New copulas based on general partitions-of-unity and their applications to risk management
Dietmar Pfeifer, Hervé Awoumlac Tsatedem, Andreas Mändle, Côme Girschig
New copulas based on general partitions-of-unity and their applications to risk management
Dietmar Pfeifer , Hervé Awoumlac Tsatedem † , Andreas Mändle , and Côme Girschig Carl von Ossietzky Universität Oldenburg, Germany and École Nationale des Ponts et Chaussées, Paris, France July 8, 2016
Abstract:
We construct new multivariate copulas on the basis of a generalized infinite parti-tion-of-unity approach. This approach allows - in contrast to finite partition-of-unity copulas - for tail-dependence as well as for asymmetry. A possibility of fitting such copulas to real data from quantitative risk management is also pointed out.
Key words: copulas, partition-of-unity, tail dependence, asymmetry
MSC:
1. Introduction
The theory of copulas and their applications has gained much interest in the recent years, especially in the field of quantitative risk management, insurance and finance (see e.g. M C N EIL , F REY AND E MBRECHTS (2005) or R
ANK (2006)). While classical approaches like elliptically contoured copulas and Archimedean copulas are widely explored, other approaches concentrate on non-standard, non-symmetric or data-driven copula constructions (see e.g. L
AUTERBACH
AND P FEIFER (2015), L
AUTERBACH (2014), C
OTTIN AND P FEIFER (2014) or J
AWORSKI , D URANTE
AND H ÄRDLE (2013) and the papers therein for a survey, especially the contributions related to vine copulas). Statistical and computational aspects of copulas have also been investigated in more detail recently (see e.g. B
LUMENTRITT (2012) and M
AI AND S CHERER (2012)). In this paper, we want to focus on a particular class of copulas and their generalizations, the so called partition-of-unity copulas (see e.g. L I , M IKUSI Ń SKI AND T AYLOR (1998) or K
ULPA (1999)). Whereas in the usual approach, only finite partitions-of-unity are considered, which do not allow for a modelling of tail-dependence, we extend this concept to infinite partitions-of-unity, which allows for tail-dependence as well as for asymmetry, and which can also be used to fit given data to a more realistic copula model. Our investigations resemble in some sense more recent approaches such as Y
ANG ET AL . (2015), G
ONZÁLEZ -B ARRIOS AND H ERNÁNDEZ -C EDILLO (2013), Z
HENG ET AL . (2011), H
UMMEL AND M ÄRKERT (2011), or G HOSH AND H ENDERSON (2009). Whereas in these papers, local modifications of known standard copulas are considered in order to obtain tail dependence or asymmetries, we focus on a closed form representation of completely new copula densities which allows for easy Monte Carlo simu-lations as well as a data driven modelling of tail dependence and asymmetries. This approach is not re-stricted to two dimensions in general, but can likewise be used in arbitrary dimensions. However, in order to illustrate our results, we will give examples in the bivariate case only. To facilitate the readability of the paper, all elaborate proofs are given in an appendix.
2. Main Results
Let { } + = denote the set of non-negative integers and suppose that { } ( ) j + Î i i u and are non-negative maps defined on the interval { y j } ( ) + Î j v ( ) each such that (2.1) ( ) ( ) 1 j y ¥ ¥= = = å å i ji j u v = + and for (2.2) ( ) 0, ( ) 0 j a y b = > = > ò ò i i j j u du v dv , . Î i j The maps and can be thought of as representing discrete distributions over the non-negative integers with parameters u and v , resp. The sequences ( ) j i u + ( ) y j v { } a + Î i i and { } b + Î j j then represent the prob-abilities of the corresponding mixed distributions each. Let further { } , + Î ij i j p represent the probabilities of an arbitrary discrete bivariate distribution over with marginal distributions given by and for Then + + ´ i j + a ¥= = = å i ijj p p b ¥= = = å j iji p p , . Î i j ( ( , ) : ( ) ( ), , 0,1 j ya b ¥ ¥= = = å å ij i ji j i j pc u v u v u v ) Î (2.3) defines the density of a bivariate copula, called generalized partition-of-unity copula . The fact that c in fact is the density of a bivariate copula can be seen as follows: ( , ) ( ) ( ) ( )( ) ( )( ) ( ) 1, (2.4) j y b ja b a bj jj a ja a a ¥ ¥ ¥ ¥= = = =¥ ¥ ¥ ¥ ¥ ¥= = = = = = = == = = = = å å å åò òå å å å å å ij iji