Laplace–Stieltjes transform

The Laplace–Stieltjes transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes, is an integral transform similar to the Laplace transform. For real-valued functions, it is the Laplace transform of a Stieltjes measure, however it is often defined for functions with values in a Banach space. It is useful in a number of areas of mathematics, including functional analysis, and certain areas of theoretical and applied probability.

The Laplace–Stieltjes transform of a real-valued function g is given by a Lebesgue–Stieltjes integral of the form

for s a complex number. As with the usual Laplace transform, one gets a slightly different transform depending on the domain of integration, and for the integral to be defined, one also needs to require that g be of bounded variation on the region of integration. The most common are:

The Laplace–Stieltjes transform in the case of a scalar-valued function is thus seen to be a special case of the Laplace transform of a Stieltjes measure. To wit,

In particular, it shares many properties with the usual Laplace transform. For instance, the convolution theorem holds:

Often only real values of the variable s are considered, although if the integral exists as a proper Lebesgue integral for a given real value s = σ, then it also exists for all complex s with re(s) ≥ σ.

The Laplace–Stieltjes transform appears naturally in the following context. If X is a random variable with cumulative distribution function F, then the Laplace–Stieltjes transform is given by the expectation:

Whereas the Laplace–Stieltjes transform of a real-valued function is a special case of the Laplace transform of a measure applied to the associated Stieltjes measure, the conventional Laplace transform cannot handle vector measures: measures with values in a Banach space. These are, however, important in connection with the study of semigroups that arise in partial differential equations, harmonic analysis, and probability theory. The most important semigroups are, respectively, the heat semigroup, Riemann-Liouville semigroup, and Brownian motion and other infinitely divisible processes.

Let g be a function from [0,∞) to a Banach space X of strongly bounded variation over every finite interval. This means that, for every fixed subinterval [0,T] one has

taken in the topology on X. The hypothesis of strong bounded variation guarantees convergence.

exists, then the value of this limit is the Laplace–Stieltjes transform of g.

The Laplace–Stieltjes transform is closely related to other integral transforms, including the Fourier transform and the Laplace transform. In particular, note the following:

If X is a continuous random variable with cumulative distribution function F(t) then moments of X can be computed using[1]

For an exponentially distributed random variable Y with rate parameter λ the LST is,

from which the first three moments can be computed as 1/λ, 2/λ2 and 6/λ3.

For Z with Erlang distribution (which is the sum of n exponential distributions) we use the fact that the probability distribution of the sum of independent random variables is equal to the convolution of their probability distributions. So if

For U with uniform distribution on the interval (a,b), the transform is given by