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Register a new userTail asymptotics for a random sign Lindley recursion
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We investigate the tail behaviour of the steady state distribution of a stochastic recursion that generalises Lindleys recursion. This recursion arises in queuing systems with dependent interarrival and service times, and includes alternating service systems and carousel storage systems as special cases. We obtain precise tail asymptotics in three qualitatively different cases, and compare these with existing results for Lindleys recursion and for alternating service systems.
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An urn contains black and red balls. Let $Z_n$ be the proportion of black balls at time $n$ and $0leq L<Uleq 1$ random barriers. At each time $n$, a ball $b_n$ is drawn. If $b_n$ is black and $Z_{n-1}<U$, then $b_n$ is replaced together with a random number $B_n$ of black balls. If $b_n$ is red and $Z_{n-1}>L$, then $b_n$ is replaced together with a random number $R_n$ of red balls. Otherwise, no additional balls are added, and $b_n$ alone is replaced. In this paper, we assume $R_n=B_n$. Then, under mild conditions, it is shown that $Z_noverset{a.s.}longrightarrow Z$ for some random variable $Z$, and begin{gather*} D_n:=sqrt{n},(Z_n-Z)longrightarrowmathcal{N}(0,sigma^2)quadtext{conditionally a.s.} end{gather*} where $sigma^2$ is a certain random variance. Almost sure conditional convergence means that begin{gather*} Pbigl(D_nincdotmidmathcal{G}_nbigr)overset{weakly}longrightarrowmathcal{N}(0,,sigma^2)quadtext{a.s.} end{gather*} where $Pbigl(D_nincdotmidmathcal{G}_nbigr)$ is a regular version of the conditional distribution of $D_n$ given the past $mathcal{G}_n$. Thus, in particular, one obtains $D_nlongrightarrowmathcal{N}(0,sigma^2)$ stably. It is also shown that $L<Z<U$ a.s. and $Z$ has non-atomic distribution.
There is a result of Diaconis and Freedman which says that, in a limiting sense, for large collections of high-dimensional data most one-dimensional projections of the data are approximately Gaussian. This paper gives quantitati
A classical result for the simple symmetric random walk with $2n$ steps is that the number of steps above the origin, the time of the last visit to the origin, and the time of the maximum height all have exactly the same distribution and converge when scaled to the arcsine law. Motivated by applications in genomics, we study the distributions of these statistics for the non-Markovian random walk generated from the ascents and descents of a uniform random permutation and a Mallows($q$) permutation and show that they have the same asymptotic distributions as for the simple random walk. We also give an unexpected conjecture, along with numerical evidence and a partial proof in special cases, for the result that the number of steps above the origin by step $2n$ for the uniform permutation generated walk has exactly the same discrete arcsine distribution as for the simple random walk, even though the other statistics for these walks have very different laws. We also give explicit error bounds to the limit theorems using Steins method for the arcsine distribution, as well as functional central limit theorems and a strong embedding of the Mallows$(q)$ permutation which is of independent interest.
We consider the sums $S_n=xi_1+cdots+xi_n$ of independent identically distributed random variables. We do not assume that the $xi$s have a finite mean. Under subexponential type conditions on distribution of the summands, we find the asymptotics of the probability ${bf P}{M>x}$ as $xtoinfty$, provided that $M=sup{S_n, nge1}$ is a proper random variable. Special attention is paid to the case of tails which are regularly varying at infinity. We provide some sufficient conditions for the integrated weighted tail distribution to be subexponential. We supplement these conditions by a number of examples which cover both the infinite- and the finite-mean cases. In particular, we show that subexponentiality of distribution $F$ does not imply subexponentiality of its integrated tail distribution $F^I$.
In this paper we obtain the limit distribution for partial sums with a random number of terms following a class of mixed Poisson distributions. The resulting weak limit is a mixing between a normal distribution and an exponential family, which we call by normal exponential family (NEF) laws. A new stability concept is introduced and a relationship between {alpha}-stable distributions and NEF laws is established. We propose estimation of the parameters of the NEF models through the method of moments and also by the maximum likelihood method, which is performed via an Expectation-Maximization algorithm. Monte Carlo simulation studies are addressed to check the performance of the proposed estimators and an empirical illustration on financial market is presented.
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