This paper investigates the estimation of the interaction function for a class of McKean-Vlasov stochastic differential equations. The estimation is based on observations of the associated particle system at time $T$, considering the scenario where both the time horizon $T$ and the number of particles $N$ tend to infinity. Our proposed method recovers polynomial rates of convergence for the resulting estimator. This is achieved under the assumption of exponentially decaying tails for the interaction function. Additionally, we conduct a thorough analysis of the transform of the associated invariant density as a complex function, providing essential insights for our main results.

翻译：本文研究了一类McKean-Vlasov随机微分方程中交互函数的估计问题。该估计基于时间$T$时刻关联粒子系统的观测值，考虑了时间范围$T$和粒子数$N$同时趋于无穷的情形。我们提出的方法使估计量恢复多项式收敛速率。这一结果是在假设交互函数具有指数衰减尾部的条件下实现的。此外，我们对关联不变密度的变换作为复函数进行了深入分析，为本文主要结果提供了关键见解。