j j ii j i ji j i jij i ii ij i ii j i j i ii i i p pc u v dv u v dv up u uu p u likewise for ( , ) . ò c u v du Note that from a „dual“ point of view, we can rewrite (2.3) as ( ( , ) ( ) ( ), , 0,1 ¥ ¥= = = å å ij i ji j c u v p f u g v u v ) Î (2.5) where ( )( )( ) , ( ) , , yja b + = = jii ji j f g i Î j denote the Lebesgue densities induced by { } ( ) j + Î i i u and This means that the copula density c can also be seen as an appropriate mixture of product densities, which possibly allows for a simple way for a stochastic simulation. { } ( ) . y + Î j j v An extension of this approach to d dimensions with is obvious: assume that > d { } ( ) j + Î ki i u for represent discrete probabilities with 1, , = kd . + Î i ( ) 1 j ¥= = å kii u for (2.6) (0,1) Î u and ( ) 0 j a = > ò ki ki u du for (2.7) Let further { } + Î d p i i represent the distribution of an arbitrary discrete d -dimensional random vector Z over where, for simplicity, we write + d ( ) , , , = d i i i i.e. ( ) , + = = Î d P p i Z i i . (2.8) Suppose further that for the marginal distributions, there holds ( ) , , 1, a + = = Î = k ki P Z i i k d , . (2.9) Then ( ) ( , 11,1 ( ) : ( ), , , 0,1 ja + =Î = = = å kd k d dk i k dd kk ik pc u u ii u u ) Î u ) Î (2.10) defines the density of a d -variate copula, which is also called generalized partition-of-unity copula . Alternatively, we can rewrite (2.10) again as ( ) ( , 11 ( ) ( ), , , 0,1 + =Î = = å kd d dk i k dk c p f u u u ii u u (2.11) where the ( )( ) , , 1, , ja + = Î = kiki ki f i k d denote the Lebesgue densities induced by the { } ( ) . j + Î ki i u
3. The symmetric case (diagonal dominance)
For simplicity, we restrict ourselves to the two-dimensional case in the sequel. The generalization to higher dimensions is obvious. i + Î Let j for i and Define y = i ( ) 0. j a = > ò i i u du , if: 0, otherwise. a ì =ïï= íïïî iij i jp (3.1) Then ( ( ) ( )( , ) : ( ) ( ), , 0,1 j j aa ¥ ¥= = = = å å i i i i ii ii u vc u v f u f v u v ) Î (3.2) defines the density of a bivariate copula, called generalized partition-of-unity copula with diagonal domi-nance . Example 1 (binomial distributions - Bernstein copula) . Consider, for a fixed integer the family of binomial distributions given by their point masses 2, ³ m j - - ìæ ö-ï ÷ïç ÷ - = -çï ÷ïç ÷= è øíïïï ³ïî i m im i m u u i mu i i m (3.3) Here we have, for 0, , 1, = - i m
1( ) (1 )( 1)! ( 1) ( ) ( 1)! !( 1 )! 1 (3.4)!( 1 )! ( 1) !( 1 )! ! a j - - æ ö- ÷ç ÷= = -ç ÷ç ÷è ø- G + G - - - -= ⋅ = ⋅ =- - G + - - ò ò i m im i m i mu du u u duim i m i m i m ii m i m i m i m m and hence ( )
21 10
1( , ) ( ) (1 )(1 ) , , 0,1 - - -= æ ö- ÷ç ÷= - -ç ÷ç ÷è ø å m m iim i mc u v m uv u v u vi ( ) Î (3.5) which corresponds to the density of a particular Bernstein copula (see e.g. C OTTIN AND P FEIFER (2014), Theorem 2.1). Especially, for we obtain 2, = m ( ) ( , ) 4 2 2 2, , 0,1 . = - - + Î c u v uv u v u v (3.6) The corresponding copula is given by C ) Î ( ( , ) ( , ) (1 )(1 ), , 0,1 = = + - - ò ò yx C x y c u v dv du xy xy x y x y (3.7) and belongs to the so called
Farlie-Gumbel-Morgenstern family (cf. e.g. N
ELSEN (2006), p. 77). For gen-eral relation (3.5) represents the density of a copula with polynomial sections of degree m in both variables (cf. N ELSEN (2006), chapter 3.2.5). The following graphs show some of these densities for dif-ferent values of m . 1, > m = = = = m m m m Clearly, all those densities are bounded by the constant m , hence the coefficients l and l of upper and lower tail dependence are zero: U L ( , ) (1 )lim lim 01 1 l -= £ =- - ò ò mt tU t t c u v du dv m tt t and
20 00 0 ( , )lim lim 0. l = £ ò ò t t mL t t c u v du dv mtt t = Î i (3.8) Example 2 (negative binomial distributions) . Consider, for fixed the family of negative binomial distributions given by their point masses b > ,
1( ) (1 ) , . bb bj + æ ö+ - ÷ç ÷= -ç ÷ç ÷è ø ii iu u ui (3.9) Here we have, for , + Î i bb b b b b ba j b b b b æ ö+ - G + G + G +÷ç ÷= = - = ⋅ =ç ÷ç ÷ G G + + + + +è ø ò ò ii i i i iu du u u dui i i i i and hence ( )( ) ( )
20 0 bb b bb bb b bb ¥= ¥= æ ö+ -- - ÷ç= + + + ÷ç ÷ç ÷çè øæ öæ ö+ - + +÷ ÷ç ç= + - - ÷ ÷ Îç ç÷ ÷ç ç÷ ÷ç çè øè ø å å ii ii iu vc u v i i uvii iu v uv u vi i
For integer choices of this expression can be explicitly evaluated as a finite sum, as can be seen from the following result. , b Lemma 1.
