scholarly journals Asymptotic behavior of some statistics in Ewens random permutations

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Valentin Feray
Keyword(s):  
Asymptotic Behavior ◽  
Method Of Moments ◽  
Large Family ◽  
Random Permutations ◽  
International Audience ◽  
General Method

International audience The purpose of this article is to present a general method to find limiting laws for some renormalized statistics on random permutations. The model considered here is Ewens sampling model, which generalizes uniform random permutations. We describe the asymptotic behavior of a large family of statistics, including the number of occurrences of any given dashed pattern. Our approach is based on the method of moments and relies on the following intuition: two events involving the images of different integers are almost independent.

With Jumps: An Introduction to Power Variations

2014 ◽  
Author(s):  
Yacine Aïıt-Sahalia ◽  
Jean Jacod
Keyword(s):  
Asymptotic Behavior ◽  
Rate Of Convergence ◽  
Method Of Moments ◽  
Generalized Method Of Moments ◽  
Parametric Estimation ◽  
Continuous Part ◽  
Moment Functions ◽  
The Law

This chapter studies the simplest possible process having both a non-trivial continuous part and jumps. It starts with the asymptotic behavior of power variations when the model is nonparametric, that is, without specifying the law of the jumps. This is done in the same spirit as in Chapter 3: the ideas for the proofs are explained in detail, but technicalities are omitted. Then, it considers the use of these variations in a parametric estimation setting based on the generalized method of moments. There, it considers the ability of certain moment functions, corresponding to power variations, to achieve identification of the parameters of the model and the resulting rate of convergence. It shows that the general nonparametric results have a parametric counterpart in terms of which values of the power p are better able to identify parameters from either the continuous or jump part of the model.

scholarly journals On the Distribution of Periods of Holomorphic Cusp Forms and Zeroes of Period Polynomials

2020 ◽  
Author(s):  
Asbjørn Christian Nordentoft
Keyword(s):  
Asymptotic Behavior ◽  
Method Of Moments ◽  
Cusp Form ◽  
Joint Distribution ◽  
Random Variables ◽  
Kloosterman Sums ◽  
Limiting Distribution ◽  
Independent Random Variables ◽  
Cusp Forms ◽  
Holomorphic Cusp Forms

Abstract In this paper, we determine the limiting distribution of the image of the Eichler–Shimura map or equivalently the limiting joint distribution of the coefficients of the period polynomials associated to a fixed cusp form. The limiting distribution is shown to be the distribution of a certain transformation of two independent random variables both of which are equidistributed on the circle $\mathbb{R}/\mathbb{Z}$, where the transformation is connected to the additive twist of the cuspidal $L$-function. Furthermore, we determine the asymptotic behavior of the zeroes of the period polynomials of a fixed cusp form. We use the method of moments and the main ingredients in the proofs are additive twists of $L$-functions and bounds for both individual and sums of Kloosterman sums.

scholarly journals A probabilistic analysis of a leader election algorithm

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Hanene Mohamed
Keyword(s):  
Asymptotic Behavior ◽  
Probabilistic Analysis ◽  
Probabilistic Approach ◽  
Leader Election ◽  
Hitting Time ◽  
Elimination Process ◽  
International Audience ◽  
Election Algorithm ◽  
The Cost ◽  
Leader Election Algorithm

International audience A leader election algorithm is an elimination process that divides recursively into tow subgroups an initial group of n items, eliminates one subgroup and continues the procedure until a subgroup is of size 1. In this paper the biased case is analyzed. We are interested in the cost of the algorithm e. the number of operations needed until the algorithm stops. Using a probabilistic approach, the asymptotic behavior of the algorithm is shown to be related to the behavior of a hitting time of two random sequences on [0,1].

scholarly journals Random permutations and their discrepancy process

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Guillaume Chapuy
Keyword(s):  
Computational Biology ◽  
Symmetric Group ◽  
Gaussian Process ◽  
Brownian Bridge ◽  
Random Permutation ◽  
Partial Sums ◽  
Random Permutations ◽  
New Process ◽  
International Audience

