**MSC:**- 60G60 Random fields
- 62G20 Asymptotic properties

measurements are taken at $t_i\in (0,1)^q$ and there is a family of

distributions $Q_{\eta},\,\eta\in(a,b)$ and $g_0:\,[0,1]^q\to (a,b)$ such

that $\mathcal{L}(Y_i)=Q_{g_0(t_i)}$. The main results of this paper is a

limit theorem for the sequence of random fields

\[ W_m(t)=\frac{1}{\sqrt{m}}\Sum_{j=1}^{m}

\left(Y_j-g_0(t_j)\right) I_{(0,t]}(t_j)\ .\]

Limit theorems for functionals of $W_m$ are used to construct an asymptotic

$\alpha$-test for $H_0:\,g=g_0$ versus $H_A:\,g\not=g_0$. For a regression

model the asymptotic power is investigated under local alternatives.