= Large deviations of a fixed sum of geometric random variables
{title2=$J(s)=s\min_i[-\log(1-p_i)]\ (s\geq0)$}
For fixed finitely many independent <geometric random variables>, the <product large-deviation principle> gives rate $\sum_i a_ix_i$ on the nonnegative orthant for their scaled vector, where $a_i=-\log(1-p_i)$. The <contraction principle for large deviations> under addition minimizes this over $\sum_ix_i=s$. The answer is $s\min_i a_i$, achieved by assigning all scaled excess to a coordinate with the slowest geometric tail. The rate is infinite for negative $s$. Deterministic coordinates with $p_i=1$ contribute zero on this scale; if all coordinates are deterministic, only zero has finite rate.
Back to article page
