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| 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 function]]s, 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 [[probability theory|theoretical]] and [[applied probability]].
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| ==Real-valued functions==
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| The Laplace–Stieltjes transform of a real-valued function ''g'' is given by a [[Lebesgue-Stieltjes integration|Lebesgue–Stieltjes integral]] of the form
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| :<math>\int\mathrm{e}^{-sx}\,dg(x)</math> | |
| 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:
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| * The bilateral (or two-sided) Laplace–Stieltjes transform is given by
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| ::<math>\{\mathcal{L}^*g\}(s) = \int_{-\infty}^{\infty} \mathrm{e}^{-sx}\,dg(x).</math>
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| * The unilateral (one-sided) Laplace–Stieltjes transform is given by
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| ::<math>\{\mathcal{L}^*g\}(s) = \int_{0^-}^{\infty} \mathrm{e}^{-sx}\,dg(x).</math>
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| :where the lower limit 0<sup>−</sup> means
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| ::<math>\lim_{\varepsilon\to 0^+}\int_{-\varepsilon}^\infty.</math>
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| :This is necessary to ensure that the transform captures a possible jump in ''g''(''x'') at ''x'' = 0, as is needed to make sense of the Laplace transform of the [[Dirac delta function]].
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| * More general transforms can be considered by integrating over a contour in the [[complex plane]]; see {{harvnb|Zhavrid|2001}}.
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| 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,
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| :<math>\mathcal{L}^*g = \mathcal{L}(dg).</math>
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| In particular, it shares many properties with the usual Laplace transform. For instance, the [[convolution theorem]] holds:
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| :<math>\{\mathcal{L}^*(g * h)\}(s) = \{\mathcal{L}^*g\}(s)\{\mathcal{L}^*h\}(s).</math>
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| 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.
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| 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 [[expected value|expectation]]:
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| :<math>\{\mathcal{L}^*F\}(s) = \mathrm{E}\left[\mathrm{e}^{-sX}\right].</math>
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| ==Vector measures==
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| 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 measure]]s: measures with values in a [[Banach space]]. These are, however, important in connection with the study of [[semigroup]]s that arise in [[partial differential equations]], [[harmonic analysis]], and [[probability theory]]. The most important semigroups are, respectively, the [[heat equation|heat semigroup]], [[Riemann-Liouville integral|Riemann-Liouville semigroup]], and [[Brownian motion]] and other [[infinitely divisible process]]es.
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| 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
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| :<math>\sup \sum_i \|g(t_i)-g(t_{i+1})\|_X < \infty</math>
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| where the [[supremum]] is taken over all partitions of [0,''T'']
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| :<math>0=t_0 < t_1<\cdots< t_n=T.</math>
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| The Stieltjes integral with respect to the vector measure d''g''
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| :<math>\int_0^T e^{-st}dg(t)</math>
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| is defined as a [[Riemann–Stieltjes integral]]. Indeed, if π is the tagged partition of the interval [0,''T''] with subdivision {{nowrap|1=0 = ''t''<sub>0</sub> ≤ ''t''<sub>1</sub> ≤ ... ≤ ''t''<sub>''n''</sub> = ''T''}}, distinguished points τ<sub>''i''</sub>∈ [''t''<sub>''i''</sub>,''t''<sub>''i''+1</sub>] and mesh size |π| = max|''t''<sub>''i''</sub>− ''t''<sub>''i''+1</sub>|, the Riemann–Stieltjes integral is defined as the value of the limit
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| :<math>\lim_{|\pi|\to 0} \sum_{i=0}^{n-1}e^{-s\tau_i}[g(t_{i+1})-g(t_i)]</math>
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| taken in the topology on ''X''. The hypothesis of strong bounded variation guarantees convergence.
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| If in the topology of ''X'' the limit
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| :<math>\lim_{T\to\infty} \int_0^T e^{-st}dg(t)</math>
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| exists, then the value of this limit is the Laplace–Stieltjes transform of ''g''.
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| ==Related transforms==
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| 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:
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| * If ''g'' has derivative ''g' '' then the Laplace–Stieltjes transform of ''g'' is the Laplace transform of ''g' ''.
