  The major contributions of this paper lie in two aspects. Firstly, we focus
on deriving Bernstein-type inequalities for both geometric and algebraic
irregularly-spaced NED random fields, which contain time series as special
case. Furthermore, by introducing the idea of "effective dimension" to the
index set of random field, our results reflect that the sharpness of
inequalities are only associated with this "effective dimension". Up to the
best of our knowledge, our paper may be the first one reflecting this
phenomenon. Hence, the first contribution of this paper can be more or less
regarded as an update of the pioneering work from \citeA{xu2018sieve}.
Additionally, as a corollary of our first contribution, a Bernstein-type
inequality for geometric irregularly-spaced $\alpha$-mixing random fields is
also obtained. The second aspect of our contributions is that, based on the
inequalities mentioned above, we show the $L_{\infty}$ convergence rate of the
many interesting kernel-based nonparametric estimators. To do this, two
deviation inequalities for the supreme of empirical process are derived under
NED and $\alpha$-mixing conditions respectively. Then, for irregularly-spaced
NED random fields, we prove the attainability of optimal rate for local linear
estimator of nonparametric regression, which refreshes another pioneering work
on this topic, \citeA{jenish2012nonparametric}. Subsequently, we analyze the
uniform convergence rate of uni-modal regression under the same NED conditions
as well. Furthermore, by following the guide of \citeA{rigollet2009optimal}, we
also prove that the kernel-based plug-in density level set estimator could be
optimal up to a logarithm factor. Meanwhile, when the data is collected from
$\alpha$-mixing random fields, we also derive the uniform convergence rate of a
simple local polynomial density estimator \cite{cattaneo2020simple}.
