Laplace-Stieltjes transform

The Laplace-Stieltjes transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes, is an transform similar to the Laplace transform. It is useful in a number of areas of mathematics, including functional analysis, and certain areas of theoretical and applied probability.

Table of contents

1 Definition 2 Properties 3 Applications 4 Related Topics 5 References

Definition

The Laplace-Stieltjes transform of a function g: R → R is the function

whenever the integral exists. The integral here is the Lebesgue-Stieltjes integral.

Often, s is a real variable, and in some cases we are interested only in a function g: [0,∞) → R, in which case the we integrate between 0 and ∞.

Properties

The Laplace-Stieltjes transform shares many properties with the Laplace transform.

One example is convolution: if g and h both map from the reals to the reals,

(where each of these transforms exists).

Applications

Laplace-Stieltjes transforms are frequently useful in theoretical and applied probability, and stochastic processes contexts. For example, if X is a random variable with distribution function F, then the Laplace-Stieltjes transform can be expressed in terms of expectation:

Specific applications include first passage times of stochastic processes such as Markov chains, and renewal theory.

Related Topics

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 g has derivative g' then the Laplace-Stieltjes transform of g is the Laplace transform of g' .

• We can obtain the Fourier-Stieltjes transform of g (and, by the above note, the Fourier transform of g' ) by

References

Common references for the Laplace-Stieltjes transform include the following,

• Apostol, T.M. (1957). Mathematical Analysis. Addison-Wesley, Reading, MA. (For 1974 2nd ed, ISBN 0201002884).
• Apostol, T.M. (1997). Modular Functions and Dirichlet Series in Number Theory, 2nd ed. Springer-Verlag, New York. ISBN 0387971270.

and in the context of probability theory and applications,

• Grimmett, G.R. and Stirzaker, D.R. (2001). Probability and Random Processes, 3nd ed. Oxford University Press, Oxford. ISBN 0198572220.