For there holds , b Î ( ) ( )
12 1 0 b bb b b bb -+ = æ öæ ö- +- - ÷ ÷ç ç= + ÷ ÷ Îç ç÷ ÷ç ç÷ ÷ç ç- è øè ø å ii u vc u v uv u vi iuv (3.12) To give an illustration of Lemma 1, we show an exemplary table for , likewise for the corre-sponding copula
1, , 6 b = ( ) ( , ) ( , ) , , 0,1 . b b = Î ò ò yx C x y c u v dv du x y b ( , ), , (0,1) b Î c u v u v ( ) (1 )(1 )2 1 - -- u vuv (1 3 )(1 ) (1 )3 (1 ) + - -- uv u vuv ( ) + + - -- uv u v u vuv ( ) + + + - -- uv u v u v u vuv ( ) + + + + - -- uv u v u v u v u vuv ( ) + + + + + - -- uv u v u v u v u v u vuv b ( , ), , (0,1) b Î C x y x y
1 (2 )1 - -- x yxy xy ( ) - - + + + - -- x y x y x y x y x yxy xy ( ) - - + + + + + + - - - - --- - + - - + + + - - - - xy x y x xy y x y x y xy x x y xy yxyx y x y x y x y x y x y x y x y x y x y x y xy The following graphs show the negative binomial copula densities for b c
1, , 6. b = b b = = b = b b = = b = Negative binomial copulas typically show an upper tail dependence, as can be seen from the following exemplary table. b
1 2 3 4 5 6 7 8 9 10 ( ) l b U
12 58 1116 93128 193256 7931024 16192048 2633332768 5338165536 215955262144 A closed formula for the tail dependence coefficients for integer values of b is given in the following result. Lemma 2.
For there holds , b Î ( , ) 2 (2 ) 4 (2 )( ) lim (1 )1 ( ) ( ) ( )2 11 1 for large . (3.13)4 t tU t c u v du dv x y dx dy u u dut x y b b b b bbb b bl b b bbb bpb -+ G G= = ⋅ = ⋅ -- G + Gæ ö÷ç ÷ç ÷ç ÷çè ø= - - ò ò ò ò ò Note that the sequence is related to certain combinatorial graph problems, see L
ISK-OVETZ AND W ALSH (2006), Table 4, p.385. The authors remark in their paper: “The latter [sequence] is also known as the enumerator of cycles of objects, where the individual objects are enumerated by the Catalan numbers.” 24 ( ) 4 U b b bl b b æ ö÷ç= -ç ÷ç ÷çè ø÷ Note that relation (3.13) also implies that lim ( ) 1. U b l b ¥ = Example 3 (Poisson distributions) . Consider the family of Poisson distributions given by their point masses , ( )( ) (1 ) ,! gg gj + = - Î i ii L uu u ii (3.14) where and Here we get, for with the substitutions and ( ) ln(1 ) 0, (0,1) = - - > Î
L u u u ( ) u (1 ) , g = + g > , + Î i = z L y z ( )( ) (1 ) ! !1 , (3.15)(1 ) ! (1 ) 1 1 i i i i zi i ii i iyi i L u zu du u du e dzi iy e dyi g gg g g ga jg g g gg g g g ¥ - +¥ -+ + = = - =æ ö æ ö÷ ÷ç ç= = = -÷ ÷ç ç÷ ÷ç ç÷ ÷+ + + +è ø è ø ò ò òò indicating that the correspond to a geometric distribution with mean and hence , g a i , g ( ) ( ) (1 ) ln(1 ) ln(1 )( , ) (1 )(1 ) (1 ) , , 0,1 .! g gg g gg ¥= + - -= + - - Î å ii u vc u v u v u vi (3.16) The following graphs show some of these copula densities for different choices of . g g = g = g = g = g = g = The corresponding copula C cannot be calculated explicitly. However, in contrast to the visual impres-sion, the coefficient of upper tail dependence is zero here for all although we have a singu-larity in the point (1,1) in all cases. ( ) l g U g > For a rigorous proof, we first remark that
20 0 0 ( , ) : exp( )! ! ! i i i ii i i x y x yh x y x yi i i ¥ ¥ ¥= = = æ ö æ ö÷ ÷ç ç÷ ÷= £ ⋅ = +ç ç÷ ÷ç ç÷ ÷è ø è ø å å å for all (3.17) , 0 x y ³ such that, with the constant : (1 g g g = - + K ), ( ) ( ) ( , ) (1 )(1 ) (1 ) (1 ) ln(1 ), (1 ) ln(1 )2(1 ) (1 ) , , 0,1 . (3.18) K K c u v u v h u vu v u v g gg g g g g g = + - - - + - - + -£ - - Î
This implies ( , ) (1 )( ) lim 2 lim 0,1 ( g l g + -= £- + ò ò Kt tU t t c u v du dv tt K = (3.19) as stated. (Note that g g g + = + - + > K Example 4 (log series distribution) . Consider the family of log series distributions given by their point masses ( ) ,( ) j = ⋅ ii uu i L u Î i (3.20) where again Here we get ( ) ln(1 ), (0,1). = - - Î L u u u
1( ) ( 1) ln( 1) a j += æ ö÷ç ÷= = -ç ÷ç ÷è ø åò i ji i j iu du jji + . Î i for (3.21) The proof of this relation requires some more sophisticated arguments, as is shown in the sequel. Lemma 3.
