I have a problem with this exercise.
Let $(X_n)$ and $(Y_n)$ be two successions of mutually independent r.v. such that $X_n$ ~ $Bern(1/n^2)$ and $Y_n$ ~ $Exp(1/n^{1/4})$. We also define
$Z_n := \begin{cases} Y_n & \text{if $X_n=1$} \\[2ex] -c_n & \text{if $X_n=0$} \end{cases}$
$S_n := Z_1 + \dots + Z_n$
I'm looking for $c_n$ values for which $S_n$ is a martingale.
My attempt:
$\mathcal{F}_0=\{\emptyset,\Omega\}$
$\mathcal{F}_n=\{X_1, \dots, X_n, Y_1, \dots, Y_n\}$, $n \ge 1$
$\mathbb{E}[Y_n] = n^{1/4}$
$\mathbb{E}[Z_n] = \mathbb{E}[Y_n]\dfrac{1}{n^2}+(-c_n)(1-\dfrac{1}{n^2})=\dfrac{1}{n^{7/4}}-c_n(1-\dfrac{1}{n^2})$
$\mathbb{E}[S_n \mid \mathcal{F}_{n-1}]=\mathbb{E}[S_{n-1} \mid \mathcal{F}_{n-1}]+\mathbb{E}[Z_n \mid \mathcal{F}_{n-1}]=S_{n-1}+\mathbb{E}[Z_n]=S_{n-1}$ iff $\mathbb{E}[Z_n]=0$, that is
$c_n = \dfrac{n^{1/4}}{n^2-1}$
Could it be correct?