Discrete problems in probability theory

Limiting size index distributions for ball-bin models with Zipf-type frequencies

“…., as t → ∞ from Karlin's results, Rouault (1978) further extended this to the case where the choices of bins are Markov dependent assuming that the frequency distribution follows a generalized Zipf-type law; that is, a distribution with a regularly varying tail. A recent survey of infinite bin models is presented in Gnedin et al (2007), which includes the case that the probability distribution of the choices of bins is randomly but independently varying, and a survey of models with finite bins is found in Ivanov et al (1985).…”

Convergence Properties in Certain Occupancy Problems Including the Karlin-Rouault Law

“…Earlier references can be found in, e.g. [9]. There are two main groups of problems associated with the spectral statistics, namely, various forms of the central limit theorem (CLT) and law of large numbers (LLNs).…”

On a Refinement of the Central Limit Theorem for Sums of Independent Random Indicators

“…Here we must point out that estimate (3) is correct in order (see [6]), therefore, the value of the coefficients of a -2 cannot be small even for very large a. Note also that the method of estimation used in the paper gives the possibility of obtaining the inequality 2A_ In the generM case, the following estimates for approximation (5) can be obtained on the basis of (3): for a k 10, =>--II-xl(x) ., < I'1 a + 3 6avr 4or 3 The possibility of improving the accuracy of an approximation by increasing the number of terms of the expansion was considered in [6]. It was shown that doing this it is impossible to obtain an approximation with a better order of accuracy.…”