I have a sequence of sequences defined as:
$$Y[k] = \alpha \prod_{n=1}^kX[k]$$
Which I want to find an $\alpha$ for which it is martingale. I have that
$$E[Y[k] \mid X[0],\dots,X[k-1]] = Y[k-1]$$
this can in turn be
$$E\left[\alpha \prod_{n=1}^kX[k] \mid X[0],\dots,X[k-1]\right] = ? $$
But I am stuck here and I'm not sure what I'm supposed to do. Any help is welcome.
edit: you know that X is iid and is a random sequence with outcomes in $R^+$. $\alpha$ is a constant.
edit: Maybe you can see that: $$Y[k-1] = \alpha \prod_{n=1}^{k-1}X[k]$$
Notice that $Y_{n+1}=X_{n+1}Y_n$ and $Y_n$ is $\sigma(X_j,0\leqslant j\leqslant n)=:\mathcal F_n$ measurable, hence $$\mathbb E[Y_{n+1}\mid \sigma(X_j,0\leqslant j\leqslant n)]=Y_n\mathbb E[X_{n+1}\mid \sigma(X_j,0\leqslant j\leqslant n)].$$ By independence, $\mathbb E[X_{n+1}\mid \sigma(X_j,0\leqslant j\leqslant n)]=\mathbb E[X_{n+1}]$, hence we need $\mathbb E[X_1]=1$.
With the version: "find $a$ such that $\left(Y_n:=a^n\prod_{j=1}^nX_j,\mathcal F_n\right)$ is a martingale", we have $\mathbb E[Y_{n+1}\mid \mathcal F_n]=a\mathbb E[Y_{n+1}]Y_n$, so we want $a\mathbb E[Y_{1}]=1$.