  We introduce fundamental bounds on achievable cumulative rate distribution
functions (CRDF) to characterize a sequential encoding process that ensures
lossless or lossy reconstruction subject to an average distortion criterion
using a non-causal decoder. The CRDF describes the rate resources spent
sequentially to compress the sequence. We also include a security constraint
that affects the set of achievable CRDF. The information leakage is defined
sequentially based on the mutual information between the source and its
compressed representation, as it evolves. To characterize the security
constraints, we introduce the concept of cumulative leakage distribution
functions (CLF), which determines the allowed information leakage as
distributed over encoded sub-blocks. Utilizing tools from majorization theory,
we derive necessary and sufficient conditions on the achievable CRDF for a
given independent and identically distributed (IID) source and CLF. One primary
result of this paper is that the concave-hull of the CRDF characterizes the
optimal achievable rate distribution.
