  We define a generalisation of the Edge-Reinforced Random Walk (ERRW)
introduced by Coppersmith and Diaconis in 1986, called *-Edge-Reinforced Random
Walk (*-ERRW), which can be seen as an extension of the r-dependent ERRW
introduced by Bacallado (2011). Similarly as for the ERRW, it has a
continuous-time counterpart, called *-Vertex-Reinforced Jump Process (*-VRJP),
which itself generalises the Vertex- Reinforced Jump Process proposed by Werner
in 2000 and initially studied by Davis and Volkov (2002, 2004). Under some
assumptions on the initial weights, the *-ERRW is partially exchangeable in the
sense of Diaconis and Freedman (1980), and thus it is a random walk in a random
environment: we compute the law of that environment.
