The short answer is that your $\delta$ is fine, but your $\gamma$ is wrong. In order to get the positive stable distribution given by your formula in R, you need to set
$$
\gamma = |1 - i \tan \left(\pi \alpha / 2\right)|^{-1/\alpha}.
$$
The earliest example I could find of the formula you gave was in (Feller, 1971), but I've only found that book in physical form. However (Hougaard, 1986) gives the same formula, along with the Laplace transform
$$
\mathrm{L}(s) = \mathrm{E}\left[\exp(-sX)\right] = \exp\left(-s^\alpha\right).
$$
From the stabledist manual (stabledist is used in fBasics), the pm=1 parameterization is from (Samorodnitsky and Taqqu, 1994), another resource whose online reproduction has eluded me. However (Weron, 2001) gives the characteristic function in Samorodnitsky and Taqqu's parameterization for $\alpha \neq 1$ to be
$$
\varphi(t) = \mathrm{E}\left[\exp(i t X) \right] = \exp\left[i \delta t - \gamma^\alpha |t|^\alpha \left(1 - i \beta \mathrm{sign}(t) \tan{\frac{\pi \alpha}{2}} \right) \right].
$$
I've renamed some parameters from Weron's paper to coinside with the notation we're using. He uses $\mu$ for $\delta$ and $\sigma$ for $\gamma$. In any case, plugging in $\beta=1$ and $\delta=0$, we get
$$
\varphi(t) = \exp\left[-\gamma^\alpha |t|^\alpha \left(1 - i \mathrm{sign}(t) \tan \frac{\pi \alpha}{2} \right) \right].
$$
Note that $(1 - i \tan (\pi \alpha / 2)) / |1 - i \tan(\pi \alpha / 2)| = \exp(-i \pi \alpha / 2)$ for $\alpha \in (0, 1)$ and that $i^\alpha = \exp(i \pi \alpha / 2)$. Formally, $\mathrm{L}(s)=\varphi(is)$, so by setting $\gamma = |1 - i \tan \left(\pi \alpha / 2\right)|^{-1/\alpha}$ in $\varphi(t)$ we get
$$
\varphi(is) = \exp\left(-s^\alpha\right) = \mathrm{L}(s).
$$
One interesting point to note is that the $\gamma$ that corresponds to $\alpha=1/2$ is also $1/2$, so if you were to try $\gamma=\alpha$ or $\gamma=1-\alpha$, which is actually not a bad approximation, you end up exactly correct for $\alpha=1/2$.
Here's an example in R to check correctness:
library(stabledist)
# Series representation of the density
PSf <- function(x, alpha, K) {
k <- 1:K
return(
-1 / (pi * x) * sum(
gamma(k * alpha + 1) / factorial(k) *
(-x ^ (-alpha)) ^ k * sin(alpha * k * pi)
)
)
}
# Derived expression for gamma
g <- function(a) {
iu <- complex(real=0, imaginary=1)
return(abs(1 - iu * tan(pi * a / 2)) ^ (-1 / a))
}
x=(1:100)/100
plot(0, xlim=c(0, 1), ylim=c(0, 2), pch='',
xlab='x', ylab='f(x)', main="Density Comparison")
legend('topright', legend=c('Series', 'gamma=g(alpha)'),
lty=c(1, 2), col=c('gray', 'black'),
lwd=c(5, 2))
text(x=c(0.1, 0.25, 0.7), y=c(1.4, 1.1, 0.7),
labels=c(expression(paste(alpha, " = 0.4")),
expression(paste(alpha, " = 0.5")),
expression(paste(alpha, " = 0.6"))))
for(a in seq(0.4, 0.6, by=0.1)) {
y <- vapply(x, PSf, FUN.VALUE=1, alpha=a, K=100)
lines(x, y, col="gray", lwd=5, lty=1)
lines(x, dstable(x, alpha=a, beta=1, gamma=g(a), delta=0, pm=1),
col="black", lwd=2, lty=2)
}
$\hskip1in$
- Feller, W. (1971). An Introduction to Probability Theory and Its Applications, 2, 2nd ed. New York: Wiley.
- Hougaard, P. (1986). Survival Models for Heterogeneous Populations Derived from Stable Distributions, Biometrika 73, 387-396.
- Samorodnitsky, G., Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes, Chapman & Hall, New York, 1994.
- Weron, R. (2001). Levy-stable distributions revisited: tail index > 2 does not exclude the Levy-stable regime, International Journal of Modern Physics C, 2001, 12(2), 209-223.
The PDF of a Normal distribution is
$$f_{\mu, \sigma}(x) = \frac{1}{\sqrt{2 \pi} \sigma} e^{-\frac{(x-\mu )^2}{2 \sigma ^2}}dx$$
but in terms of $\tau = 1/\sigma^2$ it is
$$g_{\mu, \tau}(x) = \frac{\sqrt{\tau}}{\sqrt{2 \pi}} e^{-\frac{\tau(x-\mu )^2}{2 }}dx.$$
The PDF of a Gamma distribution is
$$h_{\alpha, \beta}(\tau) = \frac{1}{\Gamma(\alpha)}e^{-\frac{\tau}{\beta }} \tau^{-1+\alpha } \beta ^{-\alpha }d\tau.$$
Their product, slightly simplified with easy algebra, is therefore
$$f_{\mu, \alpha, \beta}(x,\tau) =\frac{1}{\beta^\alpha\Gamma(\alpha)\sqrt{2 \pi}} e^{-\tau\left(\frac{(x-\mu )^2}{2 } + \frac{1}{\beta}\right)} \tau^{-1/2+\alpha}d\tau dx.$$
Its inner part evidently has the form $\exp(-\text{constant}_1 \times \tau) \times \tau^{\text{constant}_2}d\tau$, making it a multiple of a Gamma function when integrated over the full range $\tau=0$ to $\tau=\infty$. That integral therefore is immediate (obtained by knowing the integral of a Gamma distribution is unity), giving the marginal distribution
$$f_{\mu, \alpha, \beta}(x) = \frac{\sqrt{\beta } \Gamma \left(\alpha +\frac{1}{2}\right) }{\sqrt{2\pi } \Gamma (\alpha )}\frac{1}{\left(\frac{\beta}{2} (x-\mu )^2+1\right)^{\alpha +\frac{1}{2}}}.$$
Trying to match the pattern provided for the $t$ distribution shows there is an error in the question: the PDF for the Student t distribution actually is proportional to
$$\frac{1}{\sqrt{k} s }\left(\frac{1}{1+k^{-1}\left(\frac{x-l}{s}\right)^2}\right)^{\frac{k+1}{2}}$$
(the power of $(x-l)/s$ is $2$, not $1$). Matching the terms indicates $k = 2 \alpha$, $l=\mu$, and $s = 1/\sqrt{\alpha\beta}$.
Notice that no Calculus was needed for this derivation: everything was a matter of looking up the formulas of the Normal and Gamma PDFs, carrying out some trivial algebraic manipulations involving products and powers, and matching patterns in algebraic expressions (in that order).
Best Answer
One of the characterizing features of a Levy-stable distribution is that linear combinations of independent copies have the same distribution, up to location and scaling. So if this property does not hold, the distribution cannot be Levy stable. Equivalently the characteristic function isn't of the Levy form.
In the case of the student t distribution, it has a characteristic function that looks like:
$$\frac{K_{v/2}(\sqrt{v}|t|)(\sqrt{v}|t|)^{v/2}}{\Gamma(v/2)2^{v/2-1}},$$
which in general will not have the Levy form.