Geometric stable distribution - AbsoluteAstronomy.com

A geometric stable distribution or geo-stable distribution is a type of leptokurtic probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

. The geometric stable distribution may be symmetric or asymmetric. A symmetric geometric stable distribution is also referred to as a Linnik distribution. The Laplace distribution is a special case of the geometric stable distribution and of a Linnik distribution. The Mittag–Leffler distribution is also a special case of a geometric stable distribution.

The geometric stable distribution has applications in finance theory.

Characteristics

For most geometric stable distributions, the probability density function

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

and cumulative distribution function

Cumulative distribution function

In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

have no closed form solution. But a geometric stable distribution can be defined by its characteristic function

Characteristic function (probability theory)

In probability theory and statistics, the characteristic function of any random variable completely defines its probability distribution. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative...

, which has the form:

where

, which must be greater than 0 and less than or equal to 2, is the shape parameter or index of stability, which determines how heavy the tails are. Lower

corresponds to heavier tails.

, which must be greater than or equal to −1 and less than or equal to 1, is the skewness parameter. When

is negative the distribution is skewed to the left and when

is positive the distribution is skewed to the right. When

is zero the distribution is symmetric, and the characteristic function reduces to:

The symmetric geometric stable distribution with

is also referred to as a Linnik distribution. A completely skewed geometric stable distribution, that is with

, with

is also referred to as a Mittag–Leffler distribution. Although

determines the skewness of the distribution, it should not be confused with the typical skewness coefficient

Skewness

In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. The skewness value can be positive or negative, or even undefined...

or 3rd standardized moment, which in most circumstances is undefined for a geometric stable distribution.

is the scale parameter

Scale parameter

In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions...

and

is the location parameter.

When

= 2,

= 0 and

= 0 (i.e., a symmetric geometric stable distribution or Linnik distribution with

=2), the distribution becomes the symmetric Laplace distribution with mean of 0, which has a probability density function

Probability density function

of:

The Laplace distribution has a variance

Variance

In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

equal to

. However, for

the variance of the geometric stable distribution is infinite.

Relationship to the stable distribution

The stable distribution has the property that if

are independent, identically distributed random variables taken from a stable distribution, the sum

has the same distribution as the

s for some

and

.

The geometric stable distribution has a similar property, but where the number of elements in the sum is a geometrically distributed random variable. If

are independent and identically distributed random variables

Independent and identically distributed random variables

In probability theory and statistics, a sequence or other collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent....

taken from a geometric stable distribution, the limit

Limit

A limit can be:* Limit :** Limit of a function** Limit of a sequence** One-sided limit** Limit superior and limit inferior** Limit of a net** Limit point** Limit ** Direct limit and Inverse limit...

of the sum

approaches the distribution of the

s for some coefficients

and

as p approaches 0, where

is a random variable independent of the

s taken from a geometric distribution with parameter p. In other words:

The distribution is strictly geometric stable only if the sum

equals the distribution of the

s for some a.

There is also a relationship between the stable distribution characteristic function and the geometric stable distribution characteristic function. The stable distribution has a characteristic function of the form:

where

The geometric stable characteristic function can be expressed in terms of a stable characteristic function as:

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