In this work, we investigated the statistical convergence behavior of Vendi diversity scores estimated from empirical samples. We highlighted that, due to the high computational complexity of the score for datasets larger than a few tens of thousands of generated data points, the score is often calculated using sample sizes below 10,000. We demonstrated that such restricted sample sizes do not pose a problem for statistical convergence as long as the kernel feature dimension is bounded. However, our numerical results showed a lack of convergence to the population Vendi when using an infinite-dimensional kernel map, such as the Gaussian kernel. To address this gap, we introduced the truncated population Vendi as an alternative target quantity for diversity evaluation. We showed that existing Nyström and FKEA methods for approximating Vendi scores concentrate around this truncated population Vendi. An interesting future direction is to explore the relationship between other kernel approximation techniques and the truncated population Vendi. Also, a comprehensive analysis of the computational-statistical trade-offs involved in estimating the Vendi score is another relevant future direction.