A note on the ratio of the extreme to the root of the sum of squares in sequences of absolute values of Gaussian variables

Safety, Reliability and Risk Analysis: Beyond the Horizon – Steenbergen et al. (Eds) © 2014 Taylor & Francis Group, London, ISBN 978-1-138-00123-7

A note on the ratio of the extreme to the root of the sum of squares

in sequences of absolute values of Gaussian variables

Michiel A. Odijk

Independent Research, Gouda, The Netherlands

Pieter H.A.J.M. van Gelder

Faculty of Civil Engineering and Geosciences, Delft University of Technology, The Netherlands

ABSTRACT: Let Z1, ...,Zn be i.i.d. standard normal variables, Mn the extreme among their

abso-lute values, and Qn their rooted sum of squares. Here we study the ratio υn=M Qn/ . We show that n

µ υ( )n =µ(Mn)/ ( ) ~µQn bn/ n~ 2log( )/n n, where cn~ denotes that c ddn n/ →1 for n→∞. Moreover, n

convergence happens in a stable manner, i.e. υn~ ( ) in probability. For low dimensions we also derive µ υn

explicit expressions for µ υ( )n and variance σ υ2( )n . These expressions are calculated by using an appeal-ing geometrical interpretation of υn.

the extreme value distributions), it is tempting to believe that υn behaves asymptotically like b nn/ .

In this note we show that this intuition is indeed right and that in fact µ υ( )n =µ(Mn)/ ( ) ~µQn

bn/ n~ 2log( )/n n. Moreover, convergence of

happens in a stable manner, i.e. υ_n~ ( ) in µ υ_n

probability.

A noteworthy result, that we use explicitly in this note, is that by Downey & Wright (2007). They show that for non-negative i.i.d. random variables

Xi with finite second moment

µ µ µ max( ) ~ max( ) . X X X X i i i i

∑

_∑

   

(

)

(

)

In particular this implies that µ(M Sn/ ) ~n

µ(Mn)/ ( )µSn , which mimicks µ υ( )n =µ(Mn)/

µ( )Qn , though in a weaker manner.

Central element to the above is the observation that υn has the same probability distribution as

ηn: max{|= X1|, ..., |Xn|}|X=( , ...,X1 Xn) ~ ( )U Bn

where ~ ( )U Bn denotes sampling from the

uni-form distribution U over Bn, being the surface of

the n-dimensional unit hypersphere. This obser-vation follows from the fact that points can be sampled randomly from Bn by sampling from

( , ...,Z1 Zn) and normalize the samples with Qn

(see Muller 1959). In Section 2, this observation allows us to calculate µ υ( )n and σ υ2( )n explicitly

for low values of n, which is another aim of this note.

1 InTRODuCTIOn

Let {Zn} be i.i.d. standard normal variables and

define the entities

Mn _{i n} Zi Sn Zi Q Z i n n i n i = = = = ₌

∑

₌

∑

max (| |), | |, ,..., _{,..., ,...,} 1 ₁ 2 1 andυ_n n n M Q = .

For a random variable X, denote with µ(X) its mean and with σ2_{( )X its variance. If { }p}

n and

{ }qn are sequences, then qn= ( ) means that o pn

q pn n→0 (n→ ∞) . For notational convenience,

when it is clear from the context, we don’t mention explicitly that n → ∞ when we discuss asymptotic behavior. Also, we write pn~ as a shorthand qn

for p qn/ →1 or, equivalently, pn n=qn(1+o( ))1 .

note that the tilde-operator defines an equivalence relation. note that, if pn~ and cqn n~ , then dn

p cn n/ ~ / .q dn n

As Qn~ n in probability (note that for all ε > 0 Chebychev’s inequality implies

Pr(| / | ) Pr(| / )/( / ) | ) Pr(| / | ) Q n Q n Q n Q n n n n n − < = − + < ≥ − < ≥ 1 1 1 1 2 2 ε ε ε σ ((| | )/Z₁2 _ε n_→0)

and Mn is attracted by a stable law (meaning

that there are sequences { }an and { }bn such that

Also, by insensitivity of norms to scaling and translation, it follows that υn can be interpreted as

the max-distance relative to the Euclidian-distance of two arbitrary points in n-space, which is an appealing geometrical interpretation of υn. Put

dif-ferently, the ratio υn can be interpreted as the

long-est edge of a hyperrectangle (also called orthotope), normalised by the space diagonal (which is Qn).

