Concept

Characteristic function

The average of cos(tX) + i·sin(tX) for a random quantity X, viewed as a function of the frequency t. It turns sums of independent pieces into products, and it dies away as t grows whenever X has a density — which is how the absence of a density is sometimes proved.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

12 harmonic series with random signs, each settling on its own sum. Partial sums of the harmonic series with each sign chosen by a fair coin, for several independent runs, plotted against the number of terms on a logarithmic scale, beside the all-plus and alternating sign patterns.

A coin in front of every term

Put all plus signs in front of 1, 1/2, 1/3, … and the sum runs off to infinity; alternate them and it settles on log 2. Toss a fair coin for each sign instead, and the sum settles — every time, on a different number. Where it tends to settle has a smooth, flat-topped shape, and at the value 2 that shape takes a height that agrees with one eighth to forty-two decimal places and is not one eighth.

analysis · Harmonic series

Sign patterns, and the far fewer points they land on. A logarithmic plot of the number of sign patterns, two to the n, against the number of distinct values the golden geometric sum takes, which is a Fibonacci number less one and falls further behind at every step.

Two sign patterns that land together

At λ = 1/φ the sign patterns + − − and − + + land in exactly the same place, because λ² + λ = 1. That one coincidence, repeated wherever it fits, puts 2ⁿ patterns onto a Fibonacci number of points, leaves the random sum's transform ringing at the same height forever, and makes a distribution that fills a whole interval live on a set of no length.

analysis · Harmonic series

Leading digits of products of random numbers, against Benford's law. 1 factor: ones 11.1%, nines 11.1%; 2 factors: ones 24.1%, nines 3.4%; 3 factors: ones 30.1%, nines 4.2%; 6 factors: ones 30.1%, nines 4.6%; the law gives 30.1% and 4.6%.

Multiplying makes the digit one common

Multiply a few random numbers together and the product starts with 1 about 30 per cent of the time and with 9 under 5 per cent — Benford's law, which no factor contains. The logarithm of a product is a sum, the central limit theorem spreads that sum across many powers of ten, and once it is spread its fractional part is uniform. The approach is geometric, at a rate fixed by a single number for each kind of factor; sums never get there, and the powers of two get there with no randomness at all.

probability · Central limit

Named alongside it

The objects these essays reach for when they reach for this one.

Probability density Algebraic integer Almost surely Bernoulli convolution Binary expansion Central limit theorem Convergence Entropy Equidistribution Fibonacci Fourier series Fractal dimension

All concepts