Let S_n = X₁ + X₂ + · · · + X_n, where X_j = max(ξ_j, ξ_j+1), and ξ₁, ξ₂, . . . , ξ_n+1 are independent Bernoulli random variables. If all P(ξ_j = 1) are small, then we approximate Sn by a compound Poisson random variable with two matching moments. If all P(ξ_j = 1) are large, then we apply compound Poisson and negative binomial approximations to n − S_n. We estimate the accuracy of approximation in the total-variation and Kolmogorov metrics. We also show that similar results hold for sums of minima of Bernoulli variables. In the proofs, we use Heinrich’s method.

中文翻译：

---

伯努利变量极值和的复合泊松近似

令S _n = X ₁ + X ₂ + · · · + X _n，其中X _j = max( ξ _j , ξ _{j +1} )，且ξ ₁ , ξ ₂ , . . . , ξ _{n +1}是独立的伯努利随机变量。如果所有P ( ξ _j = 1) 都很小，那么我们通过具有两个匹配矩的复合泊松随机变量来近似Sn 。如果所有P ( ξ _j= 1) 很大，然后我们将复合泊松和负二项式近似应用于n - S _n。我们估计了总变差和 Kolmogorov 度量的近似精度。我们还表明，类似的结果适用于伯努利变量的最小值之和。在证明中，我们使用 Heinrich 的方法。