Sojourn time in Z+\mathbb{Z}^+ for the Bernoulli random walk on Z\mathbb{Z}

Let $(S_k)_{k\ge 1}$ be the classical Bernoulli random walk on the integer line with jump parameters $p\in(0,1)$ and $q=1-p$. The probability distribution of the sojourn time of the walk in the set of non-negative integers up to a fixed time is well-known, but its expression is not simple. By modifying slightly this sojourn time--through a particular counting process of the zeros of the walk as done by Chung & Feller ["On fluctuations in coin-tossings", Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 605-608]-, simpler representations may be obtained for its probability distribution. In the aforementioned article, only the symmetric case ($p=q=1/2$) is considered. This is the discrete counterpart to the famous Paul L\'evy's arcsine law for Brownian motion.Comment: 44 page

A simple asymmetric evolving random network

We introduce a new oriented evolving graph model inspired by biological networks. A node is added at each time step and is connected to the rest of the graph by random oriented edges emerging from older nodes. This leads to a statistical asymmetry between incoming and outgoing edges. We show that the model exhibits a percolation transition and discuss its universality. Below the threshold, the distribution of component sizes decreases algebraically with a continuously varying exponent depending on the average connectivity. We prove that the transition is of infinite order by deriving the exact asymptotic formula for the size of the giant component close to the threshold. We also present a thorough analysis of aging properties. We compute local-in-time profiles for the components of finite size and for the giant component, showing in particular that the giant component is always dense among the oldest nodes but invades only an exponentially small fraction of the young nodes close to the threshold.Comment: 33 pages, 3 figures, to appear in J. Stat. Phy

Algorithmic statistics revisited

The mission of statistics is to provide adequate statistical hypotheses (models) for observed data. But what is an "adequate" model? To answer this question, one needs to use the notions of algorithmic information theory. It turns out that for every data string $x$ one can naturally define "stochasticity profile", a curve that represents a trade-off between complexity of a model and its adequacy. This curve has four different equivalent definitions in terms of (1)~randomness deficiency, (2)~minimal description length, (3)~position in the lists of simple strings and (4)~Kolmogorov complexity with decompression time bounded by busy beaver function. We present a survey of the corresponding definitions and results relating them to each other

Efficient estimation of the cardinality of large data sets

International audienceGiroire has recently proposed an algorithm which returns the $\textit{approximate}$ number of distinct elements in a large sequence of words, under strong constraints coming from the analysis of large data bases. His estimation is based on statistical properties of uniform random variables in $[0,1]$. In this note we propose an optimal estimation, using Kullback information and estimation theory

LERW as an example of off-critical SLEs

Two dimensional loop erased random walk (LERW) is a random curve, whose continuum limit is known to be a Schramm-Loewner evolution (SLE) with parameter kappa=2. In this article we study ``off-critical loop erased random walks'', loop erasures of random walks penalized by their number of steps. On one hand we are able to identify counterparts for some LERW observables in terms of symplectic fermions (c=-2), thus making further steps towards a field theoretic description of LERWs. On the other hand, we show that it is possible to understand the Loewner driving function of the continuum limit of off-critical LERWs, thus providing an example of application of SLE-like techniques to models near their critical point. Such a description is bound to be quite complicated because outside the critical point one has a finite correlation length and therefore no conformal invariance. However, the example here shows the question need not be intractable. We will present the results with emphasis on general features that can be expected to be true in other off-critical models.Comment: 45 pages, 2 figure

Entrance and sojourn times for Markov chains. Application to (L,R)(L,R)-random walks

In this paper, we provide a methodology for computing the probability distribution of sojourn times for a wide class of Markov chains. Our methodology consists in writing out linear systems and matrix equations for generating functions involving relations with entrance times. We apply the developed methodology to some classes of random walks with bounded integer-valued jumps.Comment: 30 page