The lineage process in Galton--Watson trees and globally centered discrete snakes

0.52 MB·English

by  Jean-François Marckert

TheAnnalsofAppliedProbability
2008,Vol.18,No.1,209–244
DOI:10.1214/07-AAP450
(cid:13)c InstituteofMathematicalStatistics,2008
THE LINEAGE PROCESS IN GALTON–WATSON TREES AND
GLOBALLY CENTERED DISCRETE SNAKES
8
0
By Jean-Franc¸ois Marckert
0
2
Universit´e Bordeaux

n
a We consider branching random walks built on Galton–Watson
J trees with offspring distribution having a bounded support, condi-

2 tionedtohavennodes,andtheirrescaledconvergencestotheBrow-
2 niansnake.Weexhibitanotionof“globally centereddiscretesnake”
that extends the usual settings in which the displacements are sup-

] posedcentered.Weshowthatundersomeadditionalmomentcondi-
R
tions, when n goes to +∞, “globally centered discrete snakes” con-
P
verge to the Brownian snake. The proof relies on a precise study of
.
h thelineage ofthenodesinaGalton–Watson treeconditionedbythe
t size,andtheirlinkswithamultinomialprocess[thelineageofanode
a
u is the vector indexed by (k,j) giving the number of ancestors of
m
u having k children and for which u is a descendant of the jth one].
[
Some consequences concerning Galton–Watson trees conditioned by

1 thesize are also derived.
v
0
1. Introduction.
3
3
3 1.1. A model of centered discrete snake. We first begin with the formal
.
1 description of the notion of trees and branching random walks.
0 Let U= ∅ N⋆n be the set of finite words on the alphabet N⋆=
8 1,2,... .F{or}u∪=un≥..1.u andv=v ...v U,weletuv=u ...u v ...v
0 { } S1 n 1 m∈ 1 n 1 m
: be the concatenation of the words u and v (by convention, ∅u=u∅=u).
v
Following Neveu [22], we call planar tree T a subset of U containing the
i
X root ∅, and such that if ui T for some u U and i N⋆, then u T and
r for all j J1,iK, uj T. The∈elements of a t∈ree are cal∈led nodes or ∈vertices.
a
∈ ∈
For i=j, the nodes ui and uj are called brothers and u their father. We let
6
c (T)=sup i:ui T be the number of children of u [here c (T) will be
u u
{ ∈ }
always finite]. A node without any child is called a leaf, and we denote by
Received June 2006; revised December 2006.
AMS 2000 subject classifications. 60J80, 60F17, 60J65.
Key words and phrases. Galton–Watson trees, discrete snake, Brownian snake, limit
theorem.
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Applied Probability,
2008,Vol. 18, No. 1, 209–244. This reprint differs from the original in pagination
and typographic detail.
1
2 J.-F. MARCKERT
∂T the set of leaves of T. If v=∅, we say that uv is a descendant of u and
6
u is an ancestor of uv. An edge is a pair u,v where u is the father of v. A
{ }
path Ju,vK between the nodes u and v in a tree T is the (minimal) sequence
of nodes u:=u ,...,u :=v such that, for any i J0,j 1K, u ,u is an
0 j i i+1
∈ − { }
edge. Set also Ku,vJ=Ju,vK u,v and similar notation for Ju,vJ and for
\{ }
Ku,vK. The distance d , or simply d, is the usual graph distance. The depth
T
of u is u =d(∅,u). The cardinality of T is denoted by T , and we let
| | | | T
(resp. ) be the set of planar trees (resp. with n edges, i.e., n+1 vertices).
n
T
A branching walk is a pair (T,ℓ) where T is a tree—called the underlying
tree—and ℓ, the label function, is an application from T taking its values in
R. In other words, it is a tree in which every vertex owns a real label. We let
be the set of branching walks, and be the branching walks associated
n
B B
with trees from .
n
T
We introduce now some randomness and construct a probability distri-
bution on and on .
n
B B
The set of underlying trees is endowed with the distribution of the fam-
ily tree of a Galton–Watson (GW) process with offspring distribution µ=
(µ ) starting from one individual. In this model, all the nodes have a
k k 0
≥
random number of children, according to the distribution µ, independently
from the other individuals. We denote by T a random tree under this distri-
bution (see, e.g., [1, 10] and most of the cited papers for more information
on GW processes and trees).
