We use the method of moments to establish the limiting spectral distribution (LSD) of appropriately scaled large dimensional
random symmetric circulant, reverse circulant, Toeplitz and Hankel matrices which have suitable band structures. The input
sequence used to construct these matrices is assumed to be either i.i.d. with mean zero and variance one or independent and
appropriate finite fourth moment. The class of LSD includes the normal and the symmetrized square root of chi-square with
two degrees of freedom. In several other cases, explicit forms of the limit do not seem to be obtainable but the limits can
be shown to be symmetric and their second and the fourth moments can be calculated with some effort. Simulations suggest some
further properties of the limits.
As the variance (the square of the minimum L 2-norm, i.e., the square of the scatter) is one of the basic characteristics of the conventional statistics, it is of practical importance to know the errors of its determination for different parent distribution types. This statement is outstandingly valid for the geostatistics because the (h) variogram (called also as semi-variogram) is defined as the half variance of some quantity-difference (e.g. difference of ore concentrations) in function of the h dis- tance of the measuring points and this g (h)-curve plays a basic role in the classical geostatistics. If the scatter (s VAR) is chosen to characterize the determination uncertainties of the variance (denoted the latter by VAR), this can be easily calculate as the quotient A VAR= Ön (if the number n of the elements in the sample is large enough); for the so-called asymptotic scatter A VAR is known a simple formula (containing the fourth moment). The present paper shows that the AVAR has finite value unfortunately only for about a quarter of distribution types occurring in the earth sciences, it must be especially accentuate that A VARhas infinite value for that distribution type which most frequent occurs in the geostatistics. It is proven by the present paper that the law of large numbers is always fulfilled (i.e., the error always decreases if n increases) for the error-determinations if the semi-intersextile range is accepted (instead of the scatter); the single (quite natural) condition is the existence of the theoretical variance for the parent distribution. __