For and there holds > c , Î n ( ) ( -¥ - += æ ö- ÷ç ÷= - + -ç ÷ç ÷è ø åò nx ncx jj e ne dx j c cjx ) (3.22) Note that for the special case we obtain, by the substitution = c ln(1 ), = - - x u ( )
1: (ln(1 ) b -¥ - += æ ö- ÷ç ÷= = = -ç ÷ç ÷- - è ø åò ò nxn nx jn j e nu du e dx jju x
1) ln( 1). + (3.23) Hence with ( ) ln(1 ) j = - ⋅ - ii uu i u for this means , Î i
1( ) ( 1) ln( 1) ba j += æ ö÷ç ÷= = = -ç ÷ç ÷è ø åò i jii i j iu du jji i + . Î i for (3.24) The density of the bivariate log series copula is hence given by j ja b ¥ ¥= = = = - - å ( ) ii ii ii i uvc u v u v u v i < < u v å for (3.25) The following graph shows the corresponding copula density. plot of ( , ) c u v The log series copula does not have a positive tail dependence either, as in the case of the Poisson copula. The proof of this statement again requires some more sophisticated arguments. We proceed in the follow-ing steps.
Lemma 4.
With we have ( ) ln(1 ), = - -
L u u =- ò t t L ut dut L u (3.26)
Lemma 5.
With the given in (3.21) and the copula density given in (3.25), it holds that ( ) ln(1 ), = - -
L u u a i
1( ) : ( , ) 11 1 ( ) = £ -- - ò ò ò t t t t L utK t c u v du dv dut t < < t ( ) L u for (3.27) which in turn implies that the log series copula has no tail dependence.
4. The asymmetric case
Specifying the probabilities ij p in a non-symmetric way we obtain asymmetric copula densities even if the maps and are identical. A very simple approach to this problem is a specification of a suitable non-symmetric -matrix ( ) i j ( ) j y ( 1) n + ´ ( 1) n + , 0, , n ij i j n M p = é ù= ë û for with n + Î n nik ki ik k p p a = = = = å å for (4.1) 0, , i = n and
11 12 , . i j n > b = , if : 0, otherwise iij i jp a ì =ïï= íïïî for (4.2) Example 5 (negative binomial distributions, asymmetric case). We consider the negative binomial distri-butions from Example 2 with Then
1( ) (1 )(2 ) i i u du i i a j = = + + ò for With n and . i + Î =
18 5 5 0 210 0 0 0 01: 0 5 0 0 060 0 0 0 3 02 0 0 0 0 M é ùê úê úê úê ú= ê úê úê úê úê úë û (4.3) the conditions above are fulfilled, giving the copula density, according to (2.3), ( )
1( , ) ( ) ( ) ( ) ( ) ( ) ( ), , j j j j j ja a a a a ¥ ¥ ¥= = = = = + = = + å å å å å n nij iji j i j k ki j i j k ni j i j k p pc u v u v u v u v u v Î (4.4) or, more explicitly, ( ) ( )
26( , ) (1 )(1 ) 2 6 12 20 30 ...122030(1 )(1 )(1 )(1 ) ( 1)( 2) ( , ), , 0,1 (4.5)5(1 ) ¥= æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷= - - ⋅ ⋅ ⋅ +÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷è ø- -+ - - ⋅ + + = Î- å k kk vc u v u v u u u u M vvvu vu v k k u v H u v u vuv with the polynomial ( , ) 150 450 10 510 10 30 10 30300 30 5 80 30 94 30 30 6015 10 18 30 10 15 10 18 10 5 6. H u v u v u v u v u v u v u v u v u vu v u v u v u v u v u v u v uv u vu v u u v uv v uv v uv u v = - - + - - - + -- + - + - + + - - ++ + + - + - + - + + + (4.6) plot of plot of ( , ) c u v ( , ) ( , ) c u v c v u - The corresponding copula C again has a coefficient of upper tail dependence U l = as in the symmetric case. The following example shows an asymmetric copula composed by two different negative binomial distri-butions. Example 6.
