  We study a variant of the Generalized Excited Random Walk (GERW) on
$\mathbb{Z}^d$ introduced by Menshikov, Popov, Ram\'irez and Vachkovskaia in
[Ann. Probab. 40 (5), 2012]. It consists in a particular version of the model
studied in [arXiv preprint arXiv:2211.05715, 2022] where excitation may or may
not occur according to a time-dependent probability. Specifically, given
$\{p_n\}_{n \ge 1}$, $p_n \in (0, 1]$ for all $n \ge 1$, whenever the process
visits a site at time $n$ for the first time, with probability $p_n$ it gains a
drift in a determined direction. Otherwise, it behaves as a $d$-martingale with
zero-mean vector. We refer to the model as a GERW in time-inhomogeneous
Bernoulli environment, in short, $p_n$-GERW.
  Under bounded jumps hypothesis and with $p_n = O(n^{-\beta})$, $\beta \geq
1/2$, we show a series of results for the $p_n$-GERW depending on the value of
$\beta$ and on the dimension. Specifically, for $\beta > 1/2$, $d\geq 2$ and
$\beta=1/2$ and $d=2$ we obtain a Functional Central Limit Theorem. Finally,
for $\beta=1/2$ and $d \ge 4$ we obtain that a sequence of $p_n$-GERW is tight,
and every limit point $Y$ satisfies $W_t \cdot \ell + c_1 \sqrt{t} \preceq Y_t
\cdot \ell \preceq W_t \cdot \ell + c_2 \sqrt{t}$ where $c_1$ and $c_2$ are
positive constants, $W_{\cdot}$ is a Brownian motion and $\ell$ is the
direction of the drift.
