Concept

Polya urn

An urn from which a ball is drawn at random and returned with another of the same colour, so every draw makes its own colour likelier. Its share of red settles on a random value, and the number of red draws is exactly uniform when it starts with one ball of each colour.

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

Walks that prefer the steps they have taken. 12 Pólya-urn walks of 2000 steps, each heading off along its own straight line, with final speeds 0.99, −0.87, −0.02, −0.66, 0.01, 0.78, 0.47, 0.44, −0.04, 0.38, 0.83, 0.55, against the ±2√n band of an ordinary walk.

A walk that follows its own footsteps

Let a walk step up with probability equal to the share of up-steps it has taken so far, and every count of up-steps after twenty steps is exactly equally likely. The walk leaves along a straight line at a speed chosen by its first few steps, and although it is expected to come back to its start infinitely often, almost every run comes back only a few times.

probability · Random walk

Five urns, each drawn as walks. Simulated walks of 2000 draws for urns with replacement matrices (0,1,1,0), (2,1,1,2), (3,1,1,3), (7,1,1,7), (1,0,0,1).

An urn forgets its start only below one half

Let each draw from an urn add balls of both colours in fixed amounts, and the long run depends on a single ratio of two eigenvalues. Below one half the urn behaves like a coin, its fluctuations spread like the square root of the draws and settle into a bell. Above one half the first few draws decide most of the outcome, the spread grows faster, and the shape that results is not a bell and depends on how the urn began.

probability · Random walk

Named alongside it

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

Random walk Beta distribution Central limit theorem Eigenvalue Exchangeability Martingale Normal distribution Recurrence Scaling Variance

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