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

Let $X$ have the positive-support geometric distribution $\mathbb{P}(X=k)=p(1-p{)}^{k-1}$ with $0<p<1$. As $L\to \infty$, $X/L$ satisfies a large deviation principle of speed $L$ and good rate functionFor every real threshold $x$, the exact upper-tail formula isThe maximum with zero matters at $x=0$ and at thresholds below the first lattice point; the uncapped exponent would otherwise give a Probability greater than one.

For positive $x$, dividing the logarithm by $L$ gives the limit $-ax$. These tails bound closed sets through their leftmost nonnegative point. For an open neighborhood of a positive $x$, choose integers $k_L$ with $k_L/L\to x$; their individual masses have the same logarithmic limit. Neighborhoods of zero have Probability tending to one, supplying rate zero there. Negative-only sets have Probability zero, and the rate sublevel sets are compact intervals.

For example, with $p=1/2$, $L=100$ and $x=1/5$, the event is $X\ge 20$ and has Probability $2^{-19}$. Meanwhile $X/L$ converges in Probability to zero. The nontrivial rate describes the rare excursions away from that degenerate weak limit.

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.