In perspective, a vast amount of literature exists that deals with the properties of the ratio

M Sn/ . Application of this ratio can be found in n

the analysis of process speedup and the perform-ance of scheduling (Downey & Wright 2007), on-line bin-batching and wireless link quality (Zhang 2001; nguyen 2011) or in the modelling of high dimensional databases (Brabants 2005). Like with

υn=M Qn/ , the ratio M Sn n/ can be interpreted as n

the longest edge of a hyperrectangle, normalised by the sum of the lengths of its edges (which is Sn,

taking one edge per dimension). 2 ExACT RESuLT

In this section we calculate exact val-ues for µ υ( )n and σ υ2( )n for low

val-ues of n. To do this, like mentioned in the introduction, we use the observation that υn and

η_n: max{= X1, ..., X_n} |X=( , ...,X1 X_n) ~ ( )U B_n

have the same distribution as a consequence of the fact that points can be sampled randomly from unit hypersphere Bn by sampling from Z1, ...,Zn

and normalize the samples with Qn.

Let Bn*: {= x B x∈ n| 1≥x2≥ ≥... xn≥0 , i.e. B} n* is

the part of Bn of which all points are non-negative

and define a non-increasing mapping. It is not difficult to see that by virtue of symmetry prop-erties µ η( ) µ η( | *)

nk = nk Bn , where µ η( nk| .) denotes

the conditional k-th moment of ηn. note that

vol B( )n* =vol B( )/n 2nn!.

When integrating over Bn* we switch to

polar coordinates. That is, if x B∈ n*, then

ϕ=( , ...,ϕ1 ϕn−1) [ , / ]∈0π 2n−1 and x1=cos( )cos( )ϕ1 ϕ2 ⋅⋅⋅cos(ϕn−1)

x2=sin( )cos( ) cos(ϕ1 ϕ2 ⋅⋅⋅ ϕn−1)

x₃=sin( )cos( ) cos(ϕ₂ ϕ₃ ⋅⋅⋅ ϕ_n−₁)

….

x_n−1=sin(ϕ_n−2)cos(ϕ_n−1)

xn=sin(ϕn−1)

the Jacobian of which is well known and satisfies

Jn( )ϕ =Jn( , ...,ϕ1 ϕn−1) cos (= n−2ϕn−1)⋅Jn−1( , ...,ϕ1 ϕn−2) with J2( )ϕ = . In terms of polar coordinates we have1

Bn* { ( )x Bn| tan( ) ,tan( ) sin( ),

tan( ) sin( ), ... = ∈ ≤ ≤ ≤ ϕ ϕ ϕ ϕ ϕ3 ϕ2 1 2 1 1 ,, tan(ϕn−1) sin(≤ ϕn−2)}. Clearly max{ , ..., }x1 x_n = =x1 cos( ) cos(ϕ1 ⋅⋅⋅ ϕ_n−1) for x B∈ n*. Hence, the k-th moment of υn is

calcu-lated as µ υ µ η ϕ ϕ ϕ ( ) ( ) ! ( ) ... cos( ) cos( ) tan( ) nk nk n n n n vol B = = ⋅⋅⋅

(

≤ −

∫

2 1 1 1 1

))

−⋅⋅⋅ ≤ −

∫

− k n n J d d n n ( ) . tan( ) sin( ) ϕ ϕ ϕ ϕ ϕ 1 1 1 2

Evaluating this integral is quite cumbersome for large values of n, even for k = 1 or k = 2 (the latter being required to calculate σ υ2_{( )}

n ). In Appendix A

we derive for lower dimensions

n µ(υn) approx. σ2(υn) approx. 1 1 1 0 0 2 2 2 π 0.900 ₁ 2 1 8 2 + − π π 0.00774 3 3 2 1 2 2 π × arctan_ _ 0.831 1 3 2 3 18 1 2 2 2 2 + −    π π × arctan 0.0100 4 8 2 32 2 1 2 2 2 π π −    × arctan 0.779

For n = 2 the probability distribution has sup-port [ / , ]1 2 1 and reads

F z z z U z 2 2 1 1 0 4 1 4 ( ) Pr( ) Pr(cos( ) | ~ [ , / ]) arccos( ) = ≤ = ≤ = − υ ϕ ϕ π π

for z in the support.

