that Poisson-based approximation (that is strictly correct only for independent trials) can be remarkably accurate, and the SC method provides a bound on the total variation distance between the true and approximate PDF.展开更多
This paper discusses the asymptotic behaviors of the longest run on a countable state Markov chain. Let {Xa}a∈Z＋ be a stationary strongly ergodic reversible Markov chain on countable- state space S = {1, 2,...}. Let...This paper discusses the asymptotic behaviors of the longest run on a countable state Markov chain. Let {Xa}a∈Z＋ be a stationary strongly ergodic reversible Markov chain on countable- state space S = {1, 2,...}. Let T C S be an arbitrary finite subset of S. Denote by Ln the length of the longest run of consecutive i＇s for i E T, that occurs in the sequence X1,..., Xn. In this paper, we obtain a limit law and a week version of an Erd6s Rdnyi type law for Ln. A large deviation result of Ln is also discussed.展开更多
文摘The purpose of this application paper is to apply the Stein-Chen (SC) method to provide a Poisson-based approximation and corresponding total variation distance bounds in a time series context. The SC method that is used approximates the probability density function (PDF) defined on how many times a pattern such as I<sub>t</sub>,I<sub>t</sub><sub>+1</sub>,I<sub>t</sub><sub>+2</sub> = {1 0 1} occurs starting at position t in a time series of length N that has been converted to binary values using a threshold. The original time series that is converted to binary is assumed to consist of a sequence of independent random variables, and could, for example, be a series of residuals that result from fitting any type of time series model. Note that if {1 0 1} is known to not occur, for example, starting at position t = 1, then this information impacts the probability that {1 0 1} occurs starting at position t = 2 or t = 3, because the trials to obtain {1 0 1} are overlapping and thus not independent, so the Poisson distribution assumptions are not met. Nevertheless, the results shown in four examples demonstrate that Poisson-based approximation (that is strictly correct only for independent trials) can be remarkably accurate, and the SC method provides a bound on the total variation distance between the true and approximate PDF.
文摘This paper discusses the asymptotic behaviors of the longest run on a countable state Markov chain. Let {Xa}a∈Z＋ be a stationary strongly ergodic reversible Markov chain on countable- state space S = {1, 2,...}. Let T C S be an arbitrary finite subset of S. Denote by Ln the length of the longest run of consecutive i＇s for i E T, that occurs in the sequence X1,..., Xn. In this paper, we obtain a limit law and a week version of an Erd6s Rdnyi type law for Ln. A large deviation result of Ln is also discussed.
