Definition 1. A -field is a non-empty collection of subsets of the sample space closed under the formation of complements and countable unions (or equivalently of countable intesections – note ). Hence is a -field if
Definition 2. Set functions and measures. Let be a set and be an algebra on , and let be a non-negative set function
- is additive if and, for ,
- The map is called countably additive (or -additive) if and whenever is a sequence of disjoint sets in with union in , then
- Let be a measurable space, so that is a -algebra on .
- A map is called a measure on if is countable additive. The triple is called a measure space.
The measure is called finite if
and –finite if
, () s.th.
- Measure is called a probability measure if and is then called a probability triple.
- An element of is called -null if .
- A statement about points of is said to hold almost everywhere (a.s.) if
The phrase “Black Swan” (arising earlier in the different context of Popperian falsification) is here defined as an event characterized [p. xviii] by rarity, extreme impact, and retrospective (though not prospective) predictability, and Taleb’s thesis is that such events have much greater effect, in financial markets and the broader world of human affairs, than we usually suppose. The book is challenging to review because it requires considerable effort to separate the content from the style. The style is rambling and pugnacious—well described by one reviewer as “with few exceptions, the writers and professionals Taleb describes are knaves or fools, mostly fools. His writing is full of irrelevances, asides and colloquialisms, reading like the conversation of a raconteur rather than a tightly argued thesis”. And clearly this is perfectly deliberate. Such a book invites a review that reflects the reviewer’s opinions more than is customary in the Notices. My own overall reaction is that Taleb is sensible (going on prescient) in his discussion of financial markets and in some of his general philosophical thought but tends toward irrelevance or ridiculous exaggeration otherwise.
Let and be two integrable real-valued random variables and let be the quantile function of . Then if is integrable over we have
where is the a-mixing coefficient.
Proof: Set and then
note also that
which implies that
- Let be independent random variables with . Set . Then
Notice that and
Let be i.i.d. r.vs with and . Then
Consider a m-vector linear process
where are i.i.d. m-vector random variables with p.d.f. on , are matrices depending on a parameter vector .
Assume the following conditions are satisfied
A1 i) For some
where denotes the sum of the absolute values of the entries of .
ii) Every is continuously two times differentiable with respect to , and the derivatives satisfy
for where .
iii) for and can be expanded as follows:
where , satisfy
iv) Every is continuously two times differentiable with respect to , and the derivatives satisfy
A3 The continuous derivative of exists on .
From A1 the linear process can be expressed as