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Laplace-stieltjes Transform
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DEFINITION


The ''Laplace-Stieltjes transform'' of a function ''g'': R → R is the function
  • g\}(s) = \int_{-\infty}^{\infty} \mathrm{e}^{-sx}\,dg(x), \quad s \in \mathbb{C},

    • 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 to the reals,
  • (g --- h)\}(s) = \{\mathcal{L}^---g\}(s)\{\mathcal{L}^---h\}(s),

    • (where each of these transforms exists).



APPLICATIONS


Laplace-Stieltjes transforms are frequently useful in Theoretical and Applied Probability , and Stochastic Process es 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 :
  • F\}(s) = \mathrm{E}\left[\mathrm{e}^{-sX} ight].

    • Specific applications include first passage times of stochastic processes such as Markov Chain s, and Renewal Theory .



SEE ALSO


The Laplace-Stieltjes transform is closely related to other Integral Transform s, 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' ''.

  • g\}(s) = \{\mathcal{L}g'\}(s),

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

  • g\}(s) = \{\mathcal{L}^---g\}(\mathrm{i}s), \quad s \in \mathbb{R}.





EXAMPLES


For an exponentially distributed random variable Y the LST is,

  • (s) = \int_0^\infty e^{-st} \lambda e^{-\lambda t} dt = rac{\lambda}{\lambda+s}.



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.