The distribution of the labels is defined as follows. Consider (ν )
k k 1,2,...
a family of distributions, where ν is a distribution on Rk. The lab∈e{ls are}
k
defined conditionally on the underlying tree T: Set ℓ(∅)=0, and for any
u T ∂T, consider
∈ \
X :=(ℓ(u1) ℓ(u),...,ℓ(uc (T)) ℓ(u)),
u u
− −
the evolution-vector of the labels between u and its children. Condition-
ally on T, we assume that the r.v. X are independent, and that X has
u u
distribution νcu(T). This determines a distribution on B. For example, if
ν is the uniform distribution on 1,+1 k for any k >0, then the r.v.
k
{− }
ℓ(u1) ℓ(u),...,ℓ(uc (T)) ℓ(u) are independent with common distribu-
u
− −
tion 1(δ +δ ) (δ stands for the Dirac mass at x). In the case where ν
2 +1 1 x k
−
is the uniform distribution on (1,...,k),( 1,..., k) , the r.v. ℓ(ui) ℓ(u)
{ − − } −
and ℓ(uj) ℓ(u) are notindependentand donothave thesamedistribution.
−
Notice that a sequence of i.i.d. µ-distributed random variables indexed
by U allows to build the Galton–Watson trees, and a sequence of random
variables indexed by U N allows to build all the labels (by attaching to
×
the elements of U a list of random variables with distribution ν ,ν ,...). We
1 2
assume that we work on an underlying probability space (Ω, ,P) on which
A
are defined all the random variables and processes used in this paper.
GLOBALLY CENTERED DISCRETESNAKES 3
Fig. 1. A tree on which is indicated the depth-first traversal, its height and contour
processes.
Wedefinenowtwosetsofassumptions(H )and(H )thatwillbeassumed
1 2
to be satisfied in most of our results. (H ) is the following set of conditions:
1
µ is nondegenerate, critical and has a bounded support,
(H ):= µ +µ =1, kµ =1, there exists K>0 s.t. µ =1 .
1 0 1 k k
6 !
k 0 k K
X≥ X≤
Under (H ) the variance σ2 of µ is finiteand nonzero. Theboundedsupport
1 µ
condition is quite a strong restriction, butconsidering nonboundeddistribu-
tion leads to nontrivial complications, and we were unable to extend to that
case themostimportantresults.Thecondition on themean can beseen as a
normalization, sinceanydistribution µ˜ related to µ byµ˜ =µ ak/( akµ )
k k j k
for some a>0 induces the same distribution as µ on GW-trees conditioned
P
by the size.
Let Y(k)=(Y ,...,Y ) be ν -distributed. We denote by ν , m and
k,1 k,k k k,j k,j
σ2 the distribution, the mean and the variance of Y . We call global mean
k,j k,j
and global variance of the branching random walk,
k k
m= µ m and β2= µ E(Y2 ).
k k,j k k,j
k 1j=1 k 1j=1
X≥ X X≥ X
Let (H ) denote the conditions that the global mean is null, the global
2
variance finite and positive, and for a p>4, the centered pth moment of the
Y ’s is finite:
k,j
m=0 and β (0,+ ), there exist p>4 s.t. for
(H2):= any (k,j),1 ∈j k ∞K,E(Y m p)<+ . .
k,j k,j
(cid:18) ≤ ≤ ≤ | − | ∞ (cid:19)
Encoding of branching random walks. We denote by 4 the lexicograph-
ical order (LO) on the planar trees (and u v if u4v and u=v), and let
≺ 6
u(k) be the kth vertex in the LO [u(0)=∅].
We study the asymptotic behavior of branching random walks via their
encoding by depth-first-traversal. The depth-first traversal of a tree T
n
∈T
is a function
F : 0,...,2n vertices of T ,
T
{ }→{ }
4 J.-F. MARCKERT
Fig. 2. A branching random walk from B .On the first column, the contour process and
9
the contour label process, on the second column, the height process and the height label
process.
which we regard as a walk around T, as follows: F (0) = ∅, and given
T
F (i)=z, choose if possible and according to the LO, the smallest child
T
w of z which has not already been visited, and set F (i+1)=w. If not
T
possible, let F (i+1) be the father of z.
