Large-deviation speed ($a_N$)

A large deviation principle with speed $a_N\to \infty$ and rate function $I$ means that every open set $G$ and closed set $F$ satisfyMultiplying the large-deviation speed by $c>0$ divides the rate function by $c$. The speed must match the rare-event scale: a fixed exponential random variable divided by $N^p$ has speed $N^p$, whereas a sum of $N$ independent random variables with a normal distribution, divided by $N^p$, has speed $N^{2p-1}$ when $p>1$.

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 independent real sequences, put the faster sequence at the slower speed by slower-speed degeneration of a large deviation principle, then use the product large-deviation principle and contraction principle for large deviations for addition. Unequal speeds give a translate of the slower sequence's rate. Equal speeds give the infimal convolution of the two rates. Its unique zero is the sum of their zero points, and a translated positive finite point has positive finite cost, since compactness attains each finite fiber infimum. Thus the defining nondegeneracy is preserved.

// Target: queueing-theory.bigb