In the mathematical theory of probability, multivariate Laplace distributions are extensions of the Laplace distribution and the asymmetric Laplace distribution to multiple variables.

The marginal distributions of symmetric multivariate Laplace distribution variables are Laplace distributions.

The marginal distributions of asymmetric multivariate Laplace distribution variables are asymmetric Laplace distributions.

[1] A typical characterization of the symmetric multivariate Laplace distribution has the characteristic function: where

is the vector of means for each variable and

[2] Unlike the multivariate normal distribution, even if the covariance matrix has zero covariance and correlation the variables are not independent.

[1] The symmetric multivariate Laplace distribution is elliptical.

, the probability density function (pdf) for a k-dimensional multivariate Laplace distribution becomes: where:

is the modified Bessel function of the second kind.

[1] In the correlated bivariate case, i.e., k = 2, with

are the standard deviations of

is the correlation coefficient of

[1] For the uncorrelated bivariate Laplace case, that is k = 2,

, the pdf becomes: A typical characterization of the asymmetric multivariate Laplace distribution has the characteristic function: As with the symmetric multivariate Laplace distribution, the asymmetric multivariate Laplace distribution has mean

[3] The asymmetric multivariate Laplace distribution is not elliptical unless

, in which case the distribution reduces to the symmetric multivariate Laplace distribution with

[1] The probability density function (pdf) for a k-dimensional asymmetric multivariate Laplace distribution is: where:

is the modified Bessel function of the second kind.

[1] The asymmetric Laplace distribution, including the special case of

, is an example of a geometric stable distribution.

[3] It represents the limiting distribution for a sum of independent, identically distributed random variables with finite variance and covariance where the number of elements to be summed is itself an independent random variable distributed according to a geometric distribution.

[1] Such geometric sums can arise in practical applications within biology, economics and insurance.

[1] The distribution may also be applicable in broader situations to model multivariate data with heavier tails than a normal distribution but finite moments.

[1] The relationship between the exponential distribution and the Laplace distribution allows for a simple method for simulating bivariate asymmetric Laplace variables (including for the case of

Simulate a bivariate normal random variable vector

and covariance matrix

Independently simulate an exponential random variable

will be distributed (asymmetric) bivariate Laplace with mean

and covariance matrix