A note on the exchangeability condition in Stein’s
method
Adrian Röllin
To cite this version:
Adrian Röllin. A note on the exchangeability condition in Stein’s method. Statistics and
Probability Letters, Elsevier, 2010, 78 (13), pp.1800. .
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Accepted Manuscript
A note on the exchangeability condition in Stein’s method
Adrian Röllin
PII:
DOI:
Reference:
S0167-7152(08)00042-4
10.1016/j.spl.2008.01.043
STAPRO 4906
To appear in:
Statistics and Probability Letters
Received date: 17 October 2007
Revised date: 8 January 2008
Accepted date: 17 January 2008
Please cite this article as: Röllin, A., A note on the exchangeability condition in Stein’s
method. Statistics and Probability Letters (2008), doi:10.1016/j.spl.2008.01.043
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T
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A NOTE ON THE EXCHANGEABILITY CONDITION IN
STEIN'S METHOD
ADRIAN RÖLLIN
We show by a surprisingly simple argument that the ex-
SC
Abstract.
changeability condition, which is key to the exchangeable pair approach
in Stein's method for distributional approximation, can be omitted in
many standard settings. This is achieved by replacing the usual antisym-
AN
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metric function by a simpler one, for which only equality in distribution
is required. In the case of normal approximation we also slightly improve
the constants appearing in previous results. For Poisson approximation,
a dierent antisymmetric function is used, and additional error terms are
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needed if the bound is to be extended beyond the exchangeable setting.
1. Introduction
In the context of normal approximation, a variant of Stein's method that
is often used is the exchangeable pair coupling introduced in Diaconis (1977)
and Stein (1986). There are many applications based on this coupling, see
e.g. Rinott and Rotar (1997), Fulman (2004b), Meckes (2006) and others,
but also in the context of non-normal approximation such as Chatterjee et al.
EP
(2005), Chatterjee and Fulman (2006) and Röllin (2007).
Assume that W is a random variable which we want to approximate. The
key concept introduced by Stein (1986) is that of auxiliary randomization.
AC
C
In the context of exchangeable pairs, this means that we construct a random
variable W 0 on the same probability space such that (W, W 0 ) is exchangeable, that is L (W, W 0 ) = L (W 0 , W ) and, in general, the pair should be
constructed such that |W 0 − W | is small. One can then prove for instance
a bound as in Theorem 2.1 below (often under additional conditions on the
Supported by Swiss National Science projects 20107935/1 and PBZH2117033.
1
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2
ADRIAN RÖLLIN
exchangeable pair such as Equation (2.1)). If W and W 0 are two consec-
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utive steps of a reversible Markov chain in equilibrium, Rinott and Rotar
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(1997) note that exchangeability automatically follows. However, the pairs
constructed in some of the examples of Rinott and Rotar (1997) and Fulman
(2004a) are based on non-reversible Markov chains and therefore some ef-
fort has to be put into showing that the pairs satisfy the exchangeability
SC
condition.
The key fact to prove a bound as in Theorem 2.1 is that, for any antisym-
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metric function F , exchangeability of (W, W 0 ) implies the identity
EF (W, W 0 ) = 0
(1.1)
(see (Stein, 1986, p. 10)). In fact, this is often the only place where exchangeability is used. For example, in the case of the normal distribution,
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the standard choice for F is
¡
¢
F (w, w0 ) = (w0 − w) f (w0 ) + f (w) ,
(1.2)
where f is the solution to the Stein equation
f 0 (x) − xf (x) = h(x) − Eh(Z)
(1.3)
and Z has the standard normal distribution. We show in this paper that
(1.4)
EP
instead of the choice (1.2) we can take the simpler function
Z
0
F (w, w ) =
Z
w0
f (x)dx −
0
w
f (x)dx,
0
AC
C
which is, in particular, antisymmetric, but for which (1.1) of course holds
without exchangeability as long as L (W 0 ) = L (W ). If exchangeable pairs
are used in a discrete setting, the integrals in (1.4) of course have to be
replaced by corresponding sums.
In the following section we restate and prove results obtained by the ex-
changeable pairs approach for normal and Poisson approximation. The main
purpose is to demonstrate in detail how the exchangeability condition can be
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A NOTE ON THE EXCHANGEABILITY CONDITION IN STEIN'S METHOD
3
omitted in some standard settings, sometimes also yielding better constants.
