Approximating kth-order two-state Markov chains

1992 ◽  
Vol 29 (4) ◽  
pp. 861-868 ◽  
Author(s):  
Y. H. Wang
Keyword(s):  
Markov Chains ◽  
Total Variation ◽  
Upper Bound ◽  
Transition Probabilities ◽  
Random Variable ◽  
Poisson Random Variable ◽  
Compound Poisson ◽  
Stationary Transition

In this paper, we consider kth-order two-state Markov chains {Xi} with stationary transition probabilities. For k = 1, we construct in detail an upper bound for the total variation d(Sn, Y) = Σx |𝐏(Sn = x) − 𝐏(Y = x)|, where Sn = X1 + · ··+ Xn and Y is a compound Poisson random variable. We also show that, under certain conditions, d(Sn, Y) converges to 0 as n tends to ∞. For k = 2, the corresponding results are given without derivation. For general k ≧ 3, a conjecture is proposed.

Approximating kth-order two-state Markov chains

1992 ◽  
Vol 29 (04) ◽  
pp. 861-868 ◽  
Author(s):  
Y. H. Wang
Keyword(s):  
Markov Chains ◽  
Total Variation ◽  
Upper Bound ◽  
Transition Probabilities ◽  
Random Variable ◽  
Poisson Random Variable ◽  
Compound Poisson ◽  
Stationary Transition

In this paper, we consider kth-order two-state Markov chains {Xi } with stationary transition probabilities. For k = 1, we construct in detail an upper bound for the total variation d(Sn, Y) = Σ x |𝐏(Sn = x) − 𝐏(Y = x)|, where S n = X 1 + · ··+ Xn and Y is a compound Poisson random variable. We also show that, under certain conditions, d(Sn, Y) converges to 0 as n tends to ∞. For k = 2, the corresponding results are given without derivation. For general k ≧ 3, a conjecture is proposed.

Poisson approximation for a sum of dependent indicators: an alternative approach

2002 ◽  
Vol 34 (03) ◽  
pp. 609-625 ◽  
Author(s):  
N. Papadatos ◽  
V. Papathanasiou
Keyword(s):  
Total Variation ◽  
Upper Bound ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Birthday Problem ◽  
Alternative Approach ◽  
Variation Distance

The random variablesX1,X2, …,Xnare said to be totally negatively dependent (TND) if and only if the random variablesXiand ∑j≠iXjare negatively quadrant dependent for alli. Our main result provides, for TND 0-1 indicatorsX1,x2, …,Xnwith P[Xi= 1] =pi= 1 - P[Xi= 0], an upper bound for the total variation distance between ∑ni=1Xiand a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

Poisson approximation for a sum of dependent indicators: an alternative approach

2002 ◽  
Vol 34 (3) ◽  
pp. 609-625 ◽  
Author(s):  
N. Papadatos ◽  
V. Papathanasiou
Keyword(s):  
Total Variation ◽  
Upper Bound ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Birthday Problem ◽  
Alternative Approach ◽  
Variation Distance

The random variables X1, X2, …, Xn are said to be totally negatively dependent (TND) if and only if the random variables Xi and ∑j≠iXj are negatively quadrant dependent for all i. Our main result provides, for TND 0-1 indicators X1, x2, …, Xn with P[Xi = 1] = pi = 1 - P[Xi = 0], an upper bound for the total variation distance between ∑ni=1Xi and a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

Poisson approximation for (k1, k2)-events via the Stein-Chen method

2004 ◽  
Vol 41 (4) ◽  
pp. 1081-1092 ◽  
Author(s):  
P. Vellaisamy
Keyword(s):  
Limit Theorem ◽  
Rate Of Convergence ◽  
Upper Bound ◽  
Success Probability ◽  
Random Variable ◽  
Poisson Approximation ◽  
Bernoulli Trials ◽  
Poisson Random Variable ◽  
Markov Dependent Trials ◽  
Special Case

Consider a sequence of independent Bernoulli trials with success probability p. Let N(n; k1, k2) denote the number of times that k1 failures are followed by k2 successes among the first n Bernoulli trials. We employ the Stein-Chen method to obtain a total variation upper bound for the rate of convergence of N(n; k1, k2) to a suitable Poisson random variable. As a special case, the corresponding limit theorem is established. Similar results are obtained for Nk3(n; k1, k2), the number of times that k1 failures followed by k2 successes occur k3 times successively in n Bernoulli trials. The bounds obtained are generally sharper than, and improve upon, some of the already known results. Finally, the technique is adapted to obtain Poisson approximation results for the occurrences of the above-mentioned events under Markov-dependent trials.

Markov Chains with Stationary Transition Probabilities

1961 ◽  
Vol 45 (354) ◽  
pp. 362
Author(s):  
D. G. Kendall ◽  
Kai Lai Chung
Keyword(s):  
Markov Chains ◽  
Transition Probabilities ◽  
Stationary Transition

Uniform saddlepoint approximations

1988 ◽  
Vol 20 (3) ◽  
pp. 622-634 ◽  
Author(s):  
J. L. Jensen
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Tail Probability ◽  
Poisson Random Variable ◽  
Compound Poisson ◽  
Saddlepoint Approximations ◽  
The Mean ◽  
General Conditions

The validity of the saddlepoint expansion evaluated at the point y is considered in the limit y tending to ∞. This is done for the expansions of the density and of the tail probability of the mean of n i.i.d. random variables and also for the expansion of the tail probability of a compound Poisson sum , where N is a Poisson random variable. We consider both general conditions that ensure the validity of the expansions and study the four classes of densities for X1 introduced in Daniels (1954).

Poisson approximation for some statistics based on exchangeable trials

1983 ◽  
Vol 15 (3) ◽  
pp. 585-600 ◽  
Author(s):  
A. D. Barbour ◽  
G. K. Eagleson
Keyword(s):  
Total Variation ◽  
Limit Theorems ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Variation Distance ◽  
Image Position ◽  
Poisson Approximations

Stein's (1970) method of proving limit theorems for sums of dependent random variables is used to derive Poisson approximations for a class of statistics, constructed from finitely exchangeable random variables.Let be exchangeable random elements of a space and, for I a k-subset of , let XI be a 0–1 function. The statistics studied here are of the form where N is some collection of k -subsets of .An estimate of the total variation distance between the distributions of W and an appropriate Poisson random variable is derived and is used to give conditions sufficient for W to be asymptotically Poisson. Two applications of these results are presented.

scholarly journals On the Transition Law of Tempered Stable Ornstein–Uhlenbeck Processes

2009 ◽  
Vol 46 (3) ◽  
pp. 721-731 ◽  
Author(s):  
Shibin Zhang ◽  
Xinsheng Zhang
Keyword(s):  
Stable Distribution ◽  
Stochastic Integral ◽  
Random Variables ◽  
Transition Density ◽  
Random Variable ◽  
Independent Random Variables ◽  
Poisson Random Variable ◽  
Compound Poisson ◽  
Tempered Stable Distribution ◽  
Transition Distribution

In this paper, a stochastic integral of Ornstein–Uhlenbeck type is represented to be the sum of two independent random variables: one has a tempered stable distribution and the other has a compound Poisson distribution. In distribution, the compound Poisson random variable is equal to the sum of a Poisson-distributed number of positive random variables, which are independent and identically distributed and have a common specified density function. Based on the representation of the stochastic integral, we prove that the transition distribution of the tempered stable Ornstein–Uhlenbeck process is self-decomposable and that the transition density is a C∞-function.

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