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1.1Symmetric Random Walk 1.2Rescaled Random Walk 1.3Standard deviation 1.4Central Limit Theorem 2.1Defining the Wiener process 2.2Increment explorer 2.3Joint distributions and covariance 3.1Running maximum and minimum 3.2Reflection principle 3.3First passage times 3.4Hitting probabilities

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Curriculum/Chapter 1 · Discrete to Continuous/1.2

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1.2

Rescaled Random Walk

What happens when we compress the walk in time and scale it by √n?

Goal: watch the paths refine and the terminal distribution approach the standard normal as n grows.

Simulation controls

Refinement level nStart and end stay fixed; each notch doubles the steps, inserting new points between the existing ones

Number of steps1

1162564,096

Overlaid paths4

16

Random seed7

19999

W⁽ⁿ⁾(1) statistics

Empirical mean

0.040

Theoretical mean

0.000

Empirical variance

0.999

Theoretical variance

1.000

The terminal variance is ≈ 1 for every n and the mean ≈ 0, with the distribution approaching the standard normal (Donsker). Individual paths, however, do not converge to one function — they get rougher.

Rescaled paths W⁽ⁿ⁾(t)

n = 1

Typical step size 1/√n ≈ 1.000; the endpoints never move, refinement only adds detail between them.

Terminal distribution vs N(0,1)

Empirical densityN(0,1) density