The consistency for the estimators of semiparametric regression model based on weakly dependent errors

For the semiparametric regression model: $$Y^{(j)}(x_{in},~t_{in})=t_{in}\beta +g(x_{in})+e^{(j)}(x_{in}),~1\le j\le m,~1\le i \le n$$ Y ( j ) ( x i n , t i n ) = t i n β + g ( x i n ) + e ( j ) ( x i n ) , 1 ≤ j ≤ m , 1 ≤ i ≤ n , where $$x_{in}\in \mathbb {R}^p$$ x i n ∈ R p , $$t_{in}\in \mathbb {R}$$ t i n ∈ R are known to be nonrandom, g is an unknown continuous function on a compact set A in $$\mathbb {R}^p$$ R p , $$e^{(j)}(x_{in})$$ e ( j ) ( x i n ) are $$\tilde{\rho }$$ ρ ~ -mixing random variables with mean zero, $$Y^{(j)}(x_{in},t_{in})$$ Y ( j ) ( x i n , t i n ) are random variables which are observable at points $$x_{in}$$ x i n and $$t_{in}$$ t i n . In the paper, we establish the strong consistency, r-th ( $$r>2$$ r > 2 ) mean consistency and complete consistency for estimators $$\beta _{m,n}$$ β m , n and $$g_{m,n}(x)$$ g m , n ( x ) of $$\beta $$ β and g, respectively. The results obtained in the paper extend the corresponding ones for independent random variables and $$\varphi $$ φ -mixing random variables.