EXISTENCE OF AN EXACT SELF-AFFINE TIME FUNCTION WITH A RANDOM-WALK SCALING PROPERTY
Date
1991-07-01
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Abstract
An exact self-affine time function, unlike a fractal in two-dimensional
space, replicates exactly when scaled by differing ratios in the amplitude
and time axes. A statistical self-affine time function replicates only
statistically when scaled by differing ratios in the amplitude and time
axis, the best known example being a random walk, where the time scaling
factor is the square of the amplitude scaling factor. The existence of at
least four exact self-affine time functions, called Elliot or E(t) functions,
that allow for infinite number of exact replications of 12345abc structures,
has been demonstrated. These E(t) functions are defined algorithmically and
have no derivitive anywhere. One of these E(t) functions has the unexpected
property of scaling like a random walk. This leads to speculation that it
might be possible to construct a statistically self-affine E(t) function that
would be a random walk forgery.
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Computer Science