  We propose a sequential optimizing betting strategy in the multi-dimensional
bounded forecasting game in the framework of game-theoretic probability of
Shafer and Vovk (2001). By studying the asymptotic behavior of its capital
process, we prove a generalization of the strong law of large numbers, where
the convergence rate of the sample mean vector depends on the growth rate of
the quadratic variation process. The growth rate of the quadratic variation
process may be slower than the number of rounds or may even be zero. We also
introduce an information criterion for selecting efficient betting items. These
results are then applied to multiple asset trading strategies in discrete-time
and continuous-time games. In the case of continuous-time game we present a
measure of the jaggedness of a vector-valued continuous process. Our results
are examined by several numerical examples.