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With the approach in this paper, however, it is possible to remove the ex-
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changeability condition in many other situations, such as in Stein (1995) and
Meckes (2006), Chatterjee and Fulman (2006), and Röllin (2007).
Most ingredients in the following proofs are taken directly from the proofs
2. Main results
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of the corresponding papers; hence many details have been omitted.
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2.1. Normal approximation. For normal approximation we need some
more assumptions. Assume that W is a random variable with EW = 0 and
Var W = 1. Construct an exchangeable pair such that
EW W 0 = (1 − λ)W + R
(2.1)
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for some constant 0 < λ < 1 and some random variable R, where EW denotes
the conditional expectation with respect to W .
The following calculations are essential for the proof of Theorem 1.2 of
Rinott and Rotar (1997), but we now assume for the sake of clarity that R =
0. Let f be the solution to (1.3) for a Lipschitz-continuous test function h.
Now, with the standard choice of F as in (1.2) and exchangeability, we have
0 = EF (W, W 0 )
EP
(2.2)
©
¡
¢ª
= E (W 0 − W ) f (W 0 ) + f (W )
©
¡
¢ª
= E (W 0 − W ) 2f (W ) + f (W 0 ) − f (W )
AC
C
©
ª
©
¡
¢ª
= −2λE W f (W ) + E (W 0 − W ) f (W 0 ) − f (W ) ,
where for the last equality we used (2.1). Let now τ be a random variable
uniformly distributed on [0, 1], independent of all other random variables,
and put V = W 0 − W . Noting that the second derivative f 00 exists almost
everywhere because h is Lipschitz continuous, Taylor's expansion yields
ª
0©
f (W 0 ) = f (W ) + V f 0 (W ) + V 2 EW,W (1 − τ )f 00 (W + τ V ) .
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4
ADRIAN RÖLLIN
©
ª
ª
ª
1 © 2 0
1 © 3
E W f (W ) =
E V f (W ) +
E V (1 − τ )f 00 (W + τ V ) .
2λ
2λ
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(2.3)
T
Thus, from (2.2),
In the proof of Theorem 2.1 we show that an equation similar to (2.3) can
be deduced without exchangeability. To state the theorem, we need some
h+
ε (x) = sup{h(x + y) : |y| 6 ε},
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notation. Dene for a given function h and ε > 0,
h−
ε (x) = inf{h(x + y) : |y| 6 ε}.
Let H be a class of measurable functions on the real line such that for all
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h ∈ H, we have khk 6 1, where k·k denotes the supremum norm; for any real
numbers c and d, h ∈ H implies h(c · +d) ∈ H; for any ε > 0, h ∈ H implies
−
h+
ε , hε ∈ H and there is a constant a (depending only on the class H) such
©
ª
−
that E h+
ε (Z) − hε (Z) 6 aε where Z has standard normal distribution.
TE
DM
As in Rinott and Rotar (1997) we assume without loss of generality that
p
a > 2/π .
Theorem 2.1 (cf. Theorem 1.2 of Rinott and Rotar (1997)). Assume that
W and W 0 are random variables on the same probability space such that
L (W 0 ) = L (W ), EW = 0, Var W = 1. Assume that (2.1) holds for some
¯
¯
λ and R. Then, for δ := suph∈H ¯Eh(W ) − Eh(Z)¯,
r
√
q
2
19
E
R
aE|W 0 − W |3
Var EW (W 0 − W )2 +
+4
.
λ
λ
EP
(2.4)
6
δ6
λ
If, in addition, there is a constant A such that |W 0 − W | 6 A almost surely,
AC
C
we have
(2.5)
12
.δ 6
λ
√
q
A3
A2
37 ER2
W
0
2
+ 32
+ 6√
Var E (W − W ) +
λ
λ
λ
Proof. From Lemma 4.1 of Rinott and Rotar (1997) we have that, for any
0 < t < 1,
(2.6)
¯
¯
δ 6 2.8 sup ¯Eht (W ) − Eht (Z)¯ + 4.7at,
h∈H
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A NOTE ON THE EXCHANGEABILITY CONDITION IN STEIN'S METHOD
5
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where ht (x) = Eh(x + tZ). Let f be the solution to the Stein equation
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f 0 (x) − xf (x) = ht (x) − Eht (Z).