T
We now encode the branching random walk with the help of a pair of
processes. For any k J0, T 1K, let HT = u(k) and RT =ℓ(u(k)). The
∈ | |− k | | k
height process (HT,s [0, T 1]) and label process (RT,s [0, T 1])
s ∈ | |− s ∈ | |−
are obtained from the sequences (HT) and (RT) by linear interpolation.
k k
Alternatively, one may encode the branching random walk with a pair of
processesassociated withthedepth-firsttraversal:forany k J0,2(T 1)K,
∈ | |−
let HT = F (k) and RT =ℓ(F (k)). The processes (HT,s [0,2(T 1])])
k | T | k T s ∈ | |−
and(RT,s [0,2(T 1)]),obtainedbyinterpolation,arecalledrespectively
thecbonstour∈proces|sa|−ndbthecontourlabelprocess;thepbair(HT,RT)iscalled
b
the head of the discrete snake. See some illustrations on Figures 1 and 2.
Let d:=gcd k,k 1,µ >0 . The support of the distribubtionbof T —we
k
{ ≥ } | |
write supp(T )—is included in 1+dN [and P(T =1+kd)>0 for every
| | | |
k large enough]. For n+1 supp(T ), the distribution P under the condi-
∈ | |
tioning by T =n+1 is denoted by P , in other words
n
| |
P =P( T =n+1).
n
·|| |
Even if not recalled, each statement concerning weak convergence under P
n
is assumed to be along the subsequence (n ) for which P is well defined.
k k nk
Intheproofswewilltreat onlythecase d=1,thegeneral case beingtreated
with slight modifications.
GLOBALLY CENTERED DISCRETESNAKES 5
T T T T
We define h , h , r and r to be the processes H , H ,R and R
n n n n
under P , interpolated as follows:
n
b T b T b b
H H
h (s)= ns, h (s)= 2ns,
n n1/2 n n1/2
b
RT b RT
r (s)= ns, r (s)= 2ns for any s [0,1].
n n1/4 n n1/4 ∈
b
Theorem 1. If (H ) andb(H ) are satisfied, then
1 2
(d)
(h ,h ,r ,r ) (h,h,βr,βr)
n n n n
→n
in C([0,1],R4) endowed withbthe tobpology of uniform convergence, where h=
2e/σ and e is the normalized Brownian excursion, and where, conditionally
µ
on h, r is a centered Gaussian process with covariance function
cov(r(s),r(t))=hˇ(s,t):= min h(u) for any s,t [0,1].
u [s t,s t] ∈
∈ ∧ ∨
Notice that the same processes h and r appear twice in the limit process.
The convergence of processes associated with the contour processes (with
a ) to the same limit as the one associated with the height processes
is well understood now, and “almost” generic (Duquesne and Le Gall [9],
Sebction 2.5, and [21]), we then concentrate only on the height process. The
process (r,h) (or with a different scaling) will be called the head of the
Brownian snake with lifetime process the normalized Brownian excursion
(BSBE). We refer to the works of Le Gall (e.g., [16] and with Duquesne
[10]) for information on the Brownian snake and to the papers cited below
for discrete approaches to this object.
In the present work we deal only with the head of the snakes; this is,
in principle, different than snakes even if, thanks to the homeomorphism
theorem [20] evoked below, Theorem 1 has some direct interpretation in
terms of snakes. We refer to [13, 20] for the notion of discrete snake which
is the discrete analogue of BSBE: the discrete snake associated with the
branchingrandomwalk(T,ℓ)isthepair(HT,Φ)whereΦ=(Φ )
k k J0,2(T 1)K
and Φ is the sequence of labels on the branch J∅,F (k)K. ∈ | |−
k T
b
(d)
Related works. The convergence h h is due to Aldous [1, 2] (see also
n
→n
Marckert and Mokkadem [21] for a revisited proof, Pitman [25], Chapters
b
5 and 6 and Duquesne [9] and Duquesne and Le Gall [10], Section 2.5,
for generalization to GW trees with offspring distribution having infinite
variance).
The two first results concerning the convergence of discrete snakes to the
BSBE appeared in two independent works:
6 J.-F. MARCKERT
Chassaing and Schaeffer [7] deal with discrete snakes built on underlying
•
trees chosen uniformlyin [this correspondstothecase µ Geom(1/2)]
n
T ∼
and wherethe displacements arei.i.d., andfor any k,j, ν is theuniform
k,j
distribution in 1,0,+1 . They show the convergence of the head of
{− }
the discrete snake for the Skohorod topology, and the convergence of the
moments of the maximum of r are also given. This study was motivated
n
by the deep relation between this model of discrete snake and random
rooted quadrangulations, underlined by the authors.