We consider the negative binomial distributions from Example 2 with and Then 1 b = b =
1( ) (1 )(2 ) i i u du i i a j = = + + ò and j a = ,
2( ) 2(2 )(3 ) j j v dv j j + = = + + ò + b j for i j Let further . + Î if 2if 2 10 otherwise bb ì =ïïïï= =íïïïïî jij j j ip j i for (4.7) , , i j + Î i j + i.e. ij i j p b b b b b b b b + Î é ùê úê úê úê úé ù =ë û ê úê úê úê úê úë û (4.8) where stands for zero. Then and for since a ¥= = = å i ijj p p b ¥= = = å j iji p p , i j Î i i i i i i i i i b b a + + = + = =+ + + + + + for (4.9) . i Î + It now follows from (2.5) that ) Î ( ( , ) ( ) ( ), , 0,1 ¥ ¥= = = å å ij i ji j c u v p f u g v u v (4.10) is a copula density where (1 )( ) ii i u uf u a -= and ( 1)(1 )( ) jj j j v vg v b + -= for Us-ing (4.8), one obtains, after some tedious but straightforward calculations, that , , , (0,1). i j u v Î Î + ( )( ) ( u v v uv uvc u v u vuv - - + + += Î- ) (4.11) which obviously is asymmetric. plot of ( , ) c u v The corresponding copula C can again be calculated explicitly, giving ( ) ( ) ( , ) 2 2 2 , , (0,1).1 = - - + + - + Î- xyC x y x xy xy x y y y x yxy (4.12) This copula has a coefficient of upper tail dependence 59 U l = (4.13) which is between the coefficients of upper tail dependence for the symmetric case with and cf. the final table in Example 2. 1 b = b = e Remark 1:
Negative binomial copulas (see Examples 2 and 5) can easily be simulated through the alter-native representation formula (2.5) involving mixed Beta distributions here. Poisson copulas can be simu-lated using the transformation applied to Gamma distributed random variables Z with a ran-dom shape parameter where is generated by the geometric distribution shown in (3.15), and scale parameter - - z z a - , a g + Remark 2:
For practical applications in quantitative risk management, it seems reasonable to fit the re-quired probabilities to empirical data via their empirical copula, for instance as was proposed in P
FEIFER , S TRASSBURGER AND P HILIPPS (2009). In the particular case of Bernstein copulas (see Example 1) such a procedure can be very easily implemented, even in higher dimensions (cf. C
OTTIN AND P FEIFER (2014)). , ij i j p + Î é ùë û As a practical exercise, we refer to Example 4.2 in C
OTTIN AND P FEIFER (2014) where the empirical cop-ula from an original data set was fitted to a general Bernstein copula. The following two graphs show the scatter plot from the empirical copula (big red dots) superimposed by 1000 simulated points of that Bern-stein copula (left) and of a negative binomial copula of type (3.11), with b = Bernstein copula fit negative binomial copula fit As can be seen nicely, the Bernstein copula represents the local asymmetry of the empirical copula better, but shows no tail dependence, as the negative binomial copula does. The fit to the negative binomial copula was, for the sake of simplicity, performed by a numerical match between the theoretical correlation for the negative binomial copula and the correlation of the empirical copula, which is 0.815. Note that the theoretical correlation for the negative binomial copula of type (3.11) can be explicitly calculated as ( ) r b ( ) ( 1)( ) 12 3 3 2( 1) (1, 2) 2 1( )( 1)( 2) r b b b b b bb b b ¥= æ ö+ ÷ç ÷= - = + Yç ÷ç ÷+ + + + +è ø å i ii i i + - - (4.14) where denotes the first derivative of the digamma function, or (1, ) z Y (1, ) ln ( ), 0. dz zdz Y = G > z b
1 2 3 4 5 6 7 ( ) r b g = Bernstein copula fit Poisson copula fit Note that although the empirical plot for the Poisson copula might suggest some tail dependence here this is actually not true in the light of (3.19). More sophisticated fitting procedures - including asymmetric cases – also in higher dimensions will be investigated elsewhere. It should be finally pointed out that copula constructions as presented in this paper will have a major im-pact in the construction of Internal Models under the new Solvency II insurance supervising regime in Europe (see e.g. H
UMMEL AND M ÄRKERT (2011) or S
ANDSTRØM (2011), Chapter 13).
