  Solomonoff's uncomputable universal prediction scheme $\xi$ allows to predict
the next symbol $x_k$ of a sequence $x_1...x_{k-1}$ for any Turing computable,
but otherwise unknown, probabilistic environment $\mu$. This scheme will be
generalized to arbitrary environmental classes, which, among others, allows the
construction of computable universal prediction schemes $\xi$. Convergence of
$\xi$ to $\mu$ in a conditional mean squared sense and with $\mu$ probability 1
is proven. It is shown that the average number of prediction errors made by the
universal $\xi$ scheme rapidly converges to those made by the best possible
informed $\mu$ scheme. The schemes, theorems and proofs are given for general
finite alphabet, which results in additional complications as compared to the
binary case. Several extensions of the presented theory and results are
outlined. They include general loss functions and bounds, games of chance,
infinite alphabet, partial and delayed prediction, classification, and more
active systems.