(2.7)
Then, f satises
(2.8)
kf 0 k 6 4,
kf 00 k 6 2kh0t k 6 1.6t−1 ;
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kf k 6 2.6,
recalling that khk 6 1, the rst two bounds and the rst part of the third
one follow from Lemma 3 of Stein (1986) and, as noted by Rinott and Rotar
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(1997), the second inequality of the third bound can be deduced using the
R
equality h0t (x) = −t−1 h(x + ty)ϕ0 (y)dy , where ϕ is the standard normal
p
R
density, so that kh0t k 6 t−1 |ϕ0 (x)|dx = t−1 2/π .
Dene the function
Z
(2.9)
w
G(w) =
f (x) dx
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0
and note that |G(w)| 6 |w|kf k, so that EG(W ) exists. By Taylor's expansion, we have
(2.10)
G(W 0 ) = G(W ) + V f (W ) + 12 V 2 f 0 (W )
ª
0©
+ 21 V 3 EW,W (1 − τ )2 f 00 (W + τ V ) ,
where, again, V = W 0 − W . Thus, together with (2.1), we obtain
EP
0 = EG(W 0 ) − EG(W )
©
ª
©
ª
= −λE W f (W ) + E Rf (W )
AC
C
©
ª
©
ª
+ 21 E V 2 f 0 (W ) + 12 E V 3 (1 − τ )2 f 00 (W + τ V ) ,
which can be rearranged to obtain the following analogue of (2.3)
(2.11)
©
ª
©
ª
©
ª
λE W f (W ) = 12 E V 2 f 0 (W ) + 12 E V 3 (1 − τ )2 f 00 (W + τ V )
©
ª
+ E Rf (W ) .
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6
ADRIAN RÖLLIN
have from (2.7) and (2.11)
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¡
¢
©¡
¢
ª
λ Eht (W ) − Eht (Z) = E (λ − α) − 21 V 2 f 0 (W )
T
With α := E{RW } and noting that EV 2 = 2(λ − α) from (2.1), we thus
©
ª
+ E αf 0 (W ) − Rf (W )
SC
©
ª
− 21 E V 3 (1 − τ )2 f 00 (W + τ V )
=: J1 + J2 − 21 J3 .
Using (2.8), we obtain the estimates
√
|J2 | 6 6.6 ER2 ,
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√
|J1 | 6 2 Var EW V 2 ,
|J3 | 6
1.6
E|V |3 ;
3t
¡
¢1/2
for details see Rinott and Rotar (1997). Choosing t = 0.4 E|V |3 /(aλ)
and with (2.6), this proves (2.1).
Assume now that |V | 6 A. Note that because of (2.7), f 00 (x) = f (x) +
TE
DM
xf 0 (x)+h0t (x). Following the proof of Rinott and Rotar (1997), but recalling
that in our remainder J3 the term (1 − τ ) is squared, we have
©
¡
¢ª
|J3 | = E V 3 (1 − τ )2 f (W + τ V ) + (W + τ V )f 0 (W + τ V ) + h0t (W + τ V )
©
ª
©
ª
6 0.9A3 + 1.4E V 3 (|W | + |W 0 |) + E V 3 (1 − τ )2 h0t (W + τ V )
©
ª
6 3.7A3 + E V 3 (1 − τ )2 h0t (W + τ V ) ,
EP
where the latter expectation can be bounded by
3
3
¯ © 3
ª¯
¯E V (1 − τ )2 h0t (W + τ V ) ¯ 6 aA + A (2δ + aA);
3
3t
AC
C
see again Rinott and Rotar (1997). Collecting all the bounds on the Ji we
obtain
(2.12)
√
√
¯
¯
¯Eht (W ) − Eht (Z)¯ 6 2 Var EW V 2 + 6.6 ER2
λ
λ
3.7A3 aA3 A3 (2δ + aA)
+
+
.
+
2λ
6λ
6λt
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A NOTE ON THE EXCHANGEABILITY CONDITION IN STEIN'S METHOD
7
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p
Recalling that a >
2/π , putting (2.12) into (2.6) and with the choice
¡ A(2δ+aA) ¢1/2
t = 0.32A
, we nally have
aλ
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r
3
2
√
√
aA
aA
aδA3
δ 6 5.6 Var EW V 2 + 18.5 ER2 + 7
+ 3 √ + 4.2
.
λ
λ
λ
√
This inequality is of the form δ 6 a + b δ , for which we can show that
¤
SC
δ 6 2a + b2 and which hence proves (2.5).