5. Appendix. Proof of Lemma 1.
We will show by induction the equality of the following two expressions: b bb ¥= æ öæ+ - + +÷ ÷ç ç= ÷ç ç÷ ÷ç ç÷ ÷ç çè øè å ii i iK z zi i ö =÷ø
12 1 0 bb b bb -+ = æ öæ ö- +÷ ÷ç ç= ÷ç ç÷ ÷ç ç÷ ÷ç ç- è øè ø å ii k z zi iz ÷ < < z for (5.1) First notice that we have, for and b Î < < z
21 221 2 b b b bb b ¥ ¥- -= = æ öæ ö æ öæ ö+ - + + + - + +¶ ¶÷ ÷ ÷ ÷ç ç ç ç= ÷ ÷ = - ÷ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç¶ ¶è øè ø è øè ø å å i ii i i i i iK z K zi z i ii i i iz z ÷ z (5.2) from which we can conclude the relation ( , ) 1 ( , )( 2) ( 1, ) b bb b b æ ö¶ ¶ ÷ç= + + - ÷ç ÷çè ø¶ ¶ K z K zK zz z z or ( , ) ( , )( 1, ) ( 2) b bb b b ¶ ¶+¶ ¶+ = + K z K zz zK z . z (5.3) A similar, but more elaborate calculation shows that the latter equality remains valid if is re-placed by ( , ) b K z ( , ) : b k z ( , ) ( , )( 1, ) ( 2) b bb b b ¶ ¶+¶ ¶+ = + k z k zz zk z . z (5.4) In the first step of the induction, for we have b = ( 1)( 2) ( 1) 1 1(1, ) ''( ) (1, )2 2 2 (1 ) ¥ ¥ -= = + + -= = = = - å å i ii j i i j jK z z z h z k zz = (5.5) with
1( ) : 1 ¥= = = - å ii h z z z for < z For the second step, assume that relation (5.01) holds for some Then it follows by (5.3) and (5.4) that . b Î ( , ) ( , ) ( , ) ( , )( 1, ) ( 1,( 2) ( 2) b b b bb bb b b b ¶ ¶ ¶ ¶+ +¶ ¶ ¶ ¶+ = = = ++ + K z K z k z k zz zz z z zK z k ) z ö÷÷÷÷ø (5.6) which finishes the proof. Proof of Lemma 2.
First, note that for , b Î b b b b b b bb b - -= = æ öæ ö æ öæ ö æ- + - +÷ ÷ ÷ ÷ç ç ç ç ç÷ ÷ = ÷ ÷ =ç ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç ç- - -è øè ø è øè ø è å å i i i i i i (5.7) which is a special case of Vandermonde’s identity. This in turn implies b b b b bb b b b -= æ öæ ö æ ö- + G÷ ÷ ÷ç ç ç+ ÷ ÷ = + ÷ =ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç - Gè øè ø è ø å i i i . (5.8) Now, in the light of Lemma 1, we obtain (1 ) (1 )( , ) ( )(1 )1 1( ) lim ( 1) lim . ih h h hU h hi u vc u v du dv uv du dvuvi ih h b bb bb b bl b b +-- - - - = - -æ öæ ö -- +÷ ÷ç ç= = + ÷ ÷ç ç÷ ÷ç ç÷ ÷ç çè øè ø ò ò ò òå (5.9)
17 18 v To evaluate the last integral, we substitute and get 1 , 1 = - = - s u w (1 ) (1 )( , , ) : ( ) (1 ) (1 ) .(1 ) ( ) b b b bb b b + +- - - -= = -- + - ò ò ò ò h hi ih h u v s w - i I h i uv du dv s w ds dwuv s w sw (5.10)
In a further step, substituting we obtain , , = = s hx w hy ( , , ) (1 ) (1 ) ,( ) b b b b + = -+ - ò ò i i x y - I h i h hx hy dx dyx y hxy (5.11) giving
U hi i
I h i x y dx dyi i i ih x yx y dx dyx y b bb b bb b b b b b bbl b b bbb - - += =+ æ öæ ö æ öæ ö- + - +÷ ÷ ÷ ÷ç ç ç ç= + ÷ ÷ = + ÷ ÷ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç +è øè ø è øè øG= ⋅G + å å ò òò ò
