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• W2964175364 abstract "Abstract—The generalized binomial distribution in Tsallisstatistics (power-law system) is explicitly formulated from theprecise q-Stirling’s formula. Theα-divergence (orq-divergence)is uniquely derived from the generalized binomial distributionin the sense that when α → −1 (i.e., q → 1) it recovers KLdivergence obtained from the standard binomial distribution.Based on these combinatorial considerations, it is shown thatα-divergence (or q-divergence) is appeared as the generalizedrate function in the large deviation estimate in Tsallis statistics. I. I NTRODUCTION The large deviation principle (LDP for short) has mathemat-ically presented and quantified the asymptotic behavior of t heprobabilities of rare events in many stochastic phenomena. Ithas brought about deep significant insights for understandi ngof each phenomena [1][2][3]. The LDP covers quite broadareas ranging from the fundamentals in probability theoryand statistics to its applications such as statistical physics[4][5], risk management [6], information theory [7] and soon. In most of theoretical results in LDP, the assumption of“i.i.d. (independent and identically distributed)” for ra ndomvariables is used. This assumption leads to the discussion onthe exponential decay of rare events in stochastic phenomenawith great help of many well-established theoretical resultsbased on “i.i.d.” assumption. This strong “i.i.d.” assumpt ionhas been often tried to be weakened in many studies. Oneof the reasons is that actual observations generally do notsatisfy i.i.d. assumptions. A typical and well-known exampleis power-law behavior often observed in strongly correlatedsystems. In these cases we take Tsallis statistics as one ofsuch power-law systems because its mathematical foundationshas been widely explored [8].Along similar studies on LDP related with Tsallis statistics,there are a few papers such as [9] and [10]. The paper [9]discusses the possibility of LDP for the strongly correlatedrandom variables in Tsallis statistics. They consider the corre-lated coin tossing model based on the q-Gaussian distributionand numerically evaluate the possibility of a q-generalizationof LDP for a given q-divergence. On the other hand, ourpresent paper does not require the q-Gaussian distribution andthe q-divergence in advance for the large deviation estimate.Our approach is completely analytical starting from the funda-mental nonlinear differential equation dy/dx =y" @default.
• W2964175364 created "2019-07-30" @default.
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• W2964175364 date "2014-12-11" @default.
• W2964175364 modified "2023-09-26" @default.
• W2964175364 title "α-divergence derived as the generalized rate function in a power-law system" @default.
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