Note that if R = 0 almost surely, equation (2.11) is the same as (2.3),
except that the factor (1 − τ ) is squared; this is the only reason for the
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improved constants. If (W, W 0 ) is exchangeable, both equalities (2.3) and
(2.11) hold. At rst glance this may seem to be a contradiction, but the
remainders with the second derivatives are in fact equal. To see this, write
©
ª
E V 3 (1 − τ )2 f 00 (W + τ V )
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DM
©
ª
©
ª
= E V 3 (1 − τ )f 00 (W + τ V ) − E V 3 τ (1 − τ )f 00 (W + τ V ) .
We need only show that the second term on the right hand side is equal
to zero. Note to this end the simple fact that L (τ ) = L (1 − τ ), thus,
using this in the following calculations to obtain the rst equality and the
exchangeability of (W, W 0 ) for the third equality,
©
¡
¢ª
E (W 0 − W )3 τ (1 − τ )f 00 W + τ (W 0 − W )
EP
©
¡
¢ª
= E (W 0 − W )3 τ (1 − τ )f 00 W + (1 − τ )(W 0 − W )
©
¡
¢ª
= E (W 0 − W )3 τ (1 − τ )f 00 W 0 + τ (W − W 0 )
©
¡
¢ª
= E (W − W 0 )3 τ (1 − τ )f 00 W + τ (W 0 − W )
AC
C
©
¡
¢ª
= −E (W 0 − W )3 τ (1 − τ )f 00 W + τ (W 0 − W ) ,
which proves the claim.
2.2. Poisson approximation. The situation in the discrete setting, in which
W takes values only in Z, is more delicate. Assume that f is the solution
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8
ADRIAN RÖLLIN
to a Stein equation of a discrete distribution. Instead of (2.9), dene the
G(w) =
w
X
f (k) −
k=1
where here and in what follows
−w−1
X
f (−k),
k=0
Pb
k=a
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(2.13)
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function G as
is dened to be zero if b < a. For
SC
many standard distributions, f will be bounded, thus |G(w)| 6 |w|kf k, so
that EG(W ) exists if E|W | < ∞.
With such G, one veries that G(w) − G(w − 1) = f (w) for all w ∈ Z.
then,
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Dene Ii = I[W 0 − W = i] and ∆i G(w) := G(w + i) − G(w) for all i ∈ Z;
G(W 0 ) − G(W ) =
(2.14)
X
Ii ∆i G(W )
i∈Z
TE
DM
and thus, if L (W 0 ) = L (W ),
(2.15)
0 = EG(W 0 ) − EG(W ) =
X ©
ª X ©
ª
E Ii ∆i G(W ) =
E Pi (W )∆i G(W )
i∈Z
i∈Z
where Pi (W ) := PW [W 0 − W = i].
As mentioned in the introduction, Rinott and Rotar (1997) suggest constructing W and W 0 via an underlying stationary Markov process, which
implies L (W ) = L (W 0 ); if the chain is reversible, exchangeability follows
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immediately. However, the Markov chain that they use for the anti-voter
model is not reversible, and so exchangeability has to be proved separately.
AC
C
One way of doing this is to assume that
(2.16)
W 0 − W ∈ {−1, 0, 1},
almost surely, which is the main assumption in Röllin (2007).
Chatterjee et al. (2005) applied the method of exchangeable pairs to the
Poisson distribution. Assume that (W 0 , W ) is an exchangeable pair taking
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A NOTE ON THE EXCHANGEABILITY CONDITION IN STEIN'S METHOD
9
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values only on the non-negative integers. Dene the antisymmetric function
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F (w, w0 ) = f (W 0 )I[W 0 − W = 1] − f (W )I[W 0 − W = −1].
(2.17)
This yields
(2.18)
SC
0 = EF (W, W 0 )
©
ª
= E f (W 0 )I[W 0 − W = 1] − f (W )I[W 0 − W = −1]
©
ª
= E EW {f (W 0 )I[W 0 − W = 1]} − EW {f (W )I[W 0 − W = −1]}
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©
ª
= E f (W + 1)P[W 0 − W = 1|W ] − f (W )P[W 0 − W = −1|W ]
©
ª
= E f (W + 1)P1 (W ) − f (W )P−1 (W ) .
