Harmonic analysis of finite lamplighter random walks

0.31 MB·English

by  Fabio Scarabotti

7 Harmonic analysis of finite lamplighter random walks.
0
0
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Fabio Scarabotti, Filippo Tolli
n
a
J
February 2, 2008

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2

] Abstract
R
P
Recently, severalpapershavebeendevotedtotheanalysisoflamplighterrandom
.
h walks, in particular whenthe underlyinggraph is the infinite path Z. In the present
t
a paper,we develop a spectralanalysis for lamplighter random walks on finitegraphs.
m In the general case, we use the C -symmetry to reduce the spectral computations
2
[ to a series of eigenvalue problems on the underlying graph. In the case the graph

1 has a transitive isometry group G, we also describe the spectral analysis in terms
v of the representation theory of the wreath product C G. We apply our theory
2
3 ≀
to the lamplighter random walks on the complete graph and on the discrete circle.
0
6 These examples were already studied by Haggstrom and Jonasson by probabilistic
1 methods. 1
0
7
0
/ 1 Introduction.
h
t
a
m Let X be a simple, locally finite, connected graph. Put in each vertex a lamp, which may
: be on or off. A lamplighter performs the simple random walk on X and when he moves
v
from a vertex x to a vertex y he changes randomly the state of the lamps in x and y,
i
X
that is both the lamps may be turned on or off with equal probability. In other words, we
r
a may construct a new graph whose vertex set is (X) = (θ,x) θ : X 0,1 ,x X
L { | → { } ∈ } ≡
CX X and two vertices (θ,x) and (σ,y) are connected if x and y are connected in X and
2 ×
θ σ in X x,y . Then θ(x) is the state of the lamp in x, θ(x) = 0 if it is off, θ(x) = 1
≡ \{ }
if it is on, andthe lamplighter random walk onX is just thesimple random walk on (X).
L
Thesekindsofprocesses(thathavemanyvariants)havebeenstudiedbymanyauthors;
in particular, we mention [1, 8, 11] devoted to the spectral analysis of the lamplighter ran-
domwalk onthe infinite path; actually, the paper of L.Bartholdiand W.Woess treats the
more general case of the Distel-Leader product of two infinite homogeneous trees (see also
[31]). We also refer to [30], that contains a more general construction where the lamps are
replaced by the vertices of another graph. On the other hands, the finite case has been
treated by O. Haggstrom and J. Jonasson [12], who analyzed by probabilistic techniques
1AMS 2002 Math. Subj. Class.: Primary: 43A85; secondary: 05C05, 20C15, 20E22,60G50.
Keywords: Lamplighterrandomwalks,Markovspectrum,wreathproduct,permutationrepresentation.
1
thelamplighterprocessesonthecompletegraphandonthediscretecircle, andbyY.Peres
andD.Revelle [19], who used analytictechniques forthelamplighter process onfinite tori.
In the present paper, we develop a suitable spectral analysis for lamplighter random
walks on finite graphs. We start from the following simple observation: the lamplighter
random walk is CX-invariant. This group acts on the lamps coordinatewise: if θ CX
2 ∈ 2
and (ω,x) (X) then (θ+ω)(x) = θ(x)+ω(x) mod 2, and the action on (X) is simply
∈ L L
θ (ω,x) = (θ+ω,x). (1)
·
Clearly, this is not a transitive action. In [23], starting from the results in [3], we de-
veloped a suitable harmonic analysis for a finite Markov chain with a nontransitive group
of symmetries. In the present setting, our methods simplify noticeably: when we restrict
the Markov operator to the isotypic components of the permutation representation of CX
2
on the lamplighter graph (X), we get a series of eigenvalue problems on the graph X,
L
that lead to a complete spectral analysis of the lamplighter chain.
The plan of the paper is the following. In Section 2 we establish a series of notation
used in the paper. In Section 3 we analyze the lamplighter process described above. In
Section 4 we analyze the lamplighter process on the complete graph on n vertices. In
particular, using the standard techniques developed by P. Diaconis [7], we show that the
chain has a cut-off after k = 1nlogn steps. This result has already been obtained in [12]
2
by means of purely probabilistic techniques. In Section 5 we analyze one of the possible
variations of the lamplighter construction: we put the lamps on the edges of the graph.