Marckert and Mokkadem [20] studied also the case µ Geom(1/2), but
• ∼
with more general centered displacements that have moments of order
6+ε (the distribution ν does not depend on k,j, but ν is not assumed
k,j k
to be ν ν ). The convergence of the head of the snake holds
k,1 k,k
×···×
in (C[0,1],R2) and the convergence of the snake itself is given thanks to
a “homeomorphism theorem” which implies that the convergence of the
snake and of its tour (in space of continuous functions) are equivalent.
Here it implies that, under the hypothesis of Theorem 1, the discrete
snake associated with our model of labeled trees converges weakly to the
BSBE.
Then some generalizations appeared few months later:
Gittenberger [11] provides a generalization of a lemma from [20] allowing
•
him to consider snakes with underlying GW trees conditioned by the size
(condition equivalent to H ). The displacements must be centered and
1
have moments of order 8+ε.
Janson and Marckert [13] show that, in the i.i.d. case [ν do not depend
k,j
•
on (k,j)], moments of order 4+ε are necessary and sufficient to get the
convergence to the BSBE. If no such moment exists, the convergence to
a “hairy snake” is proved under the Hausdorff topology.
In Marckert and Miermont [19], thecase of ν dependingon k,j is inves-
k,j
•
tigated (also theunderlyingGWtreesareallowed tohavetwo types).The
hypothesis are for each k,j, m =0, condition (H ) is satisfied, and then
k,j 2
µ σ2 <+ .Amotivation was to generalize the works of Chassaing
k,j k k,j ∞
and Schaeffer [7] concerning quadrangulations to bipartite maps.
P
Another important point is the convergence of the occupation measure of
the head of the discrete snake to the one of the BSBE, the random measure
named ISE (the integrated superBrownian excursion introduced by Aldous
[3]; see also Le Gall [16] and [13, 20]). Using the convergence of the discrete
snake to the BSBE, Bousquet–M´elou [4] and Bousquet–M´elou and Janson
[5] deduce new results on the ISE and on the BSBE; for example, some
properties on the support of the ISE, and of random density of the ISE are
derived. We refer also to Le Gall [15] for the convergence of the discrete
snake conditioned to stay positive.
GLOBALLY CENTERED DISCRETESNAKES 7
The novelty in the present paper is that the condition m =0, k,j is
k,j
{ ∀ }
replaced by m= k µ m =0. This allows to consider some natu-
k 1 j=1 k k,j
ral models where, fo≥r example, the displacements are not random knowing
P P
the underlying tree (see Section 1.3). The proof of Theorem 1 relies in part
on some results from [19], and on a new approach necessary to control the
contribution of the mean of the displacements; the main point for this is
the comparison of the lineage of each node, with some multinomial r.v. This
is the aim of Theorem 2, that we think interesting in itself, since it reveals
a thin global behavior of GW trees conditioned by the size. Unfortunately,
the price of this generalization is to consider only offspringdistribution with
bounded support.The reason comes from the proof of Theorem 2. We guess
that some generalization for all families of GW trees (with finite variance)
may be found, but for this a control of an infinite sequence of processes
arising in Theorem 2 should be provided for what we were unable to do.
1.2. On the lineage of nodes. Assume that (H ) and (H ) hold, and
1 2
let K be a bound on the support of the offspring distribution. We work
again conditionally on T. For any node u= i ...i T, let u =i ...i
1 h j 1 j
∈
and J∅,uK= ∅=u ,u ,...,u be the ancestral line of u back to the root.
0 1 h
{ }
Conditionally on T, ℓ(u) owns the following representations:
u
| |
(1) ℓ(u)= ℓ(u ) ℓ(u ),
m m 1
− −
m=1
X
where ℓ(u ) ℓ(u ) is ν -distributed when c (T)=k and i =j,
m − m−1 k,j um−1 m
and where the r.v. (ℓ(u ) ℓ(u ))’s are independent (conditionally on
m m 1
T); the variables ℓ(u ) ℓ(−u )−will be often called displacements.
m m 1
− −
Consider the array
I = (k,j),1 j k K .
