Is this a martingale sequence? I have a sequence of random variables $X_1, X_2, \ldots X_N$ such that $|X_i| \leq R \ \forall \ i $,  satisfying
$$|E[X_n|X_1,X_2,\ldots X_{n-1}]| \leq |X_{n-1}|, $$
Can I construct a sub/super-martingale sequence from this? Intuitively, it makes sense that the random variables are constrained to be not from far the previous one in absolute/distance sense. But can I view this as some sort of martingale or perhaps construct a sequence (or sequences) which satisfy the definition of (sub/super)martingale(s)?
 A: It may actually not be possible. I do not have a proof but trying to use Jensen's inequality is hopeless for example.
If you aim at using Jensen's inequality, you need an even and concave function $f$ increasing on $\mathbb R_+$ to define a supermartingale. Then by Jensen's inequality:
$$E[f(X_n)|X_1,\ldots,X_{n-1}]\leq f(E[X_n|X_1,\ldots,X_{n-1}])\leq f(X_{n-1}).$$
To be more precise, since your inequality does not depend on the sign of your random variable, you cannot use anything else than even functions (you don't have an inequality for $X_{n-1}$). Then using a concave and increasing function on $\mathbb R_+$ allows you to use Jensen's inequality in the right order. A convex and decreasing function would lead to a submartingale.
Unfortunately I don't think such functions exist. The issue is in zero. An even function increasing on $\mathbb R_+$ cannot be concave on $\mathbb R$. Indeed if $f$ is even and concave on $\mathbb R_+$ for any $x>0$, 
$$f(0)=f(-\frac12 x+\frac12 x)\geq \frac12 f(-x)+ \frac12 f(x)=f(x).$$ Hence, if $f$ is monotonous on $\mathbb R_+$, it is non increasing.
