  A confidence sequence (CS) is a sequence of confidence sets that contains a
target parameter of an underlying stochastic process at any time step with high
probability. This paper proposes a new approach to constructing CSs for means
of bounded multivariate stochastic processes using a general gambling
framework, extending the recently established coin toss framework for bounded
random processes. The proposed gambling framework provides a general recipe for
constructing CSs for categorical and probability-vector-valued observations, as
well as for general bounded multidimensional observations through a simple
reduction. This paper specifically explores the use of the mixture portfolio,
akin to Cover's universal portfolio, in the proposed framework and investigates
the properties of the resulting CSs. Simulations demonstrate the tightness of
these confidence sequences compared to existing methods. When applied to the
sampling without-replacement setting for finite categorical data, it is shown
that the resulting CS based on a universal gambling strategy is provably
tighter than that of the posterior-prior ratio martingale proposed by
Waudby-Smith and Ramdas.
