On asymptotic properties of nonparametric estimators of characteristic function

Let $(X_j,j\geqslant1)$ be independent observations of a random $k$-dimensional vector $X$ with (unknown) distribution $F$ which is known to belong some family $\mathcal F$ of probability measures on Euclidean space $R^k$. Estimators $\varphi_n(\cdot)$ of a charactestic function (cf. f.) $\varphi(F_j)$ based on observations $(X_j, 1\leqslant j\leqslant n)$ are considered. Locally uniform (in $F\in\mathcal F$) weak convergence (SEE DEFINITION 1) of random functions $\omega_{nF(\cdot)}=\sqrt n[\varphi(\cdot)-\varphi(F_j)]$ to Gaussian random functions $\omega_F(\cdot)$ is established. Estimators $\varphi_n(\cdot)$ may be here both empirical ch. f. and some generally saying random of it (i. e. this transformation depends on observations $X_1, \dots, X_n$). Covariances of random function $\omega_F(\cdot)$ are obtained. Examples of nonparametric estimators of ch. f. $\varphi(F,\cdot)$ tor various families $\mathcal F$ are considered.