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| ::<math>\{\mathcal{L}^*g\}(s) = \{\mathcal{L}g'\}(s),</math>
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| * We can obtain the '''Fourier–Stieltjes transform''' of ''g'' (and, by the above note, the Fourier transform of ''g' '') by
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| ::<math>\{\mathcal{F}^*g\}(s) = \{\mathcal{L}^*g\}(\mathrm{i}s), \quad s \in \mathbb{R}.</math>
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| <!--
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| Something does need to be done about the mish-mash of transform articles. Some attempts have been made, but often these are from an applied perspective, e.g. [[Frequency transform]]. Perhaps an article on [[Integral transform]]s needs to be constructed to unify much of this? -->
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| ==Probability distributions==
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| If ''X'' is a continuous [[random variable]] with [[cumulative distribution function]] ''F''(''t'') then [[moment (mathematics)|moment]]s of ''X'' can be computed using<ref>{{cite doi|10.1017/CBO9781139226424.032}}</ref>
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| :<math>\mathbb E [X^n] = (-1)^n \left.\frac{\text{d}^n \{\mathcal{L}^*F\}(s)}{\text{d}s^n} \right|_{s=0}.</math>
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| ===Exponential distribution===
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| For an exponentially distributed random variable ''Y'' with rate parameter ''λ'' the LST is,
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| ::<math>\tilde Y(s) = \{\mathcal{L}^*F_Y\}(s) = \int_0^\infty e^{-st} \lambda e^{-\lambda t} dt = \frac{\lambda}{\lambda+s}</math>
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| from which the first three moments can be computed as 1/''λ'', 2/''λ''<sup>2</sup> and 6/''λ''<sup>3</sup>.
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| ===Erlang distribution===
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| 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 probability distributions|convolution of their probability distributions]]. So if
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| ::<math>Z = Y_1 + Y_2 + \ldots + Y_n</math>
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| with the ''Y''<sub>''i''</sub> independent then
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| ::<math>\tilde Z(s) = \tilde Y_1(s) \cdot \tilde Y_2(s) \cdot \cdots \cdot \tilde Y_n(s)</math>
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| therefore in the case where ''Z'' has an Erlang distribution,
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| ::<math>\tilde Z(s) = \left( \frac{\lambda}{\lambda+s} \right)^n.</math>
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| ===Uniform distribution===
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| For ''U'' with [[uniform distribution (continuous)|uniform distribution]] on the interval (''a'',''b''), the transform is given by
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| ::<math>\tilde U(s) = \int_0^\infty e^{-st} \frac{1}{b-a}\text{d}t = \frac{e^{-sa}-e^{-sb}}{s(b-a)}.</math>
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| ==References==
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| {{Reflist}}
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| * {{citation|last=Apostol|first=T.M.|year=1957|title=Mathematical Analysis|publisher=Addison-Wesley|publication-place=Reading, MA|edition=1st}}; 2nd ed (1974) ISBN 0-201-00288-4.
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| * {{citation|last=Apostol|first=T.M.|year=1997|title=Modular Functions and Dirichlet Series in Number Theory|edition=2nd|publisher=Springer-Verlag|publication-place=New York|isbn=0-387-97127-0}}.
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| * {{citation|last1=Grimmett|first1=G.R.|last2=Stirzaker|first2=D.R.|year=2001|title=Probability and Random Processes|edition=3rd|publisher=Oxford University Press|publication-place=Oxford|isbn=0-19-857222-0}}.
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| * {{Citation | last1=Hille | first1=Einar | authorlink1=Einar Hille | last2=Phillips | first2=Ralph S. | authorlink2=Ralph Phillips (mathematician) | title=Functional analysis and semi-groups | publisher=[[American Mathematical Society]] | location=Providence, R.I. | mr=0423094 | year=1974}}.
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| *{{springer|first=N.S.|last=Zhavrid|title=Laplace transform|id=l/l057540}}.
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| {{DEFAULTSORT:Laplace-Stieltjes Transform}}
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| [[Category:Integral transforms]]
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