论文标题
使用尺寸偏见和广义零偏置耦合的随机总和的伽马，高斯和泊松近似值
Gamma, Gaussian and Poisson approximations for random sums using size-biased and generalized zero-biased couplings
论文作者
论文摘要
令$ y = x_1+\ cdots+x_n $是随机数的可交换随机变量的总和，其中随机变量$ n $独立于$ x_j $，而$ x_j $来自Tallis（1962）引入的广义多项式模型。这放松了$ x_j $是独立的经典假设。当随机变量$ x_j $中心或$ n $具有POISSON分布时，我们使用零偏置的耦合及其概括在$ y $的近似值中给出明确的错误界限。我们进一步建立了一个明确的限制，即通过伽玛分布在$ y $的近似值中，对于$ n $是泊松的特殊情况。最后，我们简要评论了使用尺寸偏见的耦合的类似泊松近似结果。独立$ x_j $的特殊情况在整个过程中都受到特别关注。除了建立超出独立环境的结果外，我们的边界在独立案例中具有已知结果具有竞争力。
Let $Y=X_1+\cdots+X_N$ be a sum of a random number of exchangeable random variables, where the random variable $N$ is independent of the $X_j$, and the $X_j$ are from the generalized multinomial model introduced by Tallis (1962). This relaxes the classical assumption that the $X_j$ are independent. We use zero-biased coupling and its generalizations to give explicit error bounds in the approximation of $Y$ by a Gaussian random variable in Wasserstein distance when either the random variables $X_j$ are centred or $N$ has a Poisson distribution. We further establish an explicit bound for the approximation of $Y$ by a gamma distribution in stop-loss distance for the special case where $N$ is Poisson. Finally, we briefly comment on analogous Poisson approximation results that make use of size-biased couplings. The special case of independent $X_j$ is given special attention throughout. As well as establishing results which extend beyond the independent setting, our bounds are shown to be competitive with known results in the independent case.