We can use now the standard argument for Poisson approximation by Stein's
method (see (Barbour et al., 1992, p. 6)). Denoting by Po(λ) the Poisson
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distribution with mean λ, it follows that
¯ ©
¡
¢
ª¯
dTV L (W ), Po(λ) 6 sup¯E λf (W + 1) − W f (W ) ¯
(2.19)
f
¯ ©
ª¯
6 sup¯E (cP1 (W ) − λ)f (W + 1) − (cP−1 (W ) − W )f (W ) ¯ =: κc
f
for any c > 0 and where the supremum ranges over all solutions f = fA to
the Stein-equation
λf (j + 1) − jf (j) = I[j ∈ A] − Po(λ){A},
EP
(2.20)
for subsets A of the non-negative integers. For many applications investi-
AC
C
gated by Chatterjee et al. (2005), the following further bound on κc is used:
¡
¢
κc 6 λ−1/2 E|cP1 (W ) − λ| + E|cP−1 (W ) − W | ,
which results from the well known bound kf k 6 λ−1/2 (see (Barbour et al.,
1992, Lemma 1.1.1)). However, it is at times benecial to work directly with
the expression in (2.19) in the hope of re-arranging things so that the better
bound k∆1 f k 6 (1 − e−λ )λ−1 may be applied.
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ADRIAN RÖLLIN
It may seem surprising that, although in the bound (2.19) only the jump
T
probabilities of size 1 appear, no assumptions concerning the sizes of other
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jumps were made in the above calculations; we did not, for instance, assume
condition (2.16). However, with the choice f (w) = I[w = k] for k ∈ Z,
we obtain from (2.17) the detailed balance equation for reversible Markov
chains. Even if the chain makes jumps of size larger than 1, the stationary
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distribution is determined by the jump probabilities of size 1, so that in (2.19)
the full information about the distribution under consideration is actually
used, just by starting from (2.17). If exchangeability is not assumed, the
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eects of jumps of size larger that 1 have also to enter, and this is reected
in the bounds of the next theorem. The choice of F in (2.17) is fundamentally dierent from the standard choice in the continuous setting, where we
obtained the same bounds as before, but under weaker assumptions.
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Theorem 2.2 (cf. Proposition 3 of Chatterjee et al. (2005)). Let W and W 0
be non-negative random variables such that L (W 0 ) = L (W ). Then, for any
constant c > 0,
¡
¢
dTV L (W ), Po(λ) 6 κc + cρ
(2.21)
where ρ satises the bounds
ρ 6 λ−1/2
X X¯
X
¯
¯pk,k+i − pk+i,k ¯ 6 λ−1/2
i
|i|EPi (W ),
EP
(2.22)
i>2
k∈Z
|i|>2
for pk,j = P[W = k, W 0 = j] and the Pi are as before.
AC
C
Proof. Taking expectation over (2.20) with respect to W , using (2.15) and
noting that ∆1 G(W ) = f (W + 1) and ∆−1 G(W ) = −f (W ), we obtain
©
ª
E λf (W + 1) − W f (W )
©¡
¢
¡
¢
ª
λ − cP1 (W ) f (W + 1) − W − cP−1 (W ) f (W )
X ©
ª
+c
E Ii ∆i G(W ) + I−i ∆−i G(W ) ,
=E
i>2
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A NOTE ON THE EXCHANGEABILITY CONDITION IN STEIN'S METHOD
11
©
ª X
E Ii ∆i G(W ) + I−i ∆−i G(W ) =
(pk,k+i − pk+i,k )∆i G(k)
k∈Z
RI P
(2.23)
T
where, as before, Ii = I[W 0 − W = i]. Now, it is easy to see that
Recalling the bound kf k 6 λ−1/2 for all solutions f of (2.20), and hence
|∆i G(W )| 6 |i|λ−1/2 for all i ∈ Z, (2.23) yields the rst bound of (2.22).
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Now, as pk,k+i = P[W = k]Pi (k), the second bound follows from the rst.
¤
Under condition (2.16), Theorem 2.2 and estimate (2.19) clearly yield the
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same bound. If (2.16) does not hold, exchangeability is not automatically implied. Then the rst bound of (2.22) is a measure of the non-exchangeability
of (W, W 0 ); if the pair is in fact exchangeable, this term vanishes and we
regain (2.16). The second bound in (2.22) is particularly useful if the jump
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probabilities of larger jumps are small.
