Random environment on coloured trees

In this paper, we study a regular rooted coloured tree with random labels assigned to its edges, where the distribution of the label assigned to an edge depends on the colours of its endpoints. We obtain some new results relevant to this model and also show how our model generalizes many other probabilistic models, including random walk in random environment on trees, recursive distributional equations and multi-type branching random walk on $\mathbb{R}$.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ101 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Bindweeds or random walks in random environments on multiplexed trees and their asympotics

We report on the asymptotic behaviour of a new model of random walk, we term the bindweed model, evolving in a random environment on an infinite multiplexed tree. The term \textit{multiplexed} means that the model can be viewed as a nearest neighbours random walk on a tree whose vertices carry an internal degree of freedom from the finite set $\{1,...,d\}$, for some integer $d$. The consequence of the internal degree of freedom is an enhancement of the tree graph structure induced by the replacement of ordinary edges by multi-edges, indexed by the set $\{1,...,d\}\times\{1,...,d\}$. This indexing conveys the information on the internal degree of freedom of the vertices contiguous to each edge. The term \textit{random environment} means that the jumping rates for the random walk are a family of edge-indexed random variables, independent of the natural filtration generated by the random variables entering in the definition of the random walk; their joint distribution depends on the index of each component of the multi-edges. We study the large time asymptotic behaviour of this random walk and classify it with respect to positive recurrence or transience in terms of a specific parameter of the probability distribution of the jump rates. This classifying parameter is shown to coincide with the critical value of a matrix-valued multiplicative cascade on the ordinary tree (\textit{i.e.} the one without internal degrees of freedom attached to the vertices) having the same vertex set as the state space of the random walk. Only results are presented here since the detailed proofs will appear elsewhere

Dynamical systems with heavy-tailed random parameters

Motivated by the study of the time evolution of random dynamical systems arising in a vast variety of domains --- ranging from physics to ecology ---, we establish conditions for the occurrence of a non-trivial asymptotic behaviour for these systems in the absence of an ellipticity condition. More precisely, we classify these systems according to their type and --- in the recurrent case --- provide with sharp conditions quantifying the nature of recurrence by establishing which moments of passage times exist and which do not exist. The problem is tackled by mapping the random dynamical systems into Markov chains on $\mathbb{R}$ with heavy-tailed innovation and then using powerful methods stemming from Lyapunov functions to map the resulting Markov chains into positive semi-martingales.Comment: 24 page

Generalizations of forest fires with ignition at origin

We study generalizations of the Forest Fire model introduced in~\cite{BTRB} and~\cite{Volk} by allowing the rates at which the tree grow to depend on their location, introducing long-range burning, as well as continuous-space generalization of the model. We establish that in all the models in consideration the time required to reach site at distance $x$ from the origin is of order at most $(\log x)^{(\log 2)^{-1}+\delta}$ for any $\delta>0$

Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift

We study the first exit time $\tau$ from an arbitrary cone with apex at the origin by a non-homogeneous random walk (Markov chain) on $\Z^d$ ($d \geq 2$) with mean drift that is asymptotically zero. Specifically, if the mean drift at \bx \in \Z^d is of magnitude O(\| \bx\|^{-1}), we show that $\tau<\infty$ a.s. for any cone. On the other hand, for an appropriate drift field with mean drifts of magnitude \| \bx\|^{-\beta}, $\beta \in (0,1)$, we prove that our random walk has a limiting (random) direction and so eventually remains in an arbitrarily narrow cone. The conditions imposed on the random walk are minimal: we assume only a uniform bound on $2$nd moments for the increments and a form of weak isotropy. We give several illustrative examples, including a random walk in random environment model

Strong transience for one-dimensional Markov chains with asymptotically zero drifts

For near-critical, transient Markov chains on the non-negative integers in the Lamperti regime, where the mean drift at $x$ decays as $1/x$ as $x \to \infty$, we quantify degree of transience via existence of moments for conditional return times and for last exit times, assuming increments are uniformly bounded. Our proof uses a Doob $h$-transform, for the transient process conditioned to return, and we show that the conditioned process is also of Lamperti type with appropriately transformed parameters. To do so, we obtain an asymptotic expansion for the ratio of two return probabilities, evaluated at two nearby starting points; a consequence of this is that the return probability for the transient Lamperti process is a regularly-varying function of the starting point.Comment: 26 pages; v2: minor revisions, expanded discussio

Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips

We study asymptotic properties of spatially non-homogeneous random walks with non-integrable increments, including transience, almost-sure bounds, and existence and non existence of moments for first-passage and last-exit times. In our proofs we also make use of estimates for hitting probabilities and large deviations bounds. Our results are more general than existing results in the literature, which consider only the case of sums of independent (typically, identically distributed) random variables. We do not assume the Markov property. Existing results that we generalize include a circle of ideas related to the Marcinkiewicz-Zygmund strong law of large numbers, as well as more recent work of Kesten and Maller. Our proofs are robust and use martingale methods. We demonstrate the benefit of the generality of our results by applications to some non-classical models, including random walks with heavy-tailed increments on two-dimensional strips, which include, for instance, certain generalized risk processes

From Non Market Attitudes to Market Behavior: Laboratory Market Experiments in Moscow, and the Hvatat Property of Human Behavior

This paper reports on laboratory market experiments that were conducted in Moscow during the fall of 1992. Two broad concerns guided the design of the experiments. The first concern is the obvious cultural differences between people in Russia and those in the west where traditional laboratory market experiments have been conducted. Most economists who have conducted experiments would assume that cultural background would not play a major part in the equilibration process, and that the law of supply and demand would operate essentially the same in all cultures and at all times. As Russia is undergoing a dramatic social change and before an orientation to market attitudes permeates the society, a unique opportunity presented itself to test this assumption about the universal nature of the laws of the market. In the language of the experimenters, this first concern is whether or not some of the known properties of markets that operate in western culture are robust to major changes in the culture of the subject population. The properties of markets that are of interest are the fact of equilibration, the influence of price ceilings on equilibration and the influence of the asymmetries in the shapes of the demand and supply curves on the direction of convergence to equilibrium. The second concern reflects a desire to explore some more recently discovered aspects of market behavior. A transactions cost was added to the market and the question posed was whether or not the cost frustrated or biased equilibration (Jamison and Plott, 1997). The results of the experiments were not as anticipated. The standard convergence process was observed but when the data were applied to test the asymmetric rent hypothesis or the hypothesis regarding the nonbinding price controls, some surprises surfaced. In particular the data were not the same as had been observed in other experiments. This paradoxical failure of previous results to generalize prompted a detailed investigation into the behavior of individuals in the experiments. The hope was to find the causes of the discrepancies. A special section of the paper is devoted to a conjecture that resulted from the ex-post examination of the data and is offered as an explanation. The paper is organized as follows. In the first section the purposes of the project are outlined in greater detail. The second section contains the experimental design, procedures and parameters. The third section contains the predictions of alternative models as applied to the special case of the parameters of the experiments. The fourth section contains the results of the experiments. The fifth section contains the post experiment speculations about the possible explanations of the surprising dynamic behavior. Here the idea of Hvatat, which means "to grab" in Russian, is introduced and explored. The final section is a summary of conclusions