Large deviations of a scaled geometric random variable ($I(x)=x\log (1-p{)}^{-1}(x\ge 0)$)

For a geometric random variable $X$ with $\mathbb{P}(X=k)=p(1-p{)}^{k-1}$, $0<p<1$, the family $X/L$ has large-deviation speed $L$ and good rate function $I(x)=ax$ for $x\ge 0$, infinite otherwise, where $a=-\log (1-p)$. The exact tail $\mathbb{P}(X/L\ge x)=(1-p{)}^{\lceil Lx\rceil -1}$ gives the upper bound. A lattice point tending to any positive $x$ supplies the open-neighborhood lower bound; neighborhoods of zero have probability tending to one. Finite sublevel sets are compact intervals.

For fixed finitely many independent geometric random variables, the product large-deviation principle gives rate ${\sum }_i{a}_i{x}_i$ 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 ${\sum }_i{x}_i=s$. 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.