On a class of random walks in simplexes

Abstract

We study the limit behaviour of a class of random walk models taking values in the standard d-dimensional ( $d\ge 1$ ) simplex. From an interior point z, the process chooses one of the $d+1$ vertices of the simplex, with probabilities depending on z, and then the particle randomly jumps to a new location z′ on the segment connecting z to the chosen vertex. In some special cases, using properties of the Beta distribution, we prove that the limiting distributions of the Markov chain are Dirichlet. We also consider a related history-dependent random walk model in [0, 1] based on an urn-type scheme. We show that this random walk converges in distribution to an arcsine random variable.

On the asymptotic behavior of the Diaconis–Freedman chain in a multi-dimensional simplex

“…The approach we use here is based on the description of the spectrum of the transition operator associated with (X n ) n≥0 ; it is totally different from the one used in [23], based on criteria of ergodicity and stability of stochastic processes due to A. A. Borovkov [4].…”

“…We shall use the assumption that our weight functions p i are Hölder, and there is at least one of these functions with full support. This assumption and Assumption 2.2(ii) in [23] cannot be compared. Therefore, our Theorem 3.4 and their Theorem 2.4 cannot be compared, although their Assumptions 2.2(i) and (iii) for ξ cover ours for ξ = U(0, 1).…”

“…First, we give the most natural setting of this chain on a d-dimensional simplex and consider its asymptotic behavior by using techniques from random iterated function theory and quasi-compact operator theory (see [20] in dimension one). We have recently learned that this multi-dimensional setting is also considered in [23], where the authors used another approach and only considered the cases of ergodicity. We also discuss here some other possible extensions, with several invariant probability measures.…”

Dynamical Wasserstein Barycenters for Time-series Modeling

“…leaves the state at its previous value x t−1 while a full step (γ t [k] = 1) jumps to the vertex e k . Unlike prior methods (Nguyen and Volkov, 2020), we repeat this process for each of the K components and average their results to achieve the next state x t ,…”

“…Given the nature of our model, we require that the state lie in the simplex at every timestep, a constraint not respected by the Gaussian noise that drives common continuous-state processes (Welch, 1997). Building on work by Nguyen and Volkov (2020), we employ a random walk where the driving noise comes from independent, identically distributed (IID) draws from a mixture of two Beta distributions, representing stationary and transitional dynamics. By blending the current state and a mixture-of-Betas draw in a convex manner, we construct a new state that lies in the simplex.…”