Martingale difference sequence out of an arbitrary sequence by conditioning

Question

I was reading a paper where I encountered the following sentence: Let $X_1,X_2,\ldots,X_n$ Be an arbitrary sequence of random variables. Define $Y_i= X_i - \mathbf{E}(X_i \vert X_1 \ldots X_{i-1})$. It follows straightforwardly that $Y_i$‘s form a martingale difference sequence. I do not see why this holds. Can someone explain this?

Just to recall, $Y_i$s form a martingale difference sequence if $\mathbf{E}(Y_i \vert Y_1 \ldots Y_{i-1})$ is always 0, for every i.

Answer 1

You have to define $Y_1$ as $X_1$. Note that $X_1=Y_1,X_2=Y_2+E(X_2|X_1)$ etc. From this check that $\sigma (X_1,X_2,...,X_n)=\sigma (Y_1,Y_2,...,Y_n)$. Hence, $E(Y_i|Y_1,Y_2,...,Y_{i-1})=E(Y_i|X_1,X_2,...,X_{i-1})=E(X_i|X_1,X_2,...,X_{i-1})-E(X_i|X_1,X_2,...,X_{i-1})=0$.