Geometric Sums: Bounds for Rare Events with Applications

Калашников, В. В.

doi:10.1007/978-94-017-1693-2

Cited by 225 publications

(199 citation statements)

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“…It follows from the limit theory for random sums (Brown 1990, Kalashnikov 1997, Kozubowski & Panorska 1998 that if the number of terms in the summation, N, follows a geometric distribution with mean 1/p, then under certain conditions when the number of terms increases (p → 0),…”

Section: Stochastic Modelmentioning

confidence: 99%

“…In our case, this translates into the following question: How small does p have to be for the exponential approximation of the random sum to be reasonable? Like any other rate of convergence problem, this is a difficult mathematical question, and there are few results and only for special cases (Brown 1990, Kalashnikov 1997; see Appendix 1 for additional details). In practice, it is recommended to employ goodness-of-fit statistics for the exponential match to the distribution of regime magnitude.…”

Section: Stochastic Modelmentioning

confidence: 99%

“…Other extensive applications of random sums are in reliability and queuing theories (e.g. Kalashnikov 1997). We begin by quantifying multiannual time series events in terms of 3 random variables (parameters).…”

Section: Introductionmentioning

confidence: 99%

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Stochastic modeling of regime shifts

Biondi¹,

Kozubowski²,

Panorska³

2002

Clim. Res.

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Probabilistic methods for modeling the distribution of regimes and their shifts over time are developed by drawing on statistical decision and limit theory of random sums. Multi-annual episodes of opposite sign are graphically and numerically represented by their duration, magnitude, and intensity. Duration is defined as the number of consecutive years above or below a reference line, magnitude is the sum of time series values for any given duration, and intensity is the ratio between magnitude and duration. Assuming that a regime shift can occur every year, independently of prior years, the waiting times for the regime shift (or regime duration) are naturally modeled by a geometric distribution. Because magnitude can be expressed as a random sum of N random variables (where N is duration), its probability distribution is mathematically derived and can be statistically tested. Here we analyze a reconstructed time series of the Pacific Decadal Oscillation (PDO), explicitly describe the geometric, exponential, and Laplace probability distributions for regime duration and magnitude, and estimate parameters from the data obtaining a reasonably good fit. This stochastic approach to modeling duration and magnitude of multi-annual events enables the computation of probabilities of climatic episodes, and it provides a rigorous solution to deciding whether 2 regimes are significantly different from one another.

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Section: Stochastic Modelmentioning

confidence: 99%

Section: Stochastic Modelmentioning

confidence: 99%

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Stochastic modeling of regime shifts

Biondi¹,

Kozubowski²,

Panorska³

2002

Clim. Res.

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show abstract

“…Let C k denote the length of the kth cycle, where the first cycle starts at time C 0 = τ z . Let S k = k i=0 C i and let K(t) denote the current cycle at time t. Then S K(τy)−1 ≤ τ y = S K(τy)−1 + τ y , where τ y is distributed as the first hitting time of level y, starting from z and given that the process stays above z. Renyi's theorem states that if µ C = E(C 1 → denotes weak convergence and Z is an exponential random variable with unit mean (see the extended version given as Theorem 2.4 in [17]). Let ξ i = max{X t |t ∈ [S i , S i+1 ]} denote the ith cycle maximum and G(y) = P(ξ 1 ≤ y) = 1 − P(K(τ y ) = 1) the common distribution function of the ξ i .…”

mentioning

confidence: 99%

Hitting Times and the Running Maximum of Markovian Growth-Collapse Processes

Löpker

Stadje

2011

Journal of Applied Probability

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We consider a Markovian growth collapse process on the state space E = [0, ∞) which evolves as follows. Between random downward jumps the process increases with slope one. Both the jump intensity and the jump sizes depend on the current state of the process. We are interested in the behavior of the first hitting time τ y = inf{t ≥ 0|X t = y} as y becomes large and the growth of the maximum process M t = sup{X s |0 ≤ s ≤ t} as t → ∞. We consider the recursive sequence of equations Am n = m n−1 , m 0 ≡ 1, where A is the extended generator of the MGCP, and show that the solution sequence (which is essentially unique and can be given in integral form) is related to the moments of τ y . The Laplace transform of τ y can be expressed in closed form (in terms of an integral involving a certain kernel) in a similar way. We derive asymptotic results for the running maximum: (i) if m 1 (y) is of rapid variation, we have M t /m −1 (t)

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“…non-negative random variables, if P g (x CR ) is sufficient small (i.e. R << a), by applying the Rényi Theorem [19], the proof follows:…”

Section: Appendix B Proof Of Propositionmentioning

confidence: 99%

Optimal Primary-User Mobility Aware Spectrum Sensing Design for Cognitive Radio Networks

Cacciapuoti

Akyildiz

Paura

2013

IEEE J. Select. Areas Commun.

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Abstract-A key issue of the spectrum sensing functionality in Cognitive Radio (CR) networks is the ability of tuning the sensing time parameters, i.e., the sensing time and the transmission time, according to the Primary User (PU) network dynamics. In fact, these parameters influence both the spectrum sensing efficiency and the PU interference avoidance. This issue becomes even more challenging in presence of PU mobility. In this paper, an optimal spectrum sensing design for mobile PU scenarios is proposed with the aim to achieve the following important features: i) to determine the optimal mobility-aware transmission time, i.e., the transmission time value that jointly maximizes the spectrum sensing efficiency and satisfies the PU interference constraint; ii) to determine the optimal mobility-aware sensing time threshold, i.e., the maximum sensing time value assuring efficient spectrum sensing. First, closed-form expressions of both the optimal transmission time and the optimal sensing time threshold are analytically derived for a general PU mobility model. Then, the derived expressions are specialized for two widely adopted mobility models, i.e., the Random Walk mobility Model with reflection and the Random Way-Point mobility Model. Practical rules for the sensing parameter tuning are provided with reference to the considered mobility models. The analytical results are finally validated through simulations.

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Geometric Sums: Bounds for Rare Events with Applications

Cited by 225 publications

References 0 publications

Stochastic modeling of regime shifts

Stochastic modeling of regime shifts

Hitting Times and the Running Maximum of Markovian Growth-Collapse Processes

Optimal Primary-User Mobility Aware Spectrum Sensing Design for Cognitive Radio Networks

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