3 MOMEnT COnvERGEnCE AnD STABILITy OF υn

Following the previous section, since

vol B( )n = 2πn/2/ ( / )Γn 2 , Qn is chi-squared

distrib-uted with n degrees of freedom and density Χ( , ),t n

and Mnk Z Z

k n

k

µ

ϕ

ϕ ϕ ϕ

( ) ! ...

cos( )

tan( ) tan( ) sin( )

M n r nk n n n = ⋅⋅⋅ ≤ ≤ ∞

∫

∫

∫

− − 2 1 1 1 2 0 1 ccos( ) ( ) ( ) ϕ ϑ ϕ ϕ ϕ n k n n n n r J r drd d − − −

(

)

⋅⋅⋅ 1 1 1 1 = ⋅⋅⋅

(

)

⋅ ≤ − −

∫

2 1 1 1 1 1 n n n k n n n vol B J d ! ( ) ... cos( ) cos( ) ( ) tan( )ϕ ϕ ϕ ϕ ϕ ⋅⋅⋅ × − ≤ −

∫

d n n ϕ ϕ 1 ϕ 2 1 tan( ) sin( ) ... ... _{( )} ( )/ / × ∞ − − +

∫

vol B_n 1_n e r rn kdr 2 2 2 1 0 2 π = ∞ − − +

∫

µ υ π ( ) ( ) ( )/ ( ) ( / ) / ( )/ nk vol Bn n e r r n k d r 1 2 2 2 2 2 2 2 2 0 2 = ∞ − −

∫

µ υ π ( ) ( ) ( / ) ( / ) / / / / / nk nn k n t n vol B n _t n e t dt Γ Γ 2 2 1 2 2 2 2 2 2 2 1 0 =µ υ( )nk ∞

∫

tk/2 ( , )t n dt 0 Χ =µ υ µ( ) (nk Qnk)

where Γ is the Gamma function, for which it is known that Γ( ) (k = k−1 and )! Γ(k_{+1 2}/ )_{= π}(₂k_{−1 2})!!/ k_.

Furthermore, it is well-known in literature that

µ(Mn) ~ ~bn 2log( )n , where bn is defined in the

introduction and more precise in Appendix B. As to µ( )Qn , note that µ( ) ( / ) (( )/ ) ( / ) / / / Q xx e n dx n n n n x n =∞

∫

₂2 1− − 2 = + 0 2 2 2 1 2 2 Γ Γ Γ By observing that µ(Qn−1)<µ( )Qn <µ(Qn+1) , we calculate straightforwardly that

n− =1 µ( ) (Q_n µQ_n−1)<µ2( )Q_n <µ( ) (Q_n µQ_n+1)=n, so that n− <1 µ( )Qn < n. Hence µ( ) ~Qn n, implying µ υ µ µ ( ) ( ) ( ) ~ ~ log( ) n n n n M Q b n n n = 2 .

Also, from Appendix it follows directly (note that µ( )Qn2 = ) thatn σ υ( )n π n nb 2 2 2 6 0 − → so that σ υ( )n2 → .0

From Appendix B it also follows straightfor-wardly that σ υ µ υ µ υ µ υ µ µ µ µ 2 2 2 2 2 2 2 2 1 1 ( ) ( ) ( ) ( ) ( )/ ( ) ( )/ ( ) ( n n n n n n n n M Q M Q = − = − = bb n o b n o n n 2 21₁ 1₁ 1 0 / )( ( )) ( / ) ( ( )) + + − →

and so (applying Chebychev’s inequality), for all

ε > 0 Pr ( ) ( ) Pr ( ) ( ) ( ) ( ) υ µ υ µ υ ε υ µ υ ε µ υ σ υ σ υ n n n n n n n n − _≤     =  − ≤   ≥ 11−σ υ₂2 1₂→1 µ υ ε ( ) ( )nn

Thus, we conclude that υn is relatively

sta-ble, i.e. υ_n~ ( ) in probability. note that since µ υ_n µ υ( )n → 0 this is a stronger concept than absolute

stability.

