Assume for simplicity that $\bar c_i/x^{k_i-1}\leqslant\mathrm P(X_i\geqslant x)\leqslant c_i/x^{k_i-1}$ when $x\to\infty$ and that each random variable $X_i$ is almost surely nonnegative. Then, for every $1\leqslant j\leqslant N$, $ [X_j\geqslant x]\subseteq[X\geqslant x]\subseteq\bigcup_{i=1}^N[X_i\geqslant x/N]. $ This implies that $ \max\limits_{i=1}^N\mathrm P(X_i\geqslant x)\leqslant\mathrm P(X\geqslant x)\leqslant\sum_{i=1}^N\mathrm P(X_i\geqslant x/N), $ hence $X$ has exponent $k=\min\limits_{i=1}^Nk_i$ in the sense that, when $x\to+\infty$, $ \bar C_N/x^{k-1}\leqslant\mathrm P(X\geqslant x)\leqslant C_N/x^{k-1}. $ The result you mention about $\alpha$-stable distribution concerns the regime where $N\to\infty$ and one rescales $X$, hence the constants $C_N$ and $\bar C_N$ come into play and modify the exponent $k$.