The asymptotic properties of multivariate Szász-Mirakyan estimators for distribution functions supported on the nonnegative orthant are investigated. Explicit bias and variance expansions are derived on compact subsets of the interior, yielding sharp mean squared error characterizations and optimal smoothing rates. The analysis shows that the proposed Poisson smoothing yields a non-negligible variance reduction relative to the empirical distribution function, leading to asymptotic efficiency gains that can be quantified through local and global deficiency measures. The behavior of the estimator near the boundary of its support is examined separately. Under a boundary-layer scaling that preserves nondegenerate Poisson smoothing as the evaluation point approaches the boundary of $[0,\infty)^d$, bias and variance expansions are obtained that differ fundamentally from those in the interior region. In particular, the variance reduction mechanism disappears at leading order, implying that no asymptotically optimal smoothing parameter exists in the boundary regime. Central limit theorems and almost sure uniform consistency are also established. Together, these results provide a unified asymptotic theory for multivariate Szász-Mirakyan distribution estimation and clarify the distinct roles of smoothing in the interior and boundary regions.

翻译：本文研究了定义在非负象限上的分布函数的多元Szász-Mirakyan估计量的渐近性质。在内部区域的紧子集上，推导了显式的偏差与方差展开式，从而得到了精确的均方误差刻画以及最优平滑速率。分析表明，相对于经验分布函数，所提出的泊松平滑方法能够带来不可忽略的方差缩减，这一渐近效率提升可通过局部与全局缺陷度量进行量化。此外，本文单独考察了估计量在其支撑集边界附近的行为。在边界层尺度变换下，当评估点趋近于 $[0,\infty)^d$ 的边界时，泊松平滑保持非退化，此时得到的偏差与方差展开式与内部区域的结果存在本质差异。特别地，方差缩减机制在主导阶消失，这意味着在边界区域不存在渐近最优的平滑参数。本文还建立了中心极限定理与几乎必然一致收敛性。这些结果共同构成了多元Szász-Mirakyan分布估计的统一渐近理论，并阐明了平滑在内部区域与边界区域所扮演的不同角色。