Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 31 / 2 / c / Solution

Let $I_X,I_Y$ be the original rates and ${\mu }_X,{\mu }_Y$ their unique zero points. Suppose first that $H<G$. At speed $b_n=n^{2(1-G)}$, the preceding argument gives X the rate zero at ${\mu }_X$ and infinity elsewhere, while Y retains $I_Y$. Independence allows the product large-deviation principle, and addition is continuous, so the contraction principle for large deviations gives the sum rateThis is good, has unique zero ${\mu }_X+{\mu }_Y$, and inherits a strictly positive finite value by translation from $I_Y$. Thus the sum has Hurstiness G. The case $G<H$ is identical with the roles exchanged.

If $G=H$, the same product and contraction results apply to the original rates at their common speed, giving their infimal convolutionIt is good. Every finite fiber infimum is attained: a minimizing sequence lies in a product of compact sublevel sets and has a subsequence converging in the closed addition fiber. Consequently its only zero is ${\mu }_X+{\mu }_Y$. Choose $\hat{x}$ with $0<I_X(\hat{x})<\infty$ and set $\hat{z}=\hat{x}+{\mu }_Y$. Then $\hat{z}$ differs from the zero point and $0<K(\hat{z})\le I_X(\hat{x})<\infty$. The nondegeneracy requirement is therefore satisfied here too. In all cases,The general results used are the independent product large-deviation theorem and the continuous contraction principle. Checking the zero set and a positive finite value is necessary in addition to obtaining an LDP.