Transfer operator

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. Template:Probability distribution In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is known as a van der Waals profile. It is a special case of the inverse-gamma distribution.

It is one of the few distributions that are stable and that have probability density functions that can be expressed analytically, the others being the normal distribution and the Cauchy distribution. All three are special cases of the stable distributions, which do not generally have a probability density function which can be expressed analytically.

Definition

The probability density function of the Lévy distribution over the domain $x\ge \mu$ is

$f(x;\mu ,c)=\sqrt{\frac{c}{2\pi }}\frac{e^{-\frac{c}{2(x-\mu )}}}{(x-\mu {)}^{3/2}}$

where $\mu$ is the location parameter and $c$ is the scale parameter. The cumulative distribution function is

$F(x;\mu ,c)=\text{<mrow data-mjx-texclass="ORD">erfc</mrow>}\left(\sqrt{\frac{c}{2(x-\mu )}}\right)$

where $\text{<mrow data-mjx-texclass="ORD">erfc</mrow>}(z)$ is the complementary error function. The shift parameter $\mu$ has the effect of shifting the curve to the right by an amount $\mu$, and changing the support to the interval [$\mu$, $\mathrm{\infty }$). Like all stable distributions, the Levy distribution has a standard form f(x;0,1) which has the following property:

$f(x;\mu ,c)dx=f(y;0,1)dy$

where y is defined as

$y=\frac{x-\mu }{c}$

The characteristic function of the Lévy distribution is given by

$\phi (t;\mu ,c)=e^{i\mu t-\sqrt{-2ict}}.$

Note that the characteristic function can also be written in the same form used for the stable distribution with $\alpha =1/2$ and $\beta =1$:

$\phi (t;\mu ,c)=e^{i\mu t-|ct{|}^{1/2}(1-i\text{<mrow data-mjx-texclass="ORD">sign</mrow>}(t))}.$

Assuming $\mu =0$, the nth moment of the unshifted Lévy distribution is formally defined by:

$m_n\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sqrt{\frac{c}{2\pi }}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-c/2x}{x}^n}{x^{3/2}}dx$

which diverges for all n > 0 so that the moments of the Lévy distribution do not exist. The moment generating function is then formally defined by:

$M(t;c)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sqrt{\frac{c}{2\pi }}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-c/2x+tx}}{x^{3/2}}dx$

which diverges for $t>0$ and is therefore not defined in an interval around zero, so that the moment generating function is not defined per se. Like all stable distributions except the normal distribution, the wing of the probability density function exhibits heavy tail behavior falling off according to a power law:

$\underset{x\to \mathrm{\infty }}{\lim }f(x;\mu ,c)=\sqrt{\frac{c}{2\pi }}\frac{1}{x^{3/2}}.$

This is illustrated in the diagram below, in which the probability density functions for various values of c and $\mu =0$ are plotted on a log-log scale.

Probability density function for the Lévy distribution on a log-log scale.

Related distributions

• If $X\sim \text{<mrow data-mjx-texclass="ORD">Levy</mrow>}(\mu ,c)$ then $kX+b\sim \text{<mrow data-mjx-texclass="ORD">Levy</mrow>}(k\mu +b,kc)$
• If $X\sim \text{<mrow data-mjx-texclass="ORD">Levy</mrow>}(0,c)$ then $X\sim \text{<mrow data-mjx-texclass="ORD">Inv<mo>−</mo>Gamma</mrow>}(\frac{1}{2},\frac{c}{2})$ (inverse gamma distribution)
• Lévy distribution is a special case of type 5 Pearson distribution
• If $Y\sim \text{<mrow data-mjx-texclass="ORD">Normal</mrow>}(\mu ,{\sigma }^2)$ (Normal distribution) then ${(Y-\mu )}^{-2}\sim \text{<mrow data-mjx-texclass="ORD">Levy</mrow>}(0,1/{\sigma }^2)$
• If $X\sim \text{<mrow data-mjx-texclass="ORD">Normal</mrow>}(\mu ,\frac{1}{\sqrt{\sigma }})$ then ${(X-\mu )}^{-2}\sim \text{<mrow data-mjx-texclass="ORD">Levy</mrow>}(0,\sigma )$
• If $X\sim \text{<mrow data-mjx-texclass="ORD">Levy</mrow>}(\mu ,c)$ then $X\sim \text{<mrow data-mjx-texclass="ORD">Stable</mrow>}(1/2,1,c,\mu )$ (Stable distribution)
• If $X\sim \text{<mrow data-mjx-texclass="ORD">Levy</mrow>}(0,c)$ then $X\sim \text{<mrow data-mjx-texclass="ORD">Scale<mo>−</mo>inv<mo>−</mo></mrow>}{\chi }^2(1,c)$ (Scaled-inverse-chi-squared distribution)
• If $X\sim \text{<mrow data-mjx-texclass="ORD">Levy</mrow>}(\mu ,c)$ then ${(X-\mu )}^{-\frac{1}{2}}\sim \text{<mrow data-mjx-texclass="ORD">FoldedNormal</mrow>}(0,1/\sqrt{c})$ (Folded normal distribution)

Applications

• The frequency of geomagnetic reversals appears to follow a Lévy distribution
• The time of hitting a single point $\alpha$ (different from the starting point 0) by the Brownian motion has the Lévy distribution with $c={\alpha }^2$. (For a Brownian motion with drift, this time may follow an inverse Gaussian distribution, which has the Lévy distribution as a limit.)

• The length of the path followed by a photon in a turbid medium follows the Lévy distribution.

• A Cauchy process can be defined as a Brownian motion subordinated to a process associated with a Lévy distribution.

Footnotes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• Template:Cite web - John P. Nolan's introduction to stable distributions, some papers on stable laws, and a free program to compute stable densities, cumulative distribution functions, quantiles, estimate parameters, etc. See especially An introduction to stable distributions, Chapter 1

External links

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