Laplace-Stieltjes transform
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The Laplace-Stieltjes transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes, is a 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.
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Definition
The Laplace-Stieltjes transform of a function g: R → R is the function
- <math>\{\mathcal{L}^*g\}(s) = \int_{-\infty}^{\infty} \mathrm{e}^{-sx}\,dg(x), \quad s \in \mathbb{C},</math>
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,
- <math>\{\mathcal{L}^*(g * h)\}(s) = \{\mathcal{L}^*g\}(s)\{\mathcal{L}^*h\}(s),</math>
(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 cumulative distribution function F, then the Laplace-Stieltjes transform can be expressed in terms of expectation:
- <math>\{\mathcal{L}^*F\}(s) = \mathrm{E}\left[\mathrm{e}^{-sX}\right].</math>
Specific applications include first passage times of stochastic processes such as Markov chains, and renewal theory.
See also
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' .
- <math>\{\mathcal{L}^*g\}(s) = \{\mathcal{L}g'\}(s),</math>
- We can obtain the Fourier-Stieltjes transform of g (and, by the above note, the Fourier transform of g' ) by
- <math>\{\mathcal{F}^*g\}(s) = \{\mathcal{L}^*g\}(\mathrm{i}s), \quad s \in \mathbb{R}.</math>
Examples
For an exponentially distributed random variable <math>Y</math> the LST is,
- <math>\tilde F_Y(s) = f_Y^*(s) = \int_0^\infty e^{-st} \lambda e^{-\lambda t} dt = \frac{\lambda}{\lambda+s}.</math>
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.