In this paper we prove the large deviation principle for a class of weighted means of linear combinations of independent Poisson distributed random variables, which converge weakly to a normal distribution. The interest in these linear combinations is motivated by the diffusion approximation in Lansky [On approximations of Stein's neuronal model, J. Theoret. Biol. 107 (1984), pp. 631–647] of the Stein's neuronal model (see Stein [A theoretical analysis of neuronal variability, Biophys. J. 5 (1965), pp. 173–194]). We also prove an analogue result for sequences of multivariate random variables based on the diffusion approximation in Tamborrino, Sacerdote, and Jacobsen [Weak convergence of marked point processes generated by crossings of multivariate jump processes. Applications to neural network modeling, Phys. D 288 (2014), pp. 45–52]. The weighted means studied in this paper generalize the logarithmic means. We also investigate moderate deviations.

在本文中，我们证明了独立泊松分布随机变量的线性组合的一类加权均值的大偏差原理，该弱组合收敛到正态分布。对这些线性组合的兴趣是由Lansky中的扩散近似激发的[关于Stein神经元模型的近似，J。Theoret。生物学 107（1984），第631–647页]（请参见Stein [神经元变异性的理论分析，Biophys。J. 5（1965），第173–194页]）。我们还基于Tamborrino，Sacerdote和Jacobsen的扩散近似证明了多元随机变量序列的模拟结果[由多元跳跃过程的交叉产生的标记点过程的弱收敛。在神经网络建模中的应用，物理。D 288（2014），第45-52页]。本文研究的加权均值概括了对数均值。我们还研究了中等偏差。