Goodness-of-fit tests based on a measure of divergency between two scalar products

Let $X_1,\dots,X_n$ be independent random variables with values in $(\mathfrak X, \mathfrak A)$ identically distributed according to a measure $P$ and let $\varphi_1,\dots,\varphi_k$ be functions on $(\mathfrak X, \mathfrak A)$ square integrable with respect to $P$. The asymptotic distribution (as $n\to\infty$) of certain tests of hypotheses on values of the functionals $\pi_1(P)=\mathbf E_P\varphi_1,\dots,\pi_k(P)=\mathbf E_P\varphi_k$ is investigated. The tests under consideration depend only on the values of the statistics $\displaystyle\overline\varphi_1=\frac{1}{n}\sum_{\nu=1}^n\varphi_1(X_\nu)\dots$, $\displaystyle\overline\varphi_k=\frac{1}{n}\sum_{\nu=1}^n\varphi_k(X_\nu)$and are based on a measure of divergency between the two scalar products introduced in [4].