4 SIMuLATIOn OF υn

Following the previous elaboration, a Monte Carlo simulation is conducted to verify the asymptotic behaviour towards 2log( )/n n . The simulation

code as included in Appendix C is used for this purpose. Figure 1 shows the result of 1000 runs. It is seen that the realisations are symmetrically dis-tributed around the mean value and that the uncer-tainty vanishes large n.

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Downey, P.J. & Wright, P.E. 2007, The ratio of the extreme to the sum in a random sequence, Extremes 10(4): 249–266, Alternatively, see working paper TR 94-18 by the same authors and with the same title from the Dep. of Comp.Sc., university of Arizona, 1994.

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theory, Technical Report 205, Dept. Statistics, univ. north Carolina.

von Mises, R. 1954, La distribution de la plus grande de n valeurs, Amer. Math. Soc., reprinted in: Selected Papers II: 271–294.

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APPEnDIx A: DERIvATIOnS OF ExACT RESuLTS For n =2 we have µ υ ϕ ϕ π π π ( ) ! ( ) cos( ) , / 2 2 2 0 1 4 1 2 2 8 2 1 2 2 2 2 0 900 = = _ _= ≈

∫

vol B d

Likewise, for n = 3 we calculate

µ υ π ϕ ϕ ϕ ϕ ϕ ( ) ! ( ) cos( ) cos ( ) / arctan(sin( 3 3 3 0 4 1 2 2 2 1 0 3 2 1 =_{vol B}

∫

⋅ d d )))

∫

= ⋅ 48 4 1 2 1 1 1 0 4 π ϕ _ϕ ϕ π cos( ) [cos(arctan(sin( )) sin(arctan(sin( )) /

∫∫

+ +... arctan(sin( ))]ϕ₁ dϕ₁ = + ⋅ + +    6 1 1 1 1 2 1 1 2 1 1 π ϕ ϕ ϕ ϕ ϕ cos( ) sin ( ) sin( ) sin ( ) arctan(sin( ))     

∫

dϕ π 1 0 4 / = + +

∫

6 1 2 0 1 2 π y y arctan( )y dy / =6

[

]

1 2₀ π yarctan( )y / =3 2 1 ≈ 2 2 0 831 π arctan( ) , and for n = 4 µ υ ϕ ϕ π ϕ ( ) ! ( )

cos( )cos ( )cos

/ arctan(sin( )) 4 4 4 0 4 0 1 2 2 4 2 1 = ⋅

∫

∫

vol B 33 3 3 2 1 0 2 ( ) arctan(sin( )) ϕ ϕ ϕ ϕ ϕ d d d

∫

=384

∫

2 2 3 2 0 4 1 2 2 2 2 1 0 π ϕ ϕ_{ϕ ϕ ϕ} π / ar cos( )cos ( ) sin(arctan(sin( )))d d cctan(sin( )) ... ϕ1

∫

+ ... cos( )cos ( ) sin(arctan(sin( ))) / ar + 384

_∫

_⋅⋅⋅ 2 2 1₃ 0 4 ₁ 2 ₂ 2 0 π ϕ ϕ ϕ π cctan(sin( ))ϕ1

∫

⋅⋅⋅ cos (arctan(sin( )))2 2 2 1 ϕ d dϕ ϕ = +

∫

128 1 2 0 4 1 2 2 ₂2 2 2 1 0 π ϕ ϕ ϕ ϕ ϕ ϕ π / arctan(s

cos( )cos ( ) sin( ) sin ( )d d iin( )) ... ϕ1

∫

+ ...

cos( )cos ( ) sin( )

sin ( ) / / a + +

(

)

∫

64 1 2 0 4 1 2 2 ₂ 2 2 3 2 2 1 0 π ϕ ϕ ϕ ϕ ϕ ϕ π d d rrctan(sin( ))ϕ1

∫

= + +

∫

∫

128 1 2 0 4 1 2 2 2 1 0 1 π ϕ ϕ ϕ ϕ π ϕ / arctan(sin( ))

cos( )cos( )d sin ( )d ....

