The stochastic convergence of Bernstein polynomial estimators in a triangular array

In this paper, we consider the Bernstein polynomial of the empirical distribution function $ F_n $ Fn under a triangular sample, which we denote by $ \hat {F}_{m,n} $ F^m,n. For the recentered and normalised statistic $ n^{1/2}\left (\hat {F}_{m,n}(x)-E_{G_n}\hat {F}_{m,n}(x)\right ) $ n1/2(F^m,n(x)−EGnF^m,n(x)), where x is defined on the interval $ (0,1) $ (0,1), the stochastic convergence to a Brownian bridge is derived. The main technicality in proving the normality is drawn off into a stochastic equicontinuity condition. To obtain the equicontinuity, we derive the uniform law of large numbers (ULLN) over a class of functions $ \sup _{\mathscr {H}}\left |\left (P_n-E_{G_n}\right )h\right | $ supH|(Pn−EGn)h| by domination conditions of random covering numbers and covering integrals. In addition, we also derive the asymptotic covariance matrix for biavariant vector of Bernstein estimators. Finally, numerical simulations are presented to verify the validity of our main results.