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Consider two independent sequences X₁, , Xₙ and Y₁, , Yₙ. Suppose that X₁, , Xₙ are i. i. d. X and Y₁, , Yₙ are i. i. d. Y, where X and Y are distributions on finite alphabets X and Y, respectively. A score F: X Y R is assigned to each pair (Xᵢ, Yⱼ) and the maximal nonaligned segment score is Mₙ = ₀ ₈, ₉ ₍ -, ₀ \^₊=₁ F (X₈+₊, Y₉+₊) \. The limit distribution of Mₙ is derived here when X and Y are not too far apart and F is slightly constrained.
Dembo et al. (Sat,) studied this question.