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Let $I$ be a finite alphabet and $\aS\subset I$ be a nonempty strict subset.The sequences in $I^\ZZ$ are organized into connected regions which alwaysstart with a symbol in $\aS$. The regions are labelled by types $C(s)$, thus aregion starting at $s'\in C(s)$ has the same type as one starting at $s$. Let$(\aP_s: s\in \aS)$ be a family of distributions on $I^\NN$ where each $\aP_s$charges sequences starting with the symbol $s$. We can define a naturaldistribution $\PP$ on $I^\NN$, that counts the number of visits to the statesfrom $\aP_s$, properly weighted. A dynamics of interest is such that at thefirst occurrence of $s'\in \aS\setminus C(s)$ the law regenerates withdistribution $\aP_{s'}$. In this case we are able to find simple conditions for$\PP$ to be stationary. In addition, we study the following more complex model:once a symbol $s'\in \aS\setminus C(s)$ has been encountered, there is adecision to be made, either a new region of type $C(s')$ governed by $\aP_{s'}$starts or the region continues to be a $C(s)$ region. This decision is modeledas random and depends on $s'$. In this setting a similar distribution to $\PP$can be constructed and the conditions for stationarity are supplied. Thesemodels are inspired by genomic sequences where $I$ is the set of codons, theclasses $(C(s): s\in \aS)$ group codons defining similar genomic classes, e.g.in bacteria there are two classes corresponding to the start and stop codons,and the random decision to continue a region or to begin a new region of adifferent class reflects the well-known fact that not every appearance of astart codon marks the beginning of a new coding region.

A distribution weighting a set of laws whose initial states are grouped into classes