When we move form x to y, the lamp in the edge x,y is randomized. In Section 6 we
{ }
compute the spectrum of the lamplighter random walk on the discrete circle, with the
lamps on the edges. The result in this section may be considered as a finite analogous
of the computations on the infinite path; moreover, both the finite and infinite cases are
random walks on groups, namely the wreath products C C and C Z. The spec-
2 n 2
≀ ≀
tral computations in this section have a clear connection with the eigenvalue problems
on finite trees treated in [13, 22, 24]. We have written Sections 3-6 with a minimum
of group formalism, in order to make this part of the paper accessible with only a dis-
crete/probabilistic background. In the remaining part of the paper, we make a systematic
use of group representation theory. In Section 7, we prove a general decomposition theo-
rem forthe permutation representation of a groupG ona space of theformCZ X, where
2 ×
both X and Z are G-homogeneous spaces. This is more than is needed for the lamplighter
random walks; in fact, in Section 9 we show that a decomposition derived by Schoolfield
(for the Bernoulli-Laplace diffusion model with sign) may be easily deduced from our gen-
eral result. In Section 8, we revisit the spectral decomposition of the lamplighter random
walk on the discrete circle, describing the spectral decomposition in terms of irreducible
representations of the group C C . In a similar way, the lamplighter on the complete
2 n
≀
graph is revisited in Section 10, now using the action of the hyperoctahedral group C S .
2 n
≀
In our join paper with T. Ceccherini-Silberstein [4], we analyzed several constructions
2
that lead to multiplicity free permutation representations of wreath products. On the
contrary, the harmonic analysis of the lamplighter random walk with a transitive group
action leads to an example of a space with multiplicities. Moreover, in our examples the
operators are not in the center of the commutant of the permutation representation, and
therefore their diagonalization with irreducible eigenspaces requires a suitable explicit
orthogonal decomposition of each isotypic component (see Proposition 7.10). An example
that leads to an operator in the center is in [25]; see also Remark 10.1.
2 Preliminaries and notation.
If X is a finite set, L(X) will denote the space of all complex functions defined on X.
The space L(X) will be endowed with the scalar product f ,f = f (x)f (x),
h 1 2iL(X) x X 1 2
f ,f L(X). The symbol δ will denote the Dirac function centered at∈x X and if
1 2 x
∈ P ∈
A X then 1 is the characteristic function of A. Let Y be another set. We will use
A
⊆
the isomorphism L(X Y) = L(X) L(Y), where for f L(X),f L(Y), x X and
∼ 1 2
× ⊗ ∈ ∈ ∈
y Y, we have (f f )(x,y) = f (x)f (y). Let C = 0,1 be the two elements cyclic
1 2 1 2 2
∈ ⊗ { }
group written additively. Then the set 0,1 X will denote the finite abelian group of all
{ }
functions θ : X 0,1 , with addition (θ +ω)(x) = θ(x)+ω(x) mod 2. The identity of
→ { }
this group will be denoted by 0 (that is 0 0 on all X).
X X
≡
For θ,ω 0,1 X, define the scalar product θ ω = θ(x)ω(x) and set χ (ω) =
∈ { } · x X θ
∈
( 1)θω. Then χ is a character of C and the dual group is CX = χ : θ CX . We
− · θ 2 P 2 { θ ∈ 2 }
also recall the orthogonality relations χ ,χ = 1 δ .
Let (X,E) be a graph simple, unorihenθtedωainL(d{0w,1}itXh)out2l|oXo|pθs,dω. We will think of the edge
set E as a subset of x,y : x,y X,x = y and we will write x y to denote that
{{ } ∈ 6 } ∼
x,y is an edge. By deg(x) = y X : x y we will denote the degree of x X.
{ } |{ ∈ ∼ }| ∈
The Markov operator of the graph is the linear selfadjoint operator M : L(X) L(X)
→
defined by setting
1
(Mf)(x) = f(y),
deg(x)
y x
X∼
while the adjacency operator is given by
(Af)(x) = f(y),
y x
X∼
for any f L(X). If X is regular of degree k, we have M = 1A, but if X is not
∈ k
regular, in general M and A have a different spectral theory. For instance, for the path
theadjacent spectrumrequiresadiscretesinetransform[2,22], whileitsMarkovspectrum
requires a discrete cosine transform [3, 9].
If g ,g ,...,g belong to a group G, then g ,g ,...g will denote the subgroup
1 2 m 1 2 m
h i
generated by g ,g ,...,g ; if v ,v ,...v belong to a vector space V, then v ,v ...v
1 2 m 1 2 m 1 2 m
h i
will denote the subspace spanned by v ,v ...v . If G is a finite group, (ρ,V) a unitary
1 2 m
representation of G and
3
V = m W (2)
j J j j
⊕ ∈
isthedecomposition ofV into irreducible representations W where m W = W ... W
j j j j j
⊕ ⊕
m -times and W ,W are inequivalent for i = j, then we say that the m W : j J are
j i j j j
6 { ∈ }
the isotypic components of V.
