Riemann-Stieltjes integrals
نویسنده
چکیده
This short note gives an introduction to the Riemann-Stieltjes integral on R and Rn. Some natural and important applications in probability theory are discussed. The reason for discussing the Riemann-Stieltjes integral instead of the more general Lebesgue and LebesgueStieltjes integrals are that most applications in elementary probability theory are satisfactorily covered by the Riemann-Stieltjes integral. In particular there is no need for invoking the standard machinery of monotone convergence and dominated convergence that hold for the Lebesgue integrals but typically do not for the Riemann integrals. The reason for introducing Stieltjes integrals is to get a more unified approach to the theory of random variables, in particular for the expectation operator, as opposed to treating discrete and continuous random variables separately. Also it makes it possible to treat mixtures of discrete and continuous random variables: It is for instance not possible to show that the expectation of the sum of a discrete and a continuous r.v. is the sum of the expectations, without using Stieltjes integrals. There are also many advantages in inference theory, for instance in the discussion of plug-in estimators. In Section 2 we introduce the Riemann-Stieltjes integral on R. In Section 3 we discuss some important applications to probability theory. In Section ?? we introduce the RiemannStieltjes integral on Rn. Section 3 contains applications to probability theory.
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