= Solution
With $n$ fixed, the asymptotic family is the scaled sum
$$
S^L=\frac{X_1+\cdots+X_n}{L},\qquad L\to\infty.
$$
An unscaled fixed random variable alone has no diverging parameter for a <large deviation principle>; this scaling is the continuation of part (a). Suppose first $0<p_i<1$ and put $a_i=-\log(1-p_i)$. <Independence> and the <product large-deviation principle> give the vector $(X_1/L,\ldots,X_n/L)$ rate
$$
I(x_1,\ldots,x_n)=
\begin{cases}
\sum_{i=1}^na_ix_i,&x_i\geq0\ \text{for every }i,\\
+\infty,&\text{otherwise}.
\end{cases}
$$
Addition is <continuous>, so the <contraction principle for large deviations> gives
$$
J(s)=\inf_{\substack{x_i\geq0\\\sum_i x_i=s}}\sum_i a_ix_i.
$$
For $s\geq0$ every feasible vector has cost at least $s a_*$, where $a_*=\min_i a_i$. Equality holds by assigning all scaled excess to an index attaining $a_*$. For $s<0$ the constraint set is empty. Thus
$$
\boxed{J(s)=
\begin{cases}
s\min_i[-\log(1-p_i)],&s\geq0,\\
+\infty,&s<0,
\end{cases}\qquad\text{speed }L.}
$$
Equivalently, the coefficient is $-\log(1-p_*)$ with $p_*=\min_i p_i$. On the exponential scale the rare large sum is carried by a variable having the slowest geometric tail. Several equal slowest tails can change polynomial prefactors, but not this <rate function>. This is the <large deviations of a fixed sum of geometric random variables>.
As a direct check of the tail exponent, for each $0<\theta<a_*$ the <moment-generating function> of the sum is finite, so the <Chernoff bound> gives an upper rate at most $-\theta s$ for $s>0$. Letting $\theta\uparrow a_*$ gives $-a_*s$. For a lower neighborhood bound, keep all other variables at fixed finite values and let a slowest-tail variable take an integer near $Ls$; its probability has rate $-a_*s$. If some $p_i=1$, their deterministic contributions disappear after division by $L$, so minimize over the nondeterministic variables. If all $p_i=1$, the rate is zero only at zero and infinite elsewhere.
Back to article page
