= Solution
All three normalized sums converge in probability to $\mu$: the <weak law of large numbers> controls $S_L/L$, and the two fixed <random variables> vanish after division by $L$. Their rare fluctuations nevertheless differ. A fixed <normal random variable> would have to fluctuate by order $L$ to alter the mean by a fixed amount, and that costs $\exp(-\text{constant}\,L^2)$. A fixed <exponential random variable> can do the same for a cost of order $\exp(-\text{constant}\,L)$, exactly the <large-deviation speed> of the empirical mean.
Below $\mu+\lambda\sigma^2$, the optimal exponential contribution is zero, and the <Gaussian> empirical mean supplies the fluctuation. Above it, the optimal empirical mean remains at $\mu+\lambda\sigma^2$, while $B/L$ supplies $x-\mu-\lambda\sigma^2$. This explains the quadratic-to-linear change in the <rate function> and why only the upper tail is altered. \b[Convergence to the same typical value does not imply the same large-deviation behavior.]
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