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Radically Elementary Probability Theory

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2067)

Abstract

The expressive power of minIST comes from the fact that it allows for the notions of finite sets with unlimited cardinality, and finite subsets of the reals whose distance is at most an infinitesimal from every point in some non-empty open interval.

Keywords

  • Stochastic Process
  • Wiener Process
  • Finite Subset
  • Stochastic Calculus
  • Finite Probability

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    Such as the system minIST  −  discussed in Appendix A.

  2. 2.

    Proving this could be a useful exercise for students. If \(P\left \{A\right \} \simeq 1\), then clearly a.s. A (simply choose \(N = \Omega \setminus \{ A\}\)). Conversely, if a.s. A, then the set

    $$M = \left \{n \in \mathbf{N}\ :\ \exists N \subseteq \Omega \left (P(N) \leq 1/n\ \&\ \Omega \setminus \{ A\} \subseteq N\right )\right \}$$

    contains all standard elements of \(\mathbf{N}\). Since there is no set which consists of all standard natural numbers (see Remark 1.1), M must contain some nonstandard \({n}_{0} \in \mathbf{N}\), too. But then, \(P\left (\Omega \setminus \{ A\}\right ) \leq 1/{n}_{0} \simeq 0\), so \(P\{A\} \simeq 1\).

  3. 3.

    In Robinsonian nonstandard analysis, this Wiener walk is known as Anderson’s [4] construction of the Wiener process.

  4. 4.

    In Robinsonian nonstandard analysis, this Poisson walk is known as Loeb’s [51] construction of the Poisson process.

  5. 5.

    One should note that this infinitesimal version of Stirling’s formula can also be proved in radically elementary probability theory, cf. van den Berg [11, last paragraph on p. 172].

References

  1. Anderson, R.: A non-standard representation for Brownian motion and Itô integration. Israel J. Math. 25(1–2), 15–46 (1976)

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Benoît, E.: Random walks and stochastic differential equations. In: Diener, F., Diener, M. (eds.) Nonstandard Analysis In Practice. Universitext, pp. 71–90. Springer, Berlin (1995)

    CrossRef  Google Scholar 

  3. Berg, I.v.d.: An external probability order theorem with applications. In: Nonstandard Analysis In Practice. Universitext, pp. 171–183. Springer, Berlin (1995)

    Google Scholar 

  4. Lawler, G.: Internal Set Theory and infinitesimal random walks. In: Faris, W. (ed.) Diffusion, quantum theory, and radically elementary mathematics. Mathematical Notes, vol. 47, pp. 157–181. Princeton University Press, Princeton, NJ (2006)

    Google Scholar 

  5. Loeb, P.: Conversion from nonstandard to standard measure spaces and applications in probability theory. Trans. Am. Math. Soc. 211, 113–122 (1975)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Nelson, E.: Radically elementary probability theory. Annals of Mathematics Studies, vol. 117. Princeton University Press, Princeton, NJ (1987)

    Google Scholar 

  7. Nelson, E.: The virtue of simplicity. In: Berg, I.v.d., Neves, V. (eds.) The Strength of Nonstandard Analysis, pp. 27–32. Springer, Vienna (2007)

    Google Scholar 

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Herzberg, F.S. (2013). Radically Elementary Probability Theory. In: Stochastic Calculus with Infinitesimals. Lecture Notes in Mathematics, vol 2067. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33149-7_2

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