3 Vertex lamplighter random walks.
Let (X,E) be a finite graph. Set
(X) = (ω,x) : ω 0,1 X,x X 0,1 X X.
L ∈ { } ∈ ≡ { } ×
Following [19], we defin(cid:8)e a graph structure on (X(cid:9)) by declaring two vertices (ω,x),
L
(θ,y) (X) adjacent if x y (in X) and ω(z) = θ(z) for all z = x,y. In other
∈ L ∼ 6
words, x must be connected to y and ω must take the same values of θ on X x,y .
\ { }
The vertex lamplighter process on X is the simple random walk on (X). Note that
L
L( (X)) L( 0,1 X) L(X) and that the Markov operator on (X) is:
L ≡ { } ⊗ L
1
[ (F f)](ω,x) = [F(ω)+F(ω +δ )+F(ω +δ )+F(ω +δ +δ )]f(y),
X x y x y
M ⊗ 4deg(x)
y x
X∼
where F L( 0,1 X), f L(X) and (ω,x) (X).
∈ { } ∈ ∈ L
If we define V = χ f : f L(X) , then we have the orthogonal decomposition
θ θ
{ ⊗ ∈ }
L( (X)) = V . (3)
θ
L
θ 0,1 X
∈M{ }
Now we show how to reduce the spectral analysis of to a series of eigenvalues
M
problem on X. For θ 0,1 X, we set X = x X : θ(x) = 0 . We define a linear
θ
∈ { } { ∈ }
operator M : L(X) L(X) by setting, for f L(X) and x X,
θ
→ ∈ ∈
1 f(y) if θ(x) = 0
(M f)(x) = deg(x) yy∈Xxθ:
θ ∼
( P0 if θ(x) = 1.
Lemma 3.1. If f L(X) then
∈
(χ f) = χ M f.
X θ θ θ
M ⊗ ⊗
Proof.
1
[ (χ f)](x,ω) = [χ (ω)+χ (ω +δ )+χ (ω +δ )+χ (ω +δ +δ )]f(y).
X θ θ θ x θ y θ x y
M ⊗ 4deg(x)
y x
X∼
4
As the term in squared brackets equals
χ (ω)[1+χ (δ )+χ (δ )+χ (δ +δ )] = χ (ω)[1+( 1)θ(x) +( 1)θ(y) +( 1)θ(x)+θ(y)] =
θ θ x θ y θ x y θ
− − −
4χ (ω) if θ(x) = θ(y) = 0
= θ
0 otherwise,
(cid:26)
if θ(x) = 0 we have that
1
[ (χ f)](x,ω) = χ (ω) f(y),
X θ θ
M ⊗ deg(x)
yXy∈Xxθ:
∼
while if θ(x) = 1 then [ (χ f)](x,ω) = 0. Therefore, (χ f) = χ M f.
X θ X θ θ θ
M ⊗ M ⊗ ⊗
Remark 3.2. In other words, the lamplighter random walk is CX-invariant (cf.(1)).
2
Moreover, (3) is the decomposition of L( (X)) into irreducible CX representations, that
L 2
is V is the isotypic component corresponding to the character χ . Then each V is CX-
θ θ θ 2
invariant and Lemma 3.1 is just the expression of the restriction of to V . Lemma 3.1
θ
M
may be also seen as a finite generalization of Lemma 3.7 in [1].
We now give a closer look at the operator M . Let X be as before and set E =
θ θ θ
x,y E : x,y X . Clearly X , with edge set E , is a subgraph of X. If we denote
θ θ θ
{{ } ∈ ∈ }
by A the adjacency operator of X , then we have
θ θ
1
(M f)(x) = (A f)(x).
θ θ
deg(x)
In particular, if deg(x) = k (i.e it is constant on X), then (M f)(x) = 1(A f)(x) and
θ k θ
therefore one can recover the spectrum of (X) by analyzing the adjacency spectra of all
L
the subgraphs of X that may be obtained erasing some vertices of X.
