On the Variance of the Length of the Longest Common Subsequences in Random Words With an Omitted Letter

We investigate the variance of the length of the longest common subsequences of two independent random words of size $n$, where the letters of one word are i.i.d. uniformly drawn from $\{α_1, α_2, \cdots, α_m\}$, while the letters of the other word are i.i.d. drawn from $\{α_1, α_2, \cdots, α_m, α_{m+1}\}$, with probability $p > 0$ to be $α_{m+1}$, and $(1-p)/m > 0$ for all the other letters. The order of the variance of this length is shown to be linear in $n$.