  We propose an approach to construct Bernoulli trials $\{X_i, i\ge 1\}$
combining dependence and independence periods, and call it Bernoulli sequence
with random dependence (BSRD). The structure of dependence, on the past $S_i =
X_1 + \ldots + X_i$, {defines} a class of non-Markovian random walks of recent
interest in the literature. In this paper, the dependence is activated by an
auxiliary collection of Bernoulli trials $\{Y_i, i\ge 1\}$, called {\it memory
switch sequence}. We introduce the concept of {\it memory lapses property},
which {is} characterized by intervals of consecutive independent steps in BSRD.
The main results include classical limit theorems for a class of linear BSRD.
In particular, we obtain a central limit theorem for a class of BSRD which
generalizes some previous results in literature. Along the paper, several
examples of potential applications are provided.
