The main contribution of this paper lies in two aspects. First, we focus on deriving Bernstein-type inequalities for both geometrically and algebraically irregularly spaced NED random fields, including time series as a special case. Furthermore, by introducing the notion of “effective dimension” to the index set of the random number field, we show that the sharpness of the inequality is only related to this “effective dimension”. To our knowledge, our paper may be the first to reflect this phenomenon. Therefore, the first contribution of this paper can be more or less regarded as an update of \citeA{xu2018sieve}’s pioneering work. Furthermore, as a result of the first contribution, we also get the Bernstein-type inequality for geometrically irregularly spaced $\alpha$ mixed random fields. His second aspect of our contribution is to show the L_{\infty}$ convergence rates of a number of interesting kernel-based nonparametric estimators based on the above inequalities. To do this, two deviation inequalities for the maximum of the empirical process are derived under the NED and $\alpha$ mixed conditions, respectively. Next, we prove the achievability of the optimal rate of the local linear estimator for nonparametric regression for an irregularly spaced NED random field. This updates another pioneering study on this topic. Subsequently, we also analyze the uniform convergence rate of unimodal regression under the same NED conditions. In addition, following the guide from \citeA{rigollet2009optimal}, we also prove that the kernel-based plug-in density level set estimator may be optimal up to logarithmic coefficients. On the other hand, the uniform convergence rate of the simple local polynomial density estimator \cite{cattaneo2020simple} is also derived when the data are collected from the $\alpha$ mixed random field.