K
{ ≤ ≤ ≤ }
Let T and u be a node of T. For any (k,j) I , let A (T) be the
K u,k,j
∈T ∈
number of strict ancestors v of u (the nodes v J∅,uJ)such that c (T)=k,
v
∈
and such that u is adescendant of vj, the jth child of v [we write f (u)=j].
v
We say that v is an ancestor of type k,j of u, and we call the vector A =
u
(A ) the lineage of u (or the content of J∅,uK). See Figure 3.
Bu,iyi(∈1I)K, conditionally on T, the label ℓ(u) owns the following representa-
tions:
Au,k,j(T)
(d) (l)
ℓ(u)= Y ,
k,j
(k,Xj)∈IK Xl=1
(l) (l)
where the r.v. Y are independent, and where for any l, Y is ν dis-
k,j k,j k,j
tributed. In order to make more apparent the contribution of the means
8 J.-F. MARCKERT
m ’s, and using that m=0, write
k,j
Au,k,j(T)
(d) (l)
(2) ℓ(u)= (Y m )+ (A (T) µ u)m .
k,j − k,j u,k,j − k| | k,j
(k,Xj)∈IK Xl=1 (k,Xj)∈IK
Assume that T is P distributed, and that u=u(ns) for some s (0,1).
n
∈
Conditionally on u , we willsee that both partsof theright-hand side of (2)
| |
dividedbyn1/4 convergeindistribution,andthelimitr.v.areasymptotically
independent:in the firstpart, the fluctuations of A around µ u are not
u,k,j k
| |
important when they are crucial in the second sum.
Wenowconcentrateonther.v.(A )sunderP .Foranyl J0,nK,(k,j)
u ′ n
∈ ∈
I , set
K
(n)
g (l):=A µ u(l).
(k,j) u(l),k,j − k| |
(n)
For every (k,j) I , the process l g (l) encodes the evolution of the
∈ K 7→ (k,j)
number of ancestors of type k,j of u(l), when l varies. Consider G(n) =
(G(n)(s)) the process taking its values in RIK defined by, for any s,
s [0,1]
∈
G(n)(s)=(G(n)(s)) ,wheres G(n)(s)istherealcontinuousprocess
k,j (k,j)∈IK 7→ k,j
(n)
that interpolates g as follows:
k,j
g(n)( ns )+ ns (g(n)( ns+1 ) g(n)( ns ))
(3) G(n)(s):= k,j ⌊ ⌋ { } k,j ⌊ ⌋ − k,j ⌊ ⌋ , s [0,1],
k,j n1/4 ∈
where x stands for the rational part of x. The random process G(n) en-
{ }
codes the lineage of all the nodes of T; its limiting behavior is described by
the following theorem.
Theorem 2. Under (H ) and (H ) the following convergence in dis-
1 2
tribution holds in C([0,1])#IK C[0,1] endowed with the topology of the
×
uniform convergence
(G(n),h )(d)(G,h),
n
→n
Fig. 3. On this tree A =1,A =1,A =1,A =1, the others A are 0.
u,1,1 u,2,2 u,4,2 u,5,3 u,i
GLOBALLY CENTERED DISCRETESNAKES 9
where h is defined as in Theorem 1, and where conditionally on h, G=
(G (s)) is a real centered Gaussian field with the following
covka,rjianc(ek,jfu)∈nIcKt,iso∈n[0:,1f]or any (k,j) and (k ,j ) in I , s and s in [0,1],
′ ′ K ′
(4) cov(G (s),G (s))=( µ µ +µ )hˇ(s,s).
k,j k′,j′ ′ k k′ k (k,j)=(k′,j′) ′
− 1
1.3. Comments, examples and applications. (1) Theorem 2 may be con-
sideredasthestrongestresultof thispaper.Itgives verypreciseinformation
on the asymptotic behavior of the process G(n) that encodes the lineage of
all the nodes. This gives a “global asymptotic” property reminiscent of the
properties of the distinguished branch in “a size biased GW tree” (see [17],
Chapter 11).
(2) For any fixed (k,j) I , knowing h, G is a Gaussian process with
K k,j
∈
covariance function
cov(G (s),G (s))=( µ2+µ )hˇ(s,s).
k,j k,j ′ − k k ′
Inotherwords,theprocess(G ,h)hasthesamedistributionas(√ µ2+µ r,h),
k,j − k k
and then up to some multiplicative constants, (G ,h) is the head of a
k,j
BSBE.AsasimpleconsequenceofTheorem2,wehavethat(G ,h)
is a sequence of heads of BSBE, and that for any (k,j) I , k,j (k,j)∈IK
K
∈
(n) (d)
(5) (G ,h ) (G ,h).
k,j n →n k,j
The dependence between the different processes G is ruled out by (4).
k,j
For any families of real numbers (λ ) , we have
k,j (k,j)∈IK
(n) (d)
(6) λ G ,h λ G ,h .
k,j k,j n!→n k,j k,j !
