  A generalization of the Renormalization Group, which describes
order-parameter fluctuations in finite systems, is developed in the specific
context of percolation. This ``Stochastic Renormalization Group'' (SRG)
expresses statistical self-similarity through a non-stationary branching
process. The SRG provides a theoretical basis for analytical or numerical
approximations, both at and away from criticality, whenever the correlation
length is much larger than the lattice spacing (regardless of the system size).
For example, the SRG predicts order-parameter distributions and finite-size
scaling functions for the complete crossover between phases. For percolation,
the simplest SRG describes structural quantities conditional on spanning, such
as the total cluster mass or the minimum chemical distance between two
boundaries. In these cases, the Central Limit Theorem (for independent random
variables) holds at the stable, off-critical fixed points, while a ``Fractal
Central Limit Theorem'' (describing long-range correlations) holds at the
unstable, critical fixed point. This first part of a series of articles
explains these basic concepts and a general theory of crossover. Subsequent
parts will focus on limit theorems and comparisons of small-cell SRG
approximations with simulation results.

Statistical Mechanics;Disordered Systems and Neural Networks