The Kolmogorov's maximal inequality states that when $X_1,\dots,X_n$ are mutually independent random variables, each with finite variance. Set $S_j=X_1+\cdots+X_j, 1 \le j\le n.$ Then, for each $\epsilon>0$, $\Pr(\max_{1 \le j \le n}|S_j-\mathbb{E}(S_j)| \ge \epsilon) \le \frac{Var(S_n)}{\epsilon^2}$
I consider the following question, given a number $n$, a random composition (strong) of this number into $k$ positive parts. So we can get $k$ random variable $Y_1, Y_2,\dots, Y_k$ with $Y_1+Y_2+\cdots+Y_k=n$
Apparently, in my case, the random variables are dependent. So how to apply Kolmogorov's Inequality or some other related inequality for these random variables?
That is let S_j'=Y_1+\cdots+Y_j, give a bound of \Pr(\max_{1 \le j \le k} |S_j'-\mathbb{E}(S_j')| \ge \epsilon) for some $\epsilon \ge 0$.