Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 79 / 1 / e / 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. Convergence to the same typical value does not imply the same large-deviation behavior.