To understand the role of the index H in fBm, we recall the
expected value, E(Y), of a random process Y(t). |
One way to
measure the correlation of a random process Y(t) is to compute the expected value of the
product of non-overlapping increments. |
With some work, it can be shown that for index H fBm, |
E((Y(t) - Y(0))⋅(Y(t + h) - Y(t))) =
((t + h)2H - t2H - h2H)/2 |
Though some more work is required to see this, this expectation is |
positive for H > 1/2 |
0 for H = 1/2 (This one is easy.) |
negative for H < 1/2 |
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So we shall consider three examples. |
|
Note as H decreases, the graph of index H fBm appears to get rougher. |
For this reason, H is called the roughness exponent. |
It is also
called the Holder or Hurst exponent. |
With probability one, the graph of index
H fBm has box-counting dimension 2 - H. |