0000000000368126
AUTHOR
Brigitte Chauvin
Persistent random walks, variable length Markov chains and piecewise deterministic Markov processes *
A classical random walk $(S_t, t\in\mathbb{N})$ is defined by $S_t:=\displaystyle\sum_{n=0}^t X_n$, where $(X_n)$ are i.i.d. When the increments $(X_n)_{n\in\mathbb{N}}$ are a one-order Markov chain, a short memory is introduced in the dynamics of $(S_t)$. This so-called "persistent" random walk is nolonger Markovian and, under suitable conditions, the rescaled process converges towards the integrated telegraph noise (ITN) as the time-scale and space-scale parameters tend to zero (see Herrmann and Vallois, 2010; Tapiero-Vallois, Tapiero-Vallois2}). The ITN process is effectively non-Markovian too. The aim is to consider persistent random walks $(S_t)$ whose increments are Markov chains with…
Context Trees, Variable Length Markov Chains and Dynamical Sources
Infinite random sequences of letters can be viewed as stochastic chains or as strings produced by a source, in the sense of information theory. The relationship between Variable Length Markov Chains (VLMC) and probabilistic dynamical sources is studied. We establish a probabilistic frame for context trees and VLMC and we prove that any VLMC is a dynamical source for which we explicitly build the mapping. On two examples, the "comb" and the "bamboo blossom", we find a necessary and sufficient condition for the existence and the uniqueness of a stationary probability measure for the VLMC. These two examples are detailed in order to provide the associated Dirichlet series as well as the genera…
Variable Length Memory Chains: Characterization of stationary probability measures
Variable Length Memory Chains (VLMC), which are generalizations of finite order Markov chains, turn out to be an essential tool to modelize random sequences in many domains, as well as an interesting object in contemporary probability theory. The question of the existence of stationary probability measures leads us to introduce a key combinatorial structure for words produced by a VLMC: the Longest Internal Suffix. This notion allows us to state a necessary and sufficient condition for a general VLMC to admit a unique invariant probability measure. This condition turns out to get a much simpler form for a subclass of VLMC: the stable VLMC. This natural subclass, unlike the general case, enj…
Variable Length Markov Chains, Persistent Random Walks: a close encounter
This is the story of the encounter between two worlds: the world of random walks and the world of Variable Length Markov Chains (VLMC). The meeting point turns around the semi-Markov property of underlying processes.
Uncommon Suffix Tries
Common assumptions on the source producing the words inserted in a suffix trie with $n$ leaves lead to a $\log n$ height and saturation level. We provide an example of a suffix trie whose height increases faster than a power of $n$ and another one whose saturation level is negligible with respect to $\log n$. Both are built from VLMC (Variable Length Markov Chain) probabilistic sources; they are easily extended to families of sources having the same properties. The first example corresponds to a ''logarithmic infinite comb'' and enjoys a non uniform polynomial mixing. The second one corresponds to a ''factorial infinite comb'' for which mixing is uniform and exponential.
Characterization of stationary probability measures for Variable Length Markov Chains
By introducing a key combinatorial structure for words produced by a Variable Length Markov Chain (VLMC), the longest internal suffix, precise characterizations of existence and uniqueness of a stationary probability measure for a VLMC chain are given. These characterizations turn into necessary and sufficient conditions for VLMC associated to a subclass of probabilised context trees: the shift-stable context trees. As a by-product, we prove that a VLMC chain whose stabilized context tree is again a context tree has at most one stationary probability measure.