Let $(A_i)_{i \geq 0}$ be a finite-state irreducible aperiodic Markov chain and f a lattice score function such that the average score is negative and positive scores are possible. Define $S_0\coloneqq 0$ and $S_k\coloneqq \sum_{i=1}^k f(A_i)$ the successive partial sums, $S^+$ the maximal non-negative partial sum, $Q_1$ the maximal segmental score of the first excursion above 0, and $M_n\coloneqq \max_{0\leq k\leq\ell\leq n} (S_{\ell}-S_k)$ the local score, first defined by Karlin and Altschul (1990). We establish recursive formulae for the exact distribution of $S^+$ and derive a new approximation for the tail behaviour of $Q_1$ , together with an asymptotic equivalence for the distribution of $M_n$ . Computational methods are explicitly presented in a simple application case. The new approximations are compared with those proposed by Karlin and Dembo (1992) in order to evaluate improvements, both in the simple application case and on the real data examples considered by Karlin and Altschul (1990).

中文翻译：

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马尔可夫序列中最大分段分数分布的改进

让$(A_i)_{i \geq 0}$是一个有限状态不可约非周期马尔可夫链和F一个格分数函数，使得平均分数为负数，正数是可能的。定义$S_0\coloneqq 0$和$S_k\coloneqq \sum_{i=1}^kf(A_i)$连续的部分和，$S^+$最大非负部分和，$Q_1$大于 0 的第一次远足的最大分段分数，以及$M_n\coloneqq \max_{0\leq k\leq\ell\leq n} (S_{\ell}-S_k)$这本地分数，首先由 Karlin 和 Altschul (1990) 定义。我们建立了精确分布的递归公式$S^+$并推导出尾部行为的新近似值$Q_1$，以及分布的渐近等价性$M_n$. 计算方法在一个简单的应用案例中显式呈现。新的近似值与 Karlin 和 Dembo (1992) 提出的近似值进行比较，以评估在简单应用案例和 Karlin 和 Altschul (1990) 考虑的真实数据示例中的改进。