...

cos( )cos( )sin( ) sin( )

sin ( ) / ar + +

∫

64 1 2 0 4 1 2 2 ₂2 2 1 0 π ϕ ϕ ϕ ϕ ϕ ϕ π d d cctan(sin( ))ϕ1

∫

= + + −        −

∫

128 1 2 1 1 2 2 2 1 2 1 2 1 1 0 4 π ϕ ϕ ϕ ϕ π cos( ) sin ( ) sin ( ) ... / d

...−128₂[ cos( )/ 1 arctan(sin( ))

_∫

2−cos ( ) cos( )2 2 2 1 0 0 1 π ϕ ϕ ϕ ϕ ϕ π d d 44

∫

+ ] ... ... _{cos( )} sin ( ) ( sin ( )) sin ( ) + + + − − − 64 1 1 2 2 1 2 2 1 2 1 2 1 2 1 2 2 π ϕ ϕ ϕ ϕ y y dy 11 1 1 0 4 1 2 ₁ / sin ( ) / +

∫

∫

              ϕ π ϕ d =128₂ +64₂ π A π B. where A y y dy y dyd = + + − + −

∫

∫

∫

+ 1 2 1 1 2 2 2 2 2 0 1 2 1 2 1 1 1 1 0 4 2 1 / / sin ( ) / cos( )ϕ ϕ ϕ π

and B _y y dy = + + − − − + cos( ) sin ( ) ( sin ( )) sin ( ) / si ϕ ϕ ϕ ϕ 1 2 1 2 1 2 1 2 2 1 1 1 1 2 2 1 2 nn ( ) / 2 1 1 0 4 1 ϕ π ϕ

∫

∫

                d = + + − − −

∫

∫

+ y y y dy y y dy 2 2 2 0 1 2 1 2 2 1 1 1 0 41 1 2 2 1 2 2 1 ( ) cos( ) / / sin ( ) / ϕ ϕ π

∫∫

dϕ1 Since 2 1 2 2 12 2 2 1 1 1 2 1 1 2 1 − = − + _ _    + +

∫

y dy y y y / sin ( ) / sin arcsin ϕ 22 1 1 ( )ϕ = + − + + − +     1 2 1 4 1 2 2 2 1 2 2 2 1 2 1 2 1 π ϕ ϕ ϕ sin ( )

sin ( ) arcsin sin ( )

we find cos( ) / sin ( ) / ϕ ϕ π ϕ π 1 2 1 1 1 1 0 4 2 2 0 2 1 4 2 18 2 1 2 2 2 2 1 − = + − + + +

∫

∫

y dyd y y dy 11 2 2 0 1 2 ₁ 2 2 / / arcsin .

∫

−

∫

+     y dy Also, we have arcsin arcsin ( / 1 2 2 1 2 2 2 2 2 2 0 1 2 2 +     = +             + y dy y y y ++ +

∫

∫

2 2 1 2 2 0 1 2 0 1 2 y ) y dy / / = _ _{ +} + +

∫

1 2 2 1 3 3 2 2 2 1 2 2 2 2 0 1 2 arcsin ( ) / y y y dy so that A y y dy = − + − _ _ + + +

∫

1 4 2 18 2 12 2 13 3 1 2 2 2 1 2 2 0 1 2 π arcsin ( ) / . Furthermore, 2 1 2 2 4 2 3 2 2 2 1 1 1 2 2 1 1 1 2 2 1 2 1 y y dy y y dy y d − − = − − + − +

∫

+

∫

/ sin ( )ϕ / sin ( )ϕ yy 1 1 1 2 1 / +sin ( )

∫

ϕ = − − + − +

∫

+

∫

2 2 3 1 2 2 1 1 1 2 1 1 1 2 1 2 1 y dy y dy / sin ( )ϕ / sin ( )ϕ = − + − + + − +        2 1 2 1 4 1 2 2 2 1 2 2 2 1 2 1 2 1 π ϕ ϕ ϕ sin ( ) sin ( ) arcsin sin ( )                  + ... = − + + + + − +    _ 1 1 4 1 2 1 1 2 2 2 1 2 1 2 1 π ϕ ϕ ϕ sin ( ) sin ( ) arcsin sin ( ) so that B y y y y dy = + − + + + −

∫

(_{( )}) ( ) ... / 2 1 1 1 1 2 2 2 2 2 0 1 2 ... sin ( ) sin ( ) arcsin sin ( ) − − + + + + − +        1 1 4 1 2 1 1 2 2 2 1 2 1 2 1 π ϕ ϕ ϕ                 