Let
M = λ P +λ P + +λ P (4)
θ θ,1 θ,1 θ,2 θ,2 θ,h(θ) θ,h(θ)
···
be the spectral decomposition of M . That is, λ ,λ ,...,λ are the distinct
θ θ,1 θ,2 θ,h(θ)
nonzero eigenvalues and P is the orthogonal projection of L(X) onto the eigenspace of
θ,j
λ . Clearly, if X ( X, M has also the eigenspace L(X X ), with eigenvalue equal to
θ,j θ θ θ
\
zero; this is omitted in (4).
LetQ : L( (X)) V betheorthogonalprojectionontoV . Then, forF = F(ω,x)
θ θ θ
L → ∈
L( (X)), we have
L
Q F = χ Q F,
θ θ θ
⊗
where (Q F)(x) = 1 F(ω,x)χ (ω).
θ 2|X| ω 0,1 X θ e
∈{ }
Lemma 3.3. The spectralPdecomposition of the operator is given by:
e X
M
5
h(θ)
= λ χ P Q ,
X θ,j θ θ,j θ
M ⊗
θ∈X{0,1}X Xj=1 (cid:16) (cid:17)
e
where (χ P Q )F = χ P Q F for any F L( (X)). The zero eigenvalues (in
θ θ,j θ θ θ,j θ
⊗ ⊗ ∈ L
particular those corresponding to the space L(X X )) are omitted and the eigenvalues
θ
\
λ : θ 0,1 X,je= 1,2,...,h(θ) eare not necessarily distinct.
θ,j
{ ∈ { } }
Proof. It is obvious: if F L( (X)) then
∈ L
F = χ Q F =
X X θ θ
M M  ⊗ 
θ 0,1 X
∈X{ }
 e 
= χ M Q F =
θ θ θ
⊗
θ 0,1 X
∈X{ }
h(θ) e
= λ χ P Q F .
θ,j θ θ,j θ
⊗
θ∈X{0,1}X Xj=1 (cid:16) (cid:17)
e
Remark 3.4. In other words, if V is the eigenspace of M corresponding to λ , j =
θ,j θ θ,j
0,1,...,h(θ) (with V the eigenspace of λ = 0) then W = χ f : f V is the
θ,0 θ,0 θ,j θ θ,j
{ ⊗ ∈ }
eigenspace of M corresponding to λ .
θ θ,j
Corollary 3.5 (k step iterate). The probability of going from (ω,x) to (η,y) in k steps
−
is equal to
h(θ)
χ (ω) χ (η)
(λ )k θ · θ (P δ )(y). (5)
θ,j · 2X θ,j x
| |
θ 0,1 X j=1
∈X{ } X
Proof. We have
1
Q (δ δ ) = χ (ω)δ
θ ω ⊗ x 2X θ x
| |
and therefore
e
h(θ)
χ (ω)
k (δ δ ) = (λ )k θ χ P δ (6)
MX ω ⊗ x θ,j · 2X θ ⊗ θ,j x
| |
θ 0,1 X j=1
∈X{ } X
from which (5) follows immediately, since it is equal to [ k (δ δ )](η,y).
MX ω ⊗ x
Now we give the lamplighter version of the celebrated upper bound lemma of Diaconis
and Shahshahani [7].
6
Corollary 3.6 (Upper boundlemma). Suppose that X is connected. Assuming that P0X,1
is the orthogonal projector on the space of constant value functions, we have
2 h(θ)
1
(cid:13)(cid:13)(cid:13)[MkX(δω ⊗δx)− 2|X||X|1L(X)(cid:13)(cid:13)(cid:13)TV ≤ |X|θ∈θ{X6=0,01X}X:Xj=1 |λθ,j|2kkPθ,jδxk2L(X)+
(cid:13) (cid:13)
h(0X)
+ |λ0X,j|2kkP0X,jδxk2L(X).
Xj=2 
Proof. From (6) and the orthogonality relations for the characters χθ’swe get:
1 2 h(θ) λ 2k
(cid:13)(cid:13)(cid:13)[MkX(δω ⊗δx)− 2|X||X|1L(X)(cid:13)(cid:13)(cid:13)L(L(X)) = θ∈θ{X6=0,01X}X:Xj=1 |2θ2,|jX|| kχθ ⊗Pθ,jδxk2L(L(X))
(cid:13) (cid:13) +h(0X) |λ202XX,j|2kkχ0X ⊗P0X,jδxk2L( (X)) =
| | L
j=2
X
h(θ)
λ 2k
= | θ,j| P δ 2 +
2X k θ,j xkL(X)
| |
θ X0,1 X:Xj=1
∈θ{6=0X}
+h(0X) |λ02XX,j|2kkP0X,jδxk2L(X).
| |
j=2
X
Then the upper bound lemma follows immediately from the Cauchy-Schwarz inequal-
ity.