It remains to evaluate the integral term in the expression above. Therefore, we consider the one-to-one map for ( ) ( : 0,1 : ( , ) , (1 ) - g T u v uv u v ) ì üæ öï ïï ï÷ç= Î < < < < ÷í ýç ÷çï ïè -ï ïî þ T u v u v u u . ø (Note that with ( , ) : ( , ), x y g u v = we have , xuv xx y = = ++ y ( ) ( , ) 0,1 .) Î x y for By the substitution formula for multiple integrals, we now obtain, putting ,( ) x yx y b b b + ( , ) : f x y = + and observing that for the determi-nant of the Jacobian, we have here det ( g u , ) , v v D = ( ) ( , ) ( , ) det ( , )( ) 1 1(1 ) min , (1 ) 2 (1 ) (5.131 g T TT x y dx dy f x y dx dy f g u v g u v du dvx y u u du dv u u du u u duu u b b b b b b b b b + - = = ⋅ D+ æ ö÷ç= - = ⋅ - = -÷ç ÷çè ø- ò ò òò òòòò ò ò ) which proves the first line in (3.13). For the first equality in the second line, note that, by symmetry and the substitution v u = u u du u u du u u duu u u du v v v duz z z z b b b b b bb b b bb bb b bb bb bb bbb - - - -- - - -- -- é ùG G ê ú- ⋅ - = ⋅ - - -ê úG G ê úë ûG G= ⋅ - - = ⋅ - -G GG - -= ⋅G ò ò òò ò
11 2 1 2 10 b b b b bbb bb b b b b - - - æ ö÷ç ÷çé ù ÷ç ÷çG - è øê ú = = =ê ú G -ê úë û
Which proves the first equality in the second line of (3.13). The asymptotic expansion follows by Stir-ling’s formula. Proof of Lemma 3.
Define ( )
1( ) : -¥ - -= ò nx cxn eg c e dxx for c Note that > ( )
1( ) : - -= nxn ef x x for is bounded by 1 for all We can therefore apply the dominated convergence theorem where appropriate. Now > x . Î n ( ) ( ) '
1( ) : 1 ( 1)1( 1) ( 1) ( 1) -¥ ¥ ¥- - - + - += ¥¥+ - + + - += = - æ ö÷ç= - ⋅ = - - ⋅ = ÷ -ç ÷ç ÷çè øæ öæ ö æ ö æ ö÷ç÷ ÷ ÷ç ç ç÷ç= ÷ - = ÷ - - = ÷ -÷ç ç çç÷ ÷ ÷÷ç ç ç÷ ÷ ÷ç ç çç + ÷è ø è ø è øçè ø åò ò òå åò nx nncx x cx j j c xjn nj j c x j j c xj j n e ng c x e dx e e dx e dxjxn n ne dx ej j jj c += + å n jj j c for Let further > c ( ( ) : ( 1) ln( ) ln( ) += æ ö÷ç ÷= - + -ç ÷ç ÷è ø å n jn j nh c j c cj ) > c for (5.16) Then ( ) ' + += =+ += = = æ ö æ ö æ ö÷ ÷ ÷ç ç ç= ÷ - + - = ÷ - - ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç +è ø è ø è øæ ö æ ö æ ö÷ ÷ ÷ç ç ç= ÷ - + ÷ - = ÷ -ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç+ +è ø è ø è ø å åå å å n nj jj jn n nj j jj j j n n ndh c j c cj jdc j c cn n nj j jj c c j c since This implies and hence or equivalently, for all for some constant But then also = æ ö÷ç ÷= - = -ç ÷ç ÷è ø å nn j nj ( ) ( ) = - n n n K g c h c > c . j ' ' n n = n g h Î n K ( ) ( ) = + n n g c h c K . ( ) ( ) ( ) ( )
100 100 0 nx ncx jn n nc c c c jnx ncx jc cj e nK g c h c e dx j c cjxe n je dx dxjx c -¥ - +¥ ¥ ¥ ¥ =-¥ - +¥ ¥= æ ö÷- æ öç ÷ç ÷ç÷= - = - ÷ - + -ç ç÷ ÷ç ç÷ ÷çè øç ÷÷çè ø- æ ö æ öæ ö÷ ÷ç ç ÷ç= - ÷ - + =÷÷ç ç ç÷ ÷÷çç ç ÷÷ç è øè øè ø åòåò nj ¥ = - = åò for all Hence for all which proves the Lemma. . Î n = n g h n , Î n
19 20 . t Proof of Lemma 4.
