Lower Previsions: The Book
This page gives some further information about the Lower Previsions book, published by myself and Gert de Cooman (Wiley Series in Probability and Statistics, 2014).
Links
- Wiley's website has a summary, with table of contents.
- Gert's blog contains part of the preface.
- My post on the SIPTA blog has a brief history of how the book came about.
Errata
Warmest thanks to Paolo Vicig, Marshall Abrams, Jasper de Bock, Fabio Cozman, Nawapon Nakharutai, and Tathagata Basu for reporting some of these errata.
- p10, line above Definition 1.14: "μ(A)⊆μ(B)" should be "μ(A)≤μ(B)"
- p16, Lemma 1.21(iii): "so the sequence uₙ ∘ f + uₙ ∘ g converges uniformly to f + g" should be "so the sequence uₙ ∘ (f + g) converges uniformly to f + g"
- p29, line 1: "a acceptable" should be "an acceptable"
- p31: The text says that Walley's 2000 notion of desirability captures a weak preference to the zero gamble, whilst his 1991 notion captures strong preference. It must be the other way around: "Our notion of acceptability coincides with Walley's earlier (1991, Appendix F) notion of desirability, also used by Moral (2000) and Couso and Moral (2009, 2011), and aims at capturing a weak preference to the zero gamble. Walley in his later paper (2000, p. 137) and also Moral (2005) use a slightly different notion of acceptability, which is rather aimed at representing a strict preference to the zero gamble."
- p38, middle of page: "Similarly, upr(D)(f) is the *infimum* price"
- p39, proof of Theorem 4.2: The proof of the equality "sup{μ∈ℝ:f-μ∈D}=P̲(f)" is incomplete. To correct the proof, replace "P̲(f)<μ⇒P̲(f-μ)<0⇒f-μ∉D" by "f-μ∉D⇒(P̲(f-μ)≤0 and f-μ≱0)⇒P̲(f)≤μ".
- p42: The inequality just before Definition 4.4 must be reversed: "P̲(f)≤Q̲(f)".
- p44, line 2: "j∈{1, ..., n}" should be "i∈{1, ..., n}"
- p45, paragraph before 4.2.4: "transacttion" should be "transaction"
- p47, Definition 4.10(D): "bounded gambles f₀, ..., fₘ" should be "bounded gambles f₀, ..., fₙ"
- p48, 2nd line of (C)=>(D): "bounded gambles f₀, ..., fₘ" should be "bounded gambles f₀, ..., fₙ"
- p51, section 4.3.4, first paragraph: "P(-f)=P(f)" should be "P(-f)=-P(f)"
- p58, Corollary 4.17: "all all" should be "all"
- p58, paragraph above Proposition 4.18: "defined *on* a linear subspace"
- p66, paragraph above section 4.5.2: "and therefore of P̲" should be "and therefore P̲"
- p68, equation (4.10c): all indices should be k instead of i
- p105, equation in proof of Proposition 6.5: The "f∧" is not carried through: after the first equality sign we should have "P̲(r(f∧...", and after the second equality sign we should have "P̲(r(f)∧...".
- p107, bottom: "This inequality *is* trivially satisfied ..."
- p110, Proposition 6.12: "Assume that P̲ is non-negative *and monotone* ..."
- p123, line just above equation (7.2): "p̲(A)" should be "p̲(x)"
- p233, fourth last line: "... that the subject is practically *certain* will only ..."
- p283, Theorem 13.53, item (iv) should be instead: "For all non-empty events A there is a gamble f such that -∞<E̲(f|A)<+∞."
- p284-285, proof of Theorem 13.53 (iv) => (i): The last step of the proof (going from (13.28) to the equation on the page 285) is wrong. To fix the proof, choose A to be the union of C₁, ..., Cₚ and assume that E̲(f|A)>-∞. Now you can easily prove that E̲(f|A)=+∞.
- p375, last paragraph: the phrases "all measurable bounded gambles are integrable" and "equivalence of Lebesgue integration and natural extension" should appear between brackets
- p381, proof of Colollary C.4, line 2: the sum should start at k=2 instead of k=1
- p382, top line: "on the set B *of* all bounded gambles"
- p396, fourth entry: gamble -> Gamble
If you think you found any other errors, please contact me.