  We introduce the first probabilistic framework tailored for sequential random
projection, an approach rooted in the challenges of sequential decision-making
under uncertainty. The analysis is complicated by the sequential dependence and
high-dimensional nature of random variables, a byproduct of the adaptive
mechanisms inherent in sequential decision processes. Our work features a novel
construction of a stopped process, facilitating the analysis of a sequence of
concentration events that are interconnected in a sequential manner. By
employing the method of mixtures within a self-normalized process, derived from
the stopped process, we achieve a desired non-asymptotic probability bound.
This bound represents a non-trivial martingale extension of the
Johnson-Lindenstrauss (JL) lemma, marking a pioneering contribution to the
literature on random projection and sequential analysis.
