On the limiting law of the length of the longest common and increasing subsequences in random words

Let X=(Xi)i≥1 and Y=(Yi)i≥1 be two sequences of independent and identically distributed (iid) random variables taking their values, uniformly, in a common totally ordered finite alphabet. Let LCIn be the length of the longest common and (weakly) increasing subsequence of X1⋯Xn and Y1⋯Yn. As n grows without bound, and when properly centered and scaled, LCIn is shown to converge, in distribution, towards a Brownian functional that we identify.