Series Representation
The stable distribution can be restated as the real part of a simpler integral:(Peach 1981, § 4.5)
Expressing the second exponential as a Taylor series, we have:
where . Reversing the order of integration and summation, and carrying out the integration yields:
which will be valid for x ≠ μ and will converge for appropriate values of the parameters. (Note that the n = 0 term which yields a delta function in x−μ has therefore been dropped.) Expressing the first exponential as a series will yield another series in positive powers of x−μ which is generally less useful.
Read more about this topic: Stable Distribution
Famous quotes containing the word series:
“Every man sees in his relatives, and especially in his cousins, a series of grotesque caricatures of himself.”
—H.L. (Henry Lewis)