Moderate deviations for the range of planar random walks

0.46 MB·English

by  Richard F. Bass

6 Moderate deviations for the range of planar
0
0 random walks
2

n
a
J
Richard Bass Xia Chen Jay Rosen
1 ∗ † ‡
3

February 2, 2008
]
R
P
.
h
t
a Abstract
m
[ Given asymmetricrandomwalk inZ2 withfinitesecondmoments,

1 let R be the range of the random walk up to time n. We study
n
v moderate deviations for R ER and ER R . We also derive the
1 n− n n− n
corresponding laws of the iterated logarithm