3. Discussion
The key to understand the present approach is the generator interpretation
introduced in Barbour (1988). Recall that
(AG)(x) = G00 (x) − xG0 (x) = f 0 (x) − xf (x)
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is the generator of the Ornstein-Uhlenbeck diusion, applied to the function G. Now assume that a Markov chain {X(n); n ∈ Z+ } with stationary distribution L (W ) is given and let (W, W 0 ) = (X(0), X(1)), assuming
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that the chain starts in its equilibrium. Construct a Markov jump process
Z = {Z(t); t > 0} by randomising the xed steps of size 1 of the Markov
chain X and wait instead an exponentially distributed amount of time with
rate 1/λ. It is easy to see that, under the assumption of (2.1), the innites-
imal operator of Z is
¯
£
¤
E G(Z(h)) ¯ Z(0) = x − G(x)
Ex G(W 0 ) − G(x)
(BG)(x) = lim
=
h→0
h
λ
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12
ADRIAN RÖLLIN
Ex V 2 00
G (x)
2λ
©
ª
Z
Ex (V − t)2 I[V > t > 0 or V < t < 0]
+
G(3) (x + t)dt
2λ
RI P
T
= −xG0 (x) +
©
ª
where Ex denotes E · |W = x and V = W 0 − W . Thus, we in fact compare
the innitesimal operator of this jump process with the generator of the
SC
Ornstein-Uhlenbeck diusion.
The antisymmetric function F (w, w0 ) = (w0 − w)(f (w0 ) + f (w)) in fact
also (almost) calculates this innitesimal operator, but with the rst step of
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the Taylor's expansion already carried out.
In the case of Stein (1995) and Meckes (2006), we have a family of Markov
jump processes {Zm ; m ∈ N} with innitesimal operators
(Bm G)(x) =
Ex G(Wm ) − G(x)
εm ϑ
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DM
−1 are the jump rates and by letting ε → 0 we show that the
where ε−1
m λ
processes Zm converge to a diusion process with innitesimal operator
(BG)(x) = (1 + 12 λ−1 Ex E)G00 (x) − xG0 (x)
for some specic random variable E . This diusion has the same linear
drift as the Ornstein-Uhlenbeck diusion, but a non-constant diusion rate.
Stein's method now allows us to state through the approaches in Stein (1995)
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and Meckes (2006) that the less Ex E uctuates around zero, the nearer the
stationary distribution of the process is to the stationary distribution of
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Ornstein-Uhlenbeck diusion, that is, to the standard normal distribution.
Acknowledgements
I thank Andrew Barbour for helpful discussions and the anonymous referee
for useful comments on the manuscript.
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References
A. D. Barbour (1988). Stein's method and Poisson process convergence.
RI P
J. Appl. Probab. 25A, 175184.
A. D. Barbour, L. Holst, and S. Janson (1992). Poisson approximation,
volume 2 of Oxford Studies in Probability. The Clarendon Press Oxford
cations.
SC
University Press, New York. ISBN 0-19-852235-5. Oxford Science Publi-
S. Chatterjee and J. Fulman (2006). Exponential approximation by exchangeable pairs and spectral graph theory.
AN
U
http://arxiv.org/abs/math/0605552.
Preprint.
S. Chatterjee, P. Diaconis, and E. Meckes (2005).
and Poisson approximation.
Available at
Exchangeable pairs
Probab. Surv. 2, 64106.
Available at
www.arxiv.org/abs/math.PR/0411525.
TE
DM
P. Diaconis (1977). The distribution of leading digits and uniform distribution mod 1. Ann. Probability 5, 7281.
J. Fulman (2004a). Stein's method and non-reversible Markov chains. In
Stein's method: expository lectures and applications, volume 46 of IMS
Lecture Notes Monogr. Ser., pages 6977. Inst. Math. Statist.
J. Fulman (2004b). Stein's method, Jack measure, and the Metropolis algorithm. J. Combin. Theory Ser. A 108, 275296.
EP
E. Meckes (2006). Linear functions on the classical matrix groups. Preprint.
Available at www.arxiv.org/math.PR/0509441.
Y. Rinott and V. Rotar (1997). On coupling constructions and rates in the
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C
CLT for dependent summands with applications to the antivoter model
and weighted U -statistics. Ann. Appl. Probab. 7, 10801105.
A. Röllin (2007). Translated Poisson approximation using exchangeable pair
couplings. Ann. Appl. Prob. 17, 15961614.
C. Stein (1986). Approximate computation of expectations. Institute of Mathematical Statistics Lecture NotesMonograph Series, 7. Institute of Mathematical Statistics, Hayward, CA.
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C. Stein (1995). The accuracy of the normal approximation to the distri-
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bution of the traces of powers of random orthogonal matrices. Technical
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C
EP
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DM
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U
SC
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Report No. 470, Standford University Department of Statistics.