∫

0 4 1 π ϕ / sin( ) d = + − + + + − − + + + + −

∫

(_{( )}) ( ) arcsin / _{2 1 1} 1 1 2 1 1 4 1 2 1 1 2 2 2 2 0 1 2 2 2 y y y y dy y y π 22 2 2 0 1 2 +    _        

∫

_y dy / = − + + − + +            

∫

∫

1 1 2 1 1 4 1 2 2 2 0 1 2 2 0 1 2 y dy y dy / / arcsin π = − + + − +

( )

+ + +

∫

_{1 2}1 1₂ 2 1 8 2 12 2 13 3 2 2 2 1 2 2 0 1 2 2 2 2 0 y dy y y y dy / arcsin ( ) π 11 2/

∫

= − + _ _ + − + +

∫

1 2 2 18 2 12 2 1 3 3 2 2 2 2 1 2 2 0 1 2 π arcsin ( ) / y y dy _. Hence µ υ π π ( ) arcsin( ) ( ) / 4 2 2 2 0 1 2 128 1 4 2 18 2 12 2 1 3 3 1 2 2 1 2 = − + − + + +  

∫

y y dy           + ... ... arcsin ( ) / + − + _ _ + − + + 

∫

64 1 2 2 18 2 12 2 13 3 2 2 2 1 2 2 2 2 0 1 2 π π y y dy              = − _ _ = − _ _ ≈ 8 2 32 2 1 3 3 8 2 32 2 1 2 2 0 779 2 2 π π π π arcsin arctan . As to the variance σ υ2 2

( ) we calculate the sec-ond moment of µ υ₂( ) of υ₂ 2 as µ υ ϕ ϕ π π π π ( ) ! ( ) cos ( ) , / 22 2 2 2 1 0 4 1 2 2 8 2 1 4 8 1 1 2 0 = = _ + _= + ≈

∫

vol B d 8818 . so that σ υ π π 2 2 1 1₂ 82 0 00774 ( )= + − ≈ , . Also µ υ ϕ ϕ ϕ ϕ π ( ) ! ( ) cos ( ) cos ( ) / arctan(sin( 32 3 3 0 4 2 1 3 2 2 1 0 3 2 = ⋅

∫

vol B d d ϕϕ1))

∫

=48 ₋ 4 1₃ 2 1 1 3 1 π ϕ ϕ ϕ

cos ( ) sin(arctan(sin( )))_{sin (arctan(sin( )))}       

∫

dϕ π 1 0 4 / = + − +     12 1 3 1 2 1 1 2 1 3 1 2 1 3 2 π ϕ ϕ ϕ ϕ ϕ cos ( ) sin( ) sin ( ) sin ( ) ( sin ( )) /        

∫

dϕ π 1 0 4 / = − − +

∫

12 2 2 1 2 1 1 0 4 π ϕ ϕ ϕ π cos ( ) cos ( ) cos( ) ... / d ... cos ( )( cos ( )) ( cos ( ))/ cos( ) / + − −

∫

12 1 3 2 2 1 2 1 2 1 3 2 1 0 4 π ϕ ϕ ϕ ϕ π d = − − − + −

∫

∫

12 2 4 2 4 2 2 2 1 2 1 2 2 3 2 1 2 1 4 2 3 2 1 2 1 π π π y y dy y y dy y y dy / / / / / ( ) ( )

∫∫

=   − −       − 12 2 1 2 2 2 1 2 1 π arcsin y y y _/ ... ... _arcsin ... / − − −            + 4 2 2 2 1 2 1 π y_y y ... _arcsin / + − + − −            4 1 2 2 2 2 3 2 2 2 1 2 1 π y y y_y y = − − + −         4 2 2 ₂ 2 2 1 2 1 π arcsin( ) / y _{y y} y y = +1 3 2 3 π . so that σ υ π π 2 3 1₃ 2₃ 182 2 1₂ 2 0 0100 ( )= + − arctan _ _{ ≈} . .

APPEnDIx B: PROOF THAT,

µ(Mn) ~ , (bn µMn2) ~bn2 AnD bn~ 2log( ))n

Let { }Zn and Z be i.i.d. standard normal

vari-ables (with distribution Φ and density ϑ ). Let Φabs( ) Pr(x = Z x≤ and denote its density with )

ϑabs. Also, we write Φ and Φabs as a shorthand

for tail probabilities 1− Φ and 1− Φabs respectively.

