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The problem of the order of the fluctuation of the Longest Common Subsequence(LCS) of two independent sequences has been open for decades. There existcontradicting conjectures on the topic, due to Chvatal - Sankoff in 1975 andWaterman in 1994. In the present article, we consider a special model of i.i.d.sequences made out of blocks. A block is a contiguous substring consisting onlyof one type of symbol. Our model allows only three possible block lengths, eachbeen equiprobable picked up. In this context, we introduce a random operation(random modification) on the blocks of one of the sequences. In the presentarticle, we develop the techniques to prove the following: if we suppose thatthe random modification increases the length of the LCS with high probability,then the order of the fluctuation of the LCS is as conjectured by Waterman.This result is a key technical part in the study of the size of the fluctuationof the LCS for sequences of i.i.d. blocks, developed by Matzinger and Torres.

Random modification effect in the size of the fluctuation of the LCS of two sequences of i.i.d. blocks