The large deviation principle for the On/Off Weibull Sojourn Process

This article proves that the on-off renewal process with Weibull sojourn times satisfies the large deviation principle on a non-linear scale. Unusually, its rate function is not convex. Apart from on a compact set, the rate function is infinite, which enables us to construct natural processes that satisfy the LDP with non-trivial rate functions on more than one time scale

Confident decoding with GRAND

We establish that during the execution of any Guessing Random Additive Noise Decoding (GRAND) algorithm, an interpretable, useful measure of decoding confidence can be evaluated. This measure takes the form of a log-likelihood ratio (LLR) of the hypotheses that, should a decoding be found by a given query, the decoding is correct versus its being incorrect. That LLR can be used as soft output for a range of applications and we demonstrate its utility by showing that it can be used to confidently discard likely erroneous decodings in favor of returning more readily managed erasures. As an application, we show that feature can be used to compromise the physical layer security of short length wiretap codes by accurately and confidently revealing a proportion of a communication when code-rate is above capacity

Guesswork, large deviations and Shannon entropy

How hard is it to guess a password? Massey showed that a simple function of the Shannon entropy of the distribution from which the password is selected is a lower bound on the expected number of guesses, but one which is not tight in general. In a series of subsequent papers under ever less restrictive stochastic assumptions, an asymptotic relationship as password length grows between scaled moments of the guesswork and specific R´enyi entropy was identified. Here we show that, when appropriately scaled, as the password length grows the logarithm of the guesswork satisfies a Large Deviation Principle (LDP), providing direct estimates of the guesswork distribution when passwords are long. The rate function governing the LDP possesses a specific, restrictive form that encapsulates underlying structure in the nature of guesswork. Returning to Massey’s original observation, a corollary to the LDP shows that expectation of the logarithm of the guesswork is the specific Shannon entropy of the password selection process

How to estimate a cumulative process’s rate-function

Consider two sequences of bounded random variables, a value and a timing process, that satisfy the large deviation principle (LDP) with rate-function J(·,·) and whose cumulative process satisfies the LDP with rate-function I(·). Under mixing conditions, an LDP for estimates of I constructed by transforming an estimate of J is proved. For the case of cumulative renewal processes it is demonstrated that this approach is favorable to a more direct method as it ensures the laws of the estimates converge weakly to a Dirac measure at I

How to estimate a cumulative process’s rate-function

Consider two sequences of bounded random variables, a value and a timing process, that satisfy the large deviation principle (LDP) with rate-function J(·,·) and whose cumulative process satisfies the LDP with rate-function I(·). Under mixing conditions, an LDP for estimates of I constructed by transforming an estimate of J is proved. For the case of cumulative renewal processes it is demonstrated that this approach is favorable to a more direct method as it ensures the laws of the estimates converge weakly to a Dirac measure at I

Logarithmic asymptotics for unserved messages at a FIFO

We consider an infinite{buffered single server First In, First Out (FIFO) queue. Messages arrives at stochastic intervals and take random amounts of time to process. Logarithmic asymptotics are proved for the tail of the distribution of the number of messages awaiting service, under general large deviation and stability assumptions, and formulae presented for the asymptotic decay rate

Logarithmic asymptotics for unserved messages at a FIFO

We consider an infinite{buffered single server First In, First Out (FIFO) queue. Messages arrives at stochastic intervals and take random amounts of time to process. Logarithmic asymptotics are proved for the tail of the distribution of the number of messages awaiting service, under general large deviation and stability assumptions, and formulae presented for the asymptotic decay rate

Tail asymptotics for busy periods

The busy period for a queue is cast as the area swept under the random walk until it first returns to zero, $B$. Encompassing non-i.i.d. increments, the large-deviations asymptotics of $B$ is addressed, under the assumption that the increments satisfy standard conditions, including a negative drift. The main conclusions provide insight on the probability of a large busy period, and the manner in which this occurs: I) The scaled probability of a large busy period has the asymptote, for any $b>0$, \lim_{n\to\infty} \frac{1}{\sqrt{n}} \log P(B\geq bn) = -K\sqrt{b}, \hbox{where} \quad K = 2 \sqrt{-\int_0^{\lambda^*} \Lambda(\theta) d\theta}, \quad \hbox{with $\lambda^*=\sup\{\theta:\Lambda(\theta)\leq0\}$,} and with $\Lambda$ denoting the scaled cumulant generating function of the increments process. II) The most likely path to a large swept area is found to be a simple rescaling of the path on $[0,1]$ given by, [\psi^*(t) = -\Lambda(\lambda^*(1-t))/\lambda^*.] In contrast to the piecewise linear most likely path leading the random walk to hit a high level, this is strictly concave in general. While these two most likely paths have very different forms, their derivatives coincide at the start of their trajectories, and at their first return to zero. These results partially answer an open problem of Kulick and Palmowski regarding the tail of the work done during a busy period at a single server queue. The paper concludes with applications of these results to the estimation of the busy period statistics $(\lambda^*, K)$ based on observations of the increments, offering the possibility of estimating the likelihood of a large busy period in advance of observing one.Comment: 15 pages, 5 figure