(k,Xj)∈IK Xk,j
(3)Considerthecaseµ= 1(δ +δ ),ν =δ ,ofbinarytreesinwhich
2 0 2 2 (+1, 1)
the displacements are not random: ℓ(u1) ℓ(u−)=+1 and ℓ(u2) ℓ(u)=
− −
1. We have m=0 and β2 = 1(1+1)=1 and Theorems 1 and 2 apply.
− 2
Hence, the clear positive bias for R (t) for small values of t disappears at
n
the limit. Note also that this normalizing factor is exactly the same as if
ν = 1(δ +δ ) [case where (ℓ(u1) ℓ(u),ℓ(u2) ℓ(u)) is equally
li2kely2(+(+1,1,−11)) or ((−11,+,1+)1)] and as if ν =(1−(δ +δ ))2−[case where the
− − 2 2 +1 −1
ℓ(u1) ℓ(u) and ℓ(u2) ℓ(u) are i.i.d., uniform on 1,1 ]. The question of
− − {− }
the convergence of the discrete snake in the case ν =δ appears first
2 (+1, 1)
in Marckert [18] in relation with some properties of the rot−ation correspon-
dence, and the difference between left and right depth in binary trees. The
convergence of (r ) is not given in [18], but the convergence of the occupa-
n
tion measure of r , “the discrete ISE,” to ISE is established. We refer also
n
to Janson [12] for recent developments concerning the same question.
10 J.-F. MARCKERT
Further, notice that in this model, the label ℓ(u) of a vertex u is ℓ(u)=
A A , that is, the number of left steps minus the number of right
u,2,1 u,2,2
−
steps necessary to climb from the root to u in the binary tree. The conver-
gence of (r ) can be seen directly via the one of (G(n)):
n
(n) (n) (d)
(7) (G ,G ,h ) (G ,G ,h),
2,1 2,2 n →n 2,1 2,2
(n) (n)(d)
andthenr =G G G G whichis,conditionally tohandac-
n 2,1− 2,2→n 2,1− 2,2
cording to (4), a centered Gaussian process with covariance function hˇ(s,t).
Here, the convergence of (r ) appears to be a consequence of the conver-
n
genceofG andG ,encodingtherightdepthandtheleftdepthinbinary
2,1 2,2
trees.
We would like to stress on the following point: discrete snake are usu-
ally constructed with “two levels of randomness”: the underlying trees are
random and so are the displacements given the underlying tree, and then
BSBE appears to be a natural limit of these objects. Here, we provide some
objects with only “one level of randomness”that converges to the Brownian
snake. The BSBE appears as a kind of internal complexity measure in trees
measuring the difference between the number of ancestors of type k,j and
some expected quantities.
2. Proofs. The proofs rely on a precise study of the lineage of the nodes
under P and, in particular, on the comparison of A with a multinomial
n u
randomvariable.For thisreason,wefirstgive someelements onmultinomial
distributions and on their asymptotic behaviors. We then proceed to the
proof of Theorem 2, showing first the convergence of the uni-dimensional
distribution,thentheconvergenceofthefinite-dimensionaldistribution.The
proof of Theorem 1 is given afterward. We think that some points of view,
especially in the description of thedistribution of thelineages in trees under
P , should provide some new approaches to study the trees under P .
n n
2.1. Prerequisite on multinomial distributions. The contents of this sec-
tionarequiteclassical. Considerp=(p ) thedistributiononI ,defined
i i∈IK K
by
p :=µ for any (k,j) I .
k,j k K
∈
tFhoartany h≥c1=, lhet. WNIe[hs]aybethtahte set(ho)fiselaemmeunlttsinco=mi(acli)ri.∈vI.KwoitfhNp#aIrKa,mseutcehr
h and pi∈,IiKf, fior any m=(m ) M,
P i i∈IK
Q ( m ):=P( (h)=m)= h pmi (m),
h { } M (cid:18)(mi)i∈IK(cid:19)i∈YIK i 1NI[h]