Category Archives: Uncategorized

A 2016 paper by M Wilkinson in Physical Review Letters suggests that large-deviation theory is a suitable framework for studying the phenomenon of unexpectedly rapid rain formation in collector-drop collision processes. Wilkinson derives asymptotic approximation formulae for a set of exact large-deviation functions in the collector-drop model, such as the cumulant generating function and theContinue reading “Accurate analytical approximation formulae for large-deviation analysis of rain formation”

The following is a record of some notes I made on the main mathematical developments in Michael Wilkinson’s 2016 Physical Review Letter on the large deviation analysis of rapid-onset rain showers (reference: Wilkinson, M., 2016, Large Deviation Analysis of Rapid Onset of Rain Showers, Phys. Rev. Lett. 116, 018501). This publication will be referred toContinue reading “A Phys. Rev. Letter by M Wilkinson on large deviation analysis of rapid-onset rainfall”

A paper by Sen and Balakrishnan pointed out that Lagrange’s polynomial interpolation formula can be used to construct relatively simple proofs of some complicated-looking sum/product identities. (Sen, A., Balakrishnan, N., 1999, Convolution of geometrics and a reliability problem, Statistics and Probability Letters, 43, pp. 421-426). In this note I want to make a quick recordContinue reading “Using Lagrange’s interpolation formula to prove complicated sum/product identities”

I needed to find an extremum of a function of the form by choice of , ideally obtaining a closed form solution for the critical value . The precise context is not relevant here – I just want to record an asymptotic approximation trick I was able to use to solve this problem. Differentiating andContinue reading “Using an asymptotic approximation to obtain a closed form solution”

In this note I want to quickly go through the motions of applying the Bromwich integral (i.e., inverse Laplace transform) to the cumulant generating function of an exponential variate with probability density function for . The cumulant generating function is the log-Laplace transform of , obtained as Note that we need to assume for theContinue reading “Applying the Bromwich integral to a cumulant generating function”

Let the Fourier transform of a function be The corresponding inverse Fourier transform would then be As an exercise, what I want to do in this note is derive the characteristic function of an exponential random variate with probability density function for . The required characteristic function is the Fourier transform of this density function.Continue reading “Inverse Fourier transform of a characteristic function”

In this note I want to record an interesting issue I have noticed with regard to how the form of the large deviation entropy function can `flip’ depending on the way the underlying cumulant generating function is formulated. The cumulant generating function of a random variable is the natural log of its moment generating function.Continue reading “Two versions of the exponential cumulant generating function”

The Legendre transform is a mechanism for converting a relationship like into a relationship like In other words, by going from a function to its Legendre transform and vice versa, the roles of and in (1) and (2) can be reversed. In moving from (1) to (2), the required Legendre transform is because differentiating bothContinue reading “The role of the Legendre transform in large deviation theory”

Markov’s inequality is a basic tool for putting upper bounds on tail probabilities. To derive it, suppose is a non-negative random variable. We want to put an upper bound on the tail probability where . We do this by observing that where is the indicator function which takes the value 1 when and the valueContinue reading “Markov’s inequality and the Cramér-Chernoff bounding method”

In relation to a calculation involving sums of random variables, I needed to differentiate with respect to a double integral of the form The parameter appears in both upper limits, but appears on its own in the outer upper limit, and as in the inner upper limit. I went through the motions of using Liebniz’sContinue reading “Differentiating a double integral with respect to its upper limits”