Suppose $u,v,w\sim\mathcal{N}(0,I_{n})$ are iid $n$-dimensional normal Gaussian vectors. Moreover, let $X$ bet defined as: $$X=\sum_{i=1}^n \sum_{j=1}^i\sum_{k=1}^j u_i v_j w_k$$ We are interested in concentration of $X^2$. It is easy to compute $\mathbb{E} X^2 = \frac{n(n+1)(n+2)}{6}$. This results from expansion of $X^2$, the linearity of expectation, and that only terms of the form $u_i^2 v_j^2 w_k^2$ have nonzero expectation and we simply need to count them. Now, my question is, is there a way to compute a concentration bound for $X^2$ w.r.t its mean?
I thought about this approach, If we define: $$W_0 = 0, \text{ for }j\ge 1 \quad W_j = W_{j-1} + w_j$$ $$V_0 = 0, \text{ for }j\ge 1 \quad V_j = V_{j-1} + v_j W_j$$ $$U_0 = 0, \text{ for }j\ge 1 \quad U_j = U_{j-1} + u_j V_j$$ $$X = U_n$$ Therefore, we can think of these random processes, as stopped martingles, i.e., after time-step $n$ all of the processes stop evolving. Because for all of them, we have: $\mathbb{E}_{j-1} W_j = W_{j-1}, \mathbb{E}_{j-1} V_j = V_{j-1}, \mathbb{E}_{j-1} U_j = U_{j-1}$. Is there a way we can control the concentration of the final term, $X^2$, as a function of $n$?