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A sequence has Hurstiness H when it satisfies a large deviation principle at the displayed speed with a good rate function having a unique zero and at least one strictly positive finite value. This definition concerns a sequence's exponential probability scale; it does not assert a sample-path Hurst exponent or self-similarity. Larger H means slower rare-event decay. The positive finite value excludes the slower-speed degeneration of a large deviation principle.

For an open neighborhood D of mu, its complement is closed and excludes mu. The preceding upper bound is minus infinity, so, for example, eventually $\mathbb{P}(X_n\notin D)\le e^{-b_n}\to 0$. Hence the sequence concentrates in every neighborhood of mu.

If an open E contains mu, its probability tends to one. Its logarithm divided by $b_n\to \infty$ therefore tends to zero, which equals $-{\inf }_E{I}^{\prime }$. If E does not contain mu, that latter value is minus infinity and its lower bound is automatic. ThusWith the closed-set upper bound this proves the slower-speed degeneration of a large deviation principle. This degenerate rate has no strictly positive finite value, so it does not give the sequence a different Hurstiness under the stated definition.

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.