- Laplace–Stieltjes transform
The

**Laplace–Stieltjes transform**, named forPierre-Simon Laplace andThomas Joannes Stieltjes , is a transform similar to theLaplace transform . It is useful in a number of areas ofmathematics , includingfunctional analysis , and certain areas of theoretical andapplied probability .**Definition**The "Laplace–Stieltjes transform" of a function "g":

**R**→**R**is the function:$\{mathcal\{L\}^*g\}(s)\; =\; int\_\{-infty\}^\{infty\}\; mathrm\{e\}^\{-sx\},dg(x),\; quad\; s\; in\; mathbb\{C\},$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

convolution : if "g" and "h" both map from the reals to the reals,:$\{mathcal\{L\}^*(g\; *\; h)\}(s)\; =\; \{mathcal\{L\}^*g\}(s)\{mathcal\{L\}^*h\}(s),$(where each of these transforms exists).**Applications**Laplace–Stieltjes transforms are frequently useful in theoretical and

applied probability , andstochastic process es contexts. For example, if X is arandom variable withcumulative distribution function "F", then the Laplace–Stieltjes transform can be expressed in terms of expectation::$\{mathcal\{L\}^*F\}(s)\; =\; mathrm\{E\}left\; [mathrm\{e\}^\{-sX\}\; ight]\; .$Specific applications include first passage times of stochastic processes such asMarkov chain s, andrenewal theory . Inphysics , the transform is sometimes used to regularize sums inquantum field theory by means ofheat kernel regularization .**ee also**The Laplace–Stieltjes transform is closely related to other

integral transform s, including theFourier transform and theLaplace transform . In particular, note the following:

* If "g" has derivative "g' " then the Laplace–Stieltjes transform of "g" is the Laplace transform of "g' ".::$\{mathcal\{L\}^*g\}(s)\; =\; \{mathcal\{L\}g\text{'}\}(s),$

* We can obtain the**Fourier–Stieltjes transform**of "g" (and, by the above note, the Fourier transform of "g' ") by::$\{mathcal\{F\}^*g\}(s)\; =\; \{mathcal\{L\}^*g\}(mathrm\{i\}s),\; quad\; s\; in\; mathbb\{R\}.$**Examples**For an exponentially distributed random variable $Y$ the LST is,

::$ilde\; F\_Y(s)\; =\; f\_Y^*(s)\; =\; int\_0^infty\; e^\{-st\}\; lambda\; e^\{-lambda\; t\}\; dt\; =\; frac\{lambda\}\{lambda+s\}.$

**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 0-201-00288-4).

* Apostol, T.M. (1997). "Modular Functions and Dirichlet Series in Number Theory, 2nd ed". Springer-Verlag, New York. ISBN 0-387-97127-0.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 0-19-857222-0.

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