International audience Let $\sigma$ be a random permutation chosen uniformly over the symmetric group $\mathfrak{S}_n$. We study a new "process-valued" statistic of $\sigma$, which appears in the domain of computational biology to construct tests of similarity between ordered lists of genes. More precisely, we consider the following "partial sums": $Y^{(n)}_{p,q} = \mathrm{card} \{1 \leq i \leq p : \sigma_i \leq q \}$ for $0 \leq p,q \leq n$. We show that a suitable normalization of $Y^{(n)}$ converges weakly to a bivariate tied down brownian bridge on $[0,1]^2$, i.e. a continuous centered gaussian process $X^{\infty}_{s,t}$ of covariance: $\mathbb{E}[X^{\infty}_{s,t}X^{\infty}_{s',t'}] = (min(s,s')-ss')(min(t,t')-tt')$.

scholarly journals Analysis of an algorithm catching elephants on the Internet

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Yousra Chabchoub ◽  
Christine Fricker ◽  
Frédéric Meunier ◽  
Danielle Tibi
Keyword(s):  
Asymptotic Behavior ◽  
Markov Chain ◽  
Stochastic Model ◽  
Limit Theorems ◽  
Bloom Filter ◽  
Simplified Model ◽  
The Internet ◽  
Huge Amount ◽  
International Audience ◽  
On Line

International audience The paper deals with the problem of catching the elephants in the Internet traffic. The aim is to investigate an algorithm proposed by Azzana based on a multistage Bloom filter, with a refreshment mechanism (called $\textit{shift}$ in the present paper), able to treat on-line a huge amount of flows with high traffic variations. An analysis of a simplified model estimates the number of false positives. Limit theorems for the Markov chain that describes the algorithm for large filters are rigorously obtained. The asymptotic behavior of the stochastic model is here deterministic. The limit has a nice formulation in terms of a $M/G/1/C$ queue, which is analytically tractable and which allows to tune the algorithm optimally.

scholarly journals On the density and the structure of the Peirce-like formulae

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Antoine Genitrini ◽  
Jakub Kozik ◽  
Grzegorz Matecki
Keyword(s):  
Finite Number ◽  
Upper Bound ◽  
Large Family ◽  
Partial Answer ◽  
International Audience ◽  
Almost All

International audience Within the language of propositional formulae built on implication and a finite number of variables $k$, we analyze the set of formulae which are classical tautologies but not intuitionistic (we call such formulae - Peirce's formulae). We construct the large family of so called simple Peirce's formulae, whose sequence of densities for different $k$ is asymptotically equivalent to the sequence $\frac{1}{ 2 k^2}$. We prove that the densities of the sets of remaining Peirce's formulae are asymptotically bounded from above by $\frac{c}{ k^3}$ for some constant $c \in \mathbb{R}$. The result justifies the statement that in the considered language almost all Peirce's formulae are simple. The result gives a partial answer to the question stated in the recent paper by H. Fournier, D. Gardy, A. Genitrini and M. Zaionc - although we have not proved the existence of the densities for Peirce's formulae, our result gives lower and upper bound for it (if it exists) and both bounds are asymptotically equivalent to $\frac{1}{ 2 k^2}$.

scholarly journals Rational Expectations Estimation of Georgia Soybean Acreage Response

1995 ◽  
Vol 27 (2) ◽  
pp. 500-509 ◽  
Author(s):  
Nicolas B. C. Ahouissoussi ◽  
Christopher S. McIntosh ◽  
Michael E. Wetzstein
Keyword(s):  
Response Function ◽  
Method Of Moments ◽  
Rational Expectations ◽  
Futures Prices ◽  
Crop Acreage ◽  
Government Programs ◽  
Demand Shocks ◽  
General Method Of Moments ◽  
General Method ◽  
Significant Factors

AbstractThe general method of moments procedure is used for estimating a soybean acreage response function assuming that producers hold rational expectations. Results indicate that soybean, corn, and wheat futures prices, lagged acreage, and government programs are significant factors for determining soybean plantings. Implications of the results are that crop acreage selection by Georgia producers is not very responsive to demand shocks. Thus, producers in other regions are more likely to absorb impacts from these shocks on crop acreage selection.

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