Non-normal limiting distribution for optimal alignment scores of strings in binary alphabets

We consider two independent binary i.i.d. random strings $X$ and $Y$ of equal length $n$ and the optimal alignments according to a symmetric scoring functions only. We decompose the space of scoring functions into five components. Two of these components add a part to the optimal score which does not depend on the alignment and which is asymptotically normal. We show that when we restrict the number of gaps sufficiently and add them only into one sequence, then the alignment score can be decomposed into a part which is normal and has order $O(\sqrt{n})$ and a part which is on a smaller order and tends to a Tracy-Widom distribution. Adding gaps only into one sequence is equivalent to aligning a string with its descendants in case of mutations and deletes. For testing relatedness of strings, the normal part is irrelevant, since it does not depend on the alignment hence it can be safely removed from the test statistic.