Obviously, for x ≥ 0 we have ϑ_abs( )x = 2ϑ( )x and Φabs( ) = 2Φ( )x x = erfc x( / 2 , where erfc is the )

complementary error function defined by and having first order Laurent expansion

erfc x e dt e x o t x x ( ) := ∞ − = − ( + ( ))

∫

2 2 2 _{1 1} π π .

For the moments of Z we have µ π π π ( ) ( ) / / / Z t e dt t de k t e d k k t k t k t = = − = − − ∞ − − ∞ − −

∫

∫

2 2 1 2 2 2 2 2 0 1 2 0 2 2 _{tt k Z}k 0 2 1 ∞ −

∫

=( − )µ

( )

so that µ(Z2i) (= 2i−1)!! and µ(Z2 1i+ )= 2/ ( )!!π i . notably, moments exist and are finite. 2 Also, the inverse Φabs← of Φabs will be used below

and follows from

1 1 2 2 1 1 1 2 2 2 2 2 Φabs x x erfc x x e o ( )= ( / )= ( + ( )) π yielding Φabs x W x o ← ₌  ₊   ( ) 2 1 1₂( ( )) π

where W is the Lambert W function, implicitly defined by W x e_{( )}W x( )_{= .}x

With Λ we refer to the Gumbel distribution (also known as the doubly exponential distribu-tion and being one of the three limiting cases of the GEv distribution, well known in extreme value theory) defined by Λ( ) exp(x = −e−x)_{, having} mean γ and variance π2_{/ , where γ ≈ 0 5776 . is the}

Euler-Mascheroni constant. It is common knowl-edge that Φ satisfies the von Mises condition for being in the domain of attraction of Λ (von Mises 1936; Falk 1993), i.e. ϑ( ) ( ) ( ) ( ) x x _x t dt x Φ2 Φ 1 ∞

∫

→ → ∞ .

It is easily seen that Φ_abs also satisfies this condi-tion and is therefore also in the domain of attrac-tion of Λ. Say that { }an and { }bn are sequences for

which (Mn−b an)/ n→ Λ in distribution. Intuitively

we would expect that µ(Mn) (− bn+anγ)→ 0 and that µ(Mn2) ( (− a2n π2/6+γ2)+bn2+2a bn nγ)→ . 0 From Pickands (1968) (see also nair 1981) we know that this is indeed the case. Furthermore, it is widely known that { }an and { }bn are solutions to bn= abs n

←

Φ ( / )1 and a_n= ( ), where r xr b_n ( )= Φ_abs( )/x ϑ_abs( )x is the reciprocal hazard function (Smith 1987). Hence

bn= W n +o     2 2 _{1 1} π ( ( )) and a n e b n b e b n _b n n b n n n = 1 ₋ = 2 1 2 1 2 2 2 2 2 / /_π / ~ π .

Hence (note that µ(Mn)= + 1 and bbn o( ) n→∞),

µ(M_n) ~b_n and µ(M_n2) ~b_n2_.

Finally, as log( ) log log( )x − x W x≤ ( )≤ log( ) log log( )/x − x 2 for x e≥ (see Hoorfar 2008), we see that in terms of elementary functions bn~ 2log( )n . Consistently, Cramer

(1946) and Deo (1972) approximate the loca-tion and scaling parameters { }an and { }bn by

bn= −cn log( log( ))/4π n 2cn+2/cn and an= 1/ , cn

where c_n= 2log( ) .n

APPEnDIx C: MATLAB CODE

FOR SIMuLATIOn OF υn Clear clg n=1000; hold on for j=1:n, z(1)=normrnd(0,1); m(1)=abs(z(1)); s(1)=m(1); q2(1)=z(1)^2; for i=2:n, z(i)=normrnd(0,1); m(i)=max(m(i-1),abs(z(i))); s(i)=s(i-1)+abs(z(i)); q2(i)=q2(i-1)+z(i)^2; q(i)=sqrt(q2(i)); nu(i)=m(i)/q(i); ex(i)=sqrt(2*log(i)/log(2.7181)/i); end loglog([1:n],nu,′b′) end loglog([2:n],ex(2:n),′r′) xlabel(′Sample size n′) ylabel(′nu′) grid

title(′Results of 1000 simulations (blue lines) and theoretical result (red line)′)