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and taking norms yields x (I + A 1 ( A)) 1 (I + A 1 ( A)) 1 A 1 ( A 1 b + b + b A x ) A A A x ,

displaying behavior that is far, far crazier We all still remember the complaints by David Viniar, Goldman Sachs CFO, on August 2007 as the credit crisis was starting to break loose: We were seeing things that were 25 standard deviation moves, several days in a row In a Normal world such happenstance is utterly impossible The universe isn t old enough to accommodate such small probability Obviously, we must not live under Normality And that summer s travails, though particularly detectable, are not, again, by any means the only high-pro le instance of the markets yelling out loud, Don t call us Normal! The posterior, even wilder, events in the credit, equity, and interest markets implacably bear witness, but there is also plenty of historical precedent.

m E Nu Cp,m 1 + C + E X(p+1)

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During the European Exchange Rate Mechanism debacle in 1992 (whereby Europe s system of of cially managed currency rates collapsed), 50 standard deviation moves in interest rates were witnessed, while 1987 s Black Monday was a 20 standard deviation (or 20-sigma) event During the summer 1998 convolutions that eventually brought down giant Long Term Capital Management, 15-plus sigma deviations became the norm Plenty of smaller (yet still sensationally non-Normal) similar gyrations have been observed in nance So-called one in a million years events have been experienced, several times, by people whose age is way below one million years Which one is wrong, the real world or the model If you said the model, you got it right The real probability distribution has fat (not thin) tails that grant extreme events the weighty weight that they deserve In nance, rare events are not that rare.

in a square room or there could be three normal men in a weird room. My brain may be choosing an unlikely interpretation of the scene, but at least it is a possible interpretation. There is no one correct interpretation, protests the Professor of English. I argue that, though the evidence is ambiguous, this does not mean that there is no correct interpretation. But, in addition, our brain hides from us this ambiguity and only presents us with one of the possible interpretations. Furthermore, sometimes our brain takes no account of the evidence about the physical world at all.

When it comes to wheeling and dealing in the markets, we are not Normal In fact, it seems to me that the assumption of Normality when it comes to the nancial markets shows strong correlation with that other famously misguided imposition, namely Prohibition in 1920s America Just like Prohibition forbade regular folks from (legally) drowning down their sorrows, Normality forbids investors from taking the markets beyond certain levels Such probabilistic assumption denies individuals the capacity to cross certain lines, explore certain territories, discover certain realities It is, thus, a very constraining assumption A tyrannical one, you might say Just like with any form of reactionary totalitarianism, individuals are judged to be of limited capabilities, requiring pre-set, centrally imposed, stringent regulations They are assumed to be unable to reach beyond certain limits, forever con ned to a restricted existence..

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There is a series of bands in the 2150 2180 nm region and a strong sharp peak at 2460 nm. These bands are combination bands. The 2150 2180 nm bands are combinations of C H and C C stretching, while the 2460 nm band is a combination of C H stretching and C H bending. The spectrum also shows bands near 1670 nm due to the rst overtone of C H stretching. The second overtone at 1130 nm is very weak in this spectrum.

v(x, y) =

Now, we apply to the bounded separable Gaussian process Xt (s) : t W Sd0 , s S, s = t t (s) the basic inequality (2.33), which gives the bound P sup Xt (s) : t W Sd0 , s S, s = t > x t (s) C2 exp( 2 x 2 )

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