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In this paper, we consider the regression model with fixed design:Yi=g(xi)+εi{Y}_{i}=g\left({x}_{i})+{\varepsilon }_{i},1≤i≤n1\le i\le n, where{xi}\left\{{x}_{i}\right\}are the nonrandom design points, and{εi}\left\{{\varepsilon }_{i}\right\}is a sequence of martingale, andggis an unknown function. Nonparametric estimatorgn(x){g}_{n}\left(x)ofg(x)g\left(x)will be introduced and its strong convergence properties are established.

Strong consistency of regression function estimator with martingale difference errors