Limit theorems for correlated Bernoulli random variables

James, Barry R.; James, Kang Ling; Qi, Yongcheng

doi:10.1016/j.spl.2008.01.104

Cited by 37 publications

(31 citation statements)

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“…In [9], {X i } is called a correlated Bernoulli process and the distribution of H n := #{i = 1, · · · , n : X i = +1} the generalized binomial distribution with density ρ := 1 + ε 2 . In this context, various limit theorems for {H n } are obtained by [14,15,19,20]. Note that those results have a counterpart for the elephant random walk by a simple relation S n = H n − (n − H n ) = 2H n − n.…”

Section: Introductionmentioning

confidence: 97%

Gaussian Fluctuation for Superdiffusive Elephant Random Walks

Kubota

Takei

2019

J Stat Phys

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Elephant random walk is a kind of one-dimensional discretetime random walk with infinite memory: For each step, with probability α the walker adopts one of his/her previous steps uniformly chosen at random, and otherwise he/she performs like a simple random walk (possibly with bias). It admits phase transition from diffusive to superdiffusive behavior at the critical value αc = 1/2. For α ∈ (αc, 1), there is a scaling factor an of order n α such that the position Sn of the walker at time n scaled by an converges to a nondegenerate random variable W , whose distribution is not Gaussian. Our main result shows that the fluctuation of Sn around W · an is still Gaussian. We also give a description of phase transition induced by bias decaying polynomially in time.

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Section: Introductionmentioning

confidence: 97%

Gaussian Fluctuation for Superdiffusive Elephant Random Walks

Kubota

Takei

2019

J Stat Phys

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“…The simplest such example is a trivial MDP with only one action available at each decision time and with a reward process that is given by a dependent Bernoulli process. The reader is referred to James, James andQi (2008, p. 2341) for an explicit example of such a process.…”

Section: When Optimal Action Monotonicity Fails: a Counterexamplementioning

confidence: 99%

Markov Decision Problems Where Means Bound Variances

Arlotto

Gans

Steele

2014

Operations Research

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We identify a rich class of finite-horizon Markov decision problems (MDPs) for which the variance of the optimal total reward can be bounded by a simple linear function of its expected value. The class is characterized by three natural properties: reward non-negativity and boundedness, existence of a do-nothing action, and optimal action monotonicity. These properties are commonly present and typically easy to check. Implications of the class properties and of the variance bound are illustrated by examples of MDPs from operations research, operations management, financial engineering, and combinatorial optimization.

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“…When θ j > 0, the (j + 1) variate is expected to have a larger success probability than p provided that the average probability of success for the previous j variates is greater than p. James et al (2008) study limit theorems of this class of distributions. We generalize this model and specify…”

Section: Qaqish (2003) Considers the Model E(mentioning

confidence: 99%

High-dimensional generation of Bernoulli random vectors

Modarres

2011

Statistics & Probability Letters

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Limit theorems for correlated Bernoulli random variables

Cited by 37 publications

References 2 publications

Gaussian Fluctuation for Superdiffusive Elephant Random Walks

Gaussian Fluctuation for Superdiffusive Elephant Random Walks

Markov Decision Problems Where Means Bound Variances

High-dimensional generation of Bernoulli random vectors

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