Substitute Then (3.26) is equivalent to = - s ( ) ( )
10 01 0 (1 ) (1 ) (1 )1 1lim lim 1,( ) (1 ) - ⋅ - - ⋅ -= =- ò ò ss ss
L u s L w sdu dws L u s L w (5.19) with the substitution This means that we have to show that = - u . w ( ) ln1lim 1.ln( ) + - = ò ss w s ws dws w (5.20) Define ln( ) ln( (1 ))( , ) : , ( , ) : .ln( ) ln( ) + += = w s w s sF w s G w sw w - (5.21) Then ln( )( , ) ( , )ln( ) + -£ £ w s wsF w s G w sw for (5.22) 0 < £ w s (note that l for 0 Now for n( ) 0 < w < < w < < s
20 0 0 ln 1 1 ln(10 G( , ) ( , ) ln(1 )ln( ) ln( ) ln( ) æ ö÷ç ÷- -ç ÷ç ÷ç + -è ø£ - £ £ - - £- - ò ò ò s s s sw s sw s F w s dw dw s dww w ) ss (5.23) with the limit
10 lim G( , ) ( , ) lim 0.ln( ) -£ - £ ò ss s sw s F w s dws s ln(1 ) = (5.24) Hence it suffices to prove !0 00 0 - += - ò ò s ss s w sF w s dw dws s w ) = (5.25) By the substitution ln( ) = - x w we obtain the equivalent expression ( ) !0 ln( ) ln1lim 1. -¥ - - - + = ò x xs s e s e dxs x (5.26) Note that ( ) ( ) ( )( ) ( ) ln( ) ln( ) ln( )ln( ) ln( ) ln( ) ln ln (1 ) ln 1ln 1 ln 1 . (5.27) x x x xx x xs s sx xx x xs s s e s e se x see dx e dx e dxx x xse see dx e dx s e dxx x - - - -¥ ¥ ¥- - -- - -- -¥ ¥ ¥- - -- - - - + - + - += =+ += - = - ò ò òò ò ò Hence it suffices to prove ( ) !0 ln( ) ln 11lim 0. -¥ - - + = ò x xs s se e dxs x (5.28) With the substitution this is equivalent to , - = T s e ( ) ! ln 1lim 0. -¥ -¥ + = ò x TTT T ee ex x dx (5.29) Substituting finally , = - y x T this means ( ) ( ) !( )0 0 ln 1 ln 1lim lim 0. ¥ ¥- + -¥ ¥ + +=+ + ò ò y yT y T yT T e ee e dy e dy T y T = y (5.30) But this is now evident due to ( ) ln 1 1 10 lim lim lim 0 ¥ ¥ ¥- - -¥ ¥ ¥ + æ ö+ + ÷ç£ £ = ç ÷ç ÷ç+ + +è ø ò ò ò y y yT T T e y ye dy e dy e dyy T y T y T ÷ = y (5.31) by Lebesgue’s dominated convergence theorem. (For an integrable majorant is given by 1, ³ T .) - y e This proves Lemma 4. Proof of Lemma 5.
First notice that by the relation 1 1( ) ln(1 ) ln 11 1 1 æ ö÷ç= - - = £ - =÷ç ÷çè ø- - - uL u u u u u
1, for we obtain 0 < < u a - = ³ - = - =⋅ + ò ò ò i i ii u udu u du u u dui L u i u i i i . Î i for all (5.32) Now a a ¥ ¥= = ì üï ïï ï= = í ýï ï- - ï ïî þ å åò ò ò ò i ii ii it t t t uv v uK t du dv du dvt i L u L v t i L v L u i (5.33) by Lebesgue’s dominated convergence theorem since for fixed by (5.32), (0,1), Î v
212 3 31 1 1 ( ) 2 ( ) 3 3( 1) ( 1)( )( ) ( ) ( ) ( ) (1 ) (1 ) a ¥ ¥ ¥ -= = = + -£ + ⋅ £ + = £- - å å å i i i ii i ii uv u v uv uvi i uvi L u L v L u L v uv uv , (5.34) the r.h.s being integrable w.r.t. u with value -=- - ò vduuv v ) < < t Now for 0 we have ( ) ( ) ( ) + = ³ ò ò ò t i i i i u v t vdu t dv t dvL u L vt L v i (5.35) and hence ( ) ( ) a + = - £ - = - ò ò ò ò ti i i ii it u u u vdu du du t dv t iL u L u L u L v + i (5.36) for all Thus we get, from (5.33), . Î i ( ) i i i ii iit t ti ii it t t v u vK t du dv t dvt i L v L u t iL vv vt L v tL vt t L vtt dv dv dvt L v i i t L v t L v a ¥ ¥ += =¥ ¥= = ì üï ïï ï= £ -í ýï ï- -ï ïî þì üï ï -ï ï= - = = -í ýï ï- - -ï ïî þ å åò ò òå åò ò ò which proves relation (3.27). From Lemma 4 we thus obtain the final result l £ = £ - = U t
K t and hence (5.38) 0, l = U which proves Lemma 5. Acknowledgements.
We would like thank the referees for several helpful comments which improved the presentation of the paper substantially, and especially for pointing out the relationship to Catalan numbers in Lemma 2 and the corresponding references given in
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