Neighbour-dependent point shifts and random exchange models: invariance and attractors

Consider a stationary renewal point process on the real line and divide each of the segments it defines in a proportion given by \iid realisations of a fixed distribution $G$ supported by [0,1]. We ask ourselves for which interpoint distribution $F$ and which division distributions $G$, the division points is again a renewal process with the same $F$? An evident case is that of degenerate $F$ and $G$. Interestingly, the only other possibility is when $F$ is Gamma and $G$ is Beta with related parameters. In particular, the division points of a Poisson process is again Poisson, if the division distribution is Beta: B$(r,1-r)$ for some $0<r<1$. We show a similar behaviour of random exchange models when a countable number of `agents' exchange randomly distributed parts of their `masses' with neighbours. More generally, a Dirichlet distribution arises in these models as a fixed point distribution preserving independence of the masses at each step. We also show that for each $G$ there is a unique attractor, a distribution of the infinite sequence of masses, which is a fixed point of the random exchange and to which iterations of a non-equilibrium configuration of masses converge weakly. In particular, iteratively applying B$(r,1-r)$-divisions to a realisation of any renewal process with finite second moment of $F$ yields a Poisson process of the same intensity in the limit.Comment: 16 page

On explicit form of the stationary distributions for a class of bounded Markov chains

We consider a class of discrete time Markov chains with state space [0,1] and the following dynamics. At each time step, first the direction of the next transition is chosen at random with probability depending on the current location. Then the length of the jump is chosen independently as a random proportion of the distance to the respective end point of the unit interval, the distributions of the proportions being fixed for each of the two directions. Chains of that kind were subjects of a number of studies and are of interest for some applications. Under simple broad conditions, we establish the ergodicity of such Markov chains and then derive closed form expressions for the stationary densities of the chains when the proportions are beta distributed with the first parameter equal to 1. Examples demonstrating the range of stationary distributions for processes described by this model are given, and an application to a robot coverage algorithm is discussed.Comment: 14 pages, 4 figure

Random weighted averages, partition structures and generalized arcsine laws

This article offers a simplified approach to the distribution theory of randomly weighted averages or $P$-means $M_P(X):= \sum_{j} X_j P_j$, for a sequence of i.i.d.random variables $X, X_1, X_2, \ldots$, and independent random weights $P:= (P_j)$ with $P_j \ge 0$ and $\sum_{j} P_j = 1$. The collection of distributions of $M_P(X)$, indexed by distributions of $X$, is shown to encode Kingman's partition structure derived from $P$. For instance, if $X_p$ has Bernoulli$(p)$ distribution on $\{0,1\}$, the $n$th moment of $M_P(X_p)$ is a polynomial function of $p$ which equals the probability generating function of the number $K_n$ of distinct values in a sample of size $n$ from $P$: $E (M_P(X_p))^n = E p^{K_n}$. This elementary identity illustrates a general moment formula for $P$-means in terms of the partition structure associated with random samples from $P$, first developed by Diaconis and Kemperman (1996) and Kerov (1998) in terms of random permutations. As shown by Tsilevich (1997) if the partition probabilities factorize in a way characteristic of the generalized Ewens sampling formula with two parameters $(\alpha,\theta)$, found by Pitman (1992), then the moment formula yields the Cauchy-Stieltjes transform of an $(\alpha,\theta)$ mean. The analysis of these random means includes the characterization of $(0,\theta)$-means, known as Dirichlet means, due to Von Neumann (1941), Watson (1956) and Cifarelli and Regazzini (1990) and generalizations of L\'evy's arcsine law for the time spent positive by a Brownian motion, due to Darling (1949) Lamperti (1958) and Barlow, Pitman and Yor (1989).Comment: 76 pages, 2 figure