Remark 3.7. The hypothesis that X is connected guarantees that the multiplicity of the
eigenvalue 1 is equal to 1.
4 The vertex lamplighter random walk on the com-
plete graph.
Supposethat(X,E) isthecomplete graphonnvertices. Weidentify X with 1,2,...,n .
{ }
Now for any θ 0,1 X, the graph (X ,E ) is the complete graph on X vertices. We
θ θ θ
∈ { } | |
recall that the eigenspaces of the adjacency operator on the complete graph on m vertices
are the space of constant functions and its orthogonal complement, with corresponding
eigenvalues m 1 and 1.
− −
7
For any θ 0,1 X, define the projector P : L(X) L(X) by setting
θ
∈ { } →
1 f(y) if x X
Pθf(x) = |Xθ| y∈Xθ ∈ θ for any f L(X).
0 if x X , ∈
(cid:26) P 6∈ θ
For X > 1, the spectral decomposition of the operator M is given by
θ θ
| |
X 1 1
θ
M = | |− P (R P )
θ θ θ θ
n 1 − n 1 −
− −
where R : L(X) L(X ) is the orthogonal projection from L(X) onto L(X ). In
θ θ θ
→
the notation introduced above, we have: h(θ) = 2, λθ,1 = |Xnθ|−11, λθ,2 = −n11, Pθ,1 = Pθ
and P = R P . − −
θ,2 θ θ
−
Clearly, if X = 1, then X = x for some x X and M 0.
θ θ θ
| | { } ∈ ≡
If x X we have
θ
∈ 1 if y X
(Pθδx)(y) = |Xθ| ∈ θ
0 if y / X
(cid:26) ∈ θ
and therefore, kPθδxk2L(X) = |X1θ| and k(Rθ −Pθ)δxk2L(X) = |X|Xθ|θ−|1.
If x / X then (δ δ ) 0.
θ X ω x
∈ M ⊗ ≡
Denote by
= θ 0,1 X : X = i+1
i θ
A { ∈ { } | | }
and observe that = n . Now we are in position to estimate the rate of convergence
|Ai| i+1
to the stationary distribution.
(cid:0) (cid:1)
Proposition 4.1. There exists C > 0 such that if k n(logn+c) with c 0, we have
≥ 2 ≥
2
1
MkX(δω0 ⊗δy)− 2nn1L(X) ≤ Cexp(−c).
(cid:13) (cid:13)TV
(cid:13) (cid:13)
(cid:13) (cid:13)
Proof. By the upper bound lemma, we have
(cid:13) (cid:13)
2
1
MkX(δω0 ⊗δx0)− 2nn1L(X) ≤
(cid:13) (cid:13)TV
(cid:13) (cid:13)
n 2 2
(cid:13) − (cid:13)
(cid:13) ≤ |X| |(cid:13)λθ,j|2kkPθ,jδx0k2L(X) +|λ0X,2|2kkP0X,2δx0k2L(X) ≤
( )
Xi=1 θX∈AiXj=1
n 2 2k
− n i 1
n
≤ i+1 n 1 i+1
( " #
i=1 (cid:18) (cid:19)(cid:18) − (cid:19)
X
n 2 2k 2k
− n 1 i 1 n 1
+ + − . (7)
i+1 n 1 i+1 n 1 n
" # )
i=1 (cid:18) (cid:19)(cid:18) − (cid:19) (cid:18) − (cid:19)
X
8
Note that in the third step we have an inequality as we have to take into account the
cases when x / X .
0 θ
∈
Thelargestnontrivialeigenvalueis n 2 andthecorrespondingtermin(7)is n2 1 1 2k <
−
n 1 n 1 − n 1
n2 exp 2k , which becomes < 1 wh−en k > n 1 log n2 n logn. It remain−s to show−
n 1 −n 1 −2 n 1 ∼ 2 (cid:0) (cid:1)
th−at the oth−er part of (7) goes to zero faster. −
(cid:0) (cid:1)
Suppose that k = 1n(logn+c) with c > 0. The last term in (7) is clearly smaller than
2
e c if n is sufficiently large. Moreover, it is obvious that the second sum is dominated
−
by the first sum, and therefore we are left to estimate the first sum. With the change of
variable i n i 1, we have:
→ − −
n 2 2k n 2 2k
− n i n − n i n
= 1
n 1 i n 1 i+1 i − n 1 n i ≤
i=1 (cid:18) − − (cid:19)(cid:18) − (cid:19) i=1 (cid:18) (cid:19)(cid:18) − (cid:19) −
X X
n−2 ni 2ki n
exp
≤ i! −n 1 n i ≤
i=1 (cid:18) − (cid:19) −
X
n 2
n − n
setting k = (logn+c) exp log(i!) ic+log
2 ≤ − − n i ≤
i=1 (cid:18) − (cid:19)
X
n 2
−
exp( c) exp[ ilog(i)+i 1+logn log(n i)],
≤ − − − − −
i=1
X
since log(i!) ilogi i+ 1 and ic c. In order to complete the proof, we just
≥ − − ≤ −
need to bound the last sum by a constant independent of n. Observe that i h(i) =
7→
[ ilog(i)+i 1+logn log(n i)]+i has derivative equal to log(i)+1+ 1 , which
− − − − − n i
is negative if i 10. Moreover, if n 20 then h(10) 0 and therefore we can−conclude
≥ ≥ ≤
that
n 2
−
exp( ilog(i)+i 1+logn log(n i))
− − − −
i=1
X
10 +
∞
exp( ilog(i)+i 1+logn log(n i))+ exp( i) C. (8)
≤ − − − − − ≤
i=1 i=11
X X
Now we give the corresponding lower bound, showing that the random walk has a
cut-off at k = n logn.
2
Proposition 4.2. Let p(k) = k (δ δ ) be the probability after k steps starting from
MX ω0 ⊗ x0
the point (ω ,x ) and let π be the uniform distribution. Then for k = 1n(logn c),
0 0 2 −
0 < c < logn and n large we have
p(k) π 1 20e c.
TV −
k − k ≥ −
9
Proof. For x = 1,2,...,n let f : X C be the characteristic function of X . Moreover,
x → δx
for x = y set θ = δ +δ and let f be the characteristic function of X . We have
6 x,y x y x,y θx,y
the following equality:
(χ f ) χ f = χ0X ⊗fx if x = y (9)
δx ⊗ x δy ⊗ y χ f if x = y.
(cid:26) θx,y ⊗ x,y 6
(cid:0) (cid:1)
Givenaprobabilitydistribution pon (X)andafunctionf : (X) C, theexpected
L L →
value of f with respect to p is E (f) = p(ω,x)f(ω,x), while the variance of f
p (ω,x) (X)
is Var (f) = E (f2) E (f)2. ∈L
p p − p P
Consider the function
F = χ f +χ f + +χ f
δ1 ⊗ 1 δ2 ⊗ 2 ··· δn ⊗ n
which is an eigenvector of the operator , with eigenvalues n 2. In what follows,
MX n−1
we suppose that ω = 0 ; this implies F(ω ,x ) = n 1. In virtue o−f (9), we have
0 X 0 0
−
F2 = χ0X⊗f1+···+χ0X⊗fn+ χθx,y⊗fx,y = (n−1)χ0X⊗1X+ χθx,y⊗fx,y. (10)
x=y x=y
X6 X6
But E (f) = [ k (f)](ω ,x ) and therefore
p(k) MX 0 0
k k
n 2 n 2
E (F) = − (f (x )+f (x )+ +f (x )) = (n 1) − . (11)
p(k) n 1 1 0 2 0 ··· n 0 − n 1
(cid:18) − (cid:19) (cid:18) − (cid:19)
Similarly, by (10) we have,
k k
n 3 n 3
E (F2) = n 1+ − f (x ) = n 1+(n 1)(n 2) − .
p(k) − n 1 x,y 0 − − − n 1
(cid:18) − (cid:19) x=y (cid:18) − (cid:19)
X6
and therefore
k 2k
n 3 n 2
Var (F) = n 1+(n 1)(n 2) − (n 1)2 − n 1. (12)
p(k) − − − n 1 − − n 1 ≤ −
(cid:18) − (cid:19) (cid:18) − (cid:19)
Since π = 2n1n1 (X) is the uniform distribution, we have Eπ(F) = 0 and Varπ(F) =
L
n 1.
−
Now define A = (ω,x) (X) : F(ω,x) < β√n 1 , where β is a constant
β
{ ∈ L | | − }
0 < β < 1 E (F) that will be suitably chosen later. From Markov’s inequality it
√n 1 p(k)
follows that −
π(A ) = 1 π (ω,x) : F(ω,x) β√n 1
β
− { | | ≥ − } ≥
1 1 (13)
1 E (F2) = 1 .
≥ − β2(n 1) π − β2
−
10