The Laplace–Stieltjes transform of a real-valued function g is given by a Lebesgue–Stieltjes integral of the form for s a complex number.
To wit, In particular, it shares many properties with the usual Laplace transform.
For instance, the convolution theorem holds: 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.
If X is a random variable with cumulative distribution function F, then the Laplace–Stieltjes transform is given by the expectation: The Laplace-Stieltjes transform of a real random variable's cumulative distribution function is therefore equal to the random variable's moment-generating function, but with the sign of the argument reversed.
These are, however, important in connection with the study of semigroups that arise in partial differential equations, harmonic analysis, and probability theory.
Let g be a function from [0,∞) to a Banach space X of strongly bounded variation over every finite interval.
Indeed, if π is the tagged partition of the interval [0,T] with subdivision 0 = t0 ≤ t1 ≤ ... ≤ tn = T, distinguished points
The hypothesis of strong bounded variation guarantees convergence.
In particular, note the following: If X is a continuous random variable with cumulative distribution function F(t) then moments of X can be computed using[1] For an exponentially distributed random variable Y with rate parameter λ the LST is, from which the first three moments can be computed as 1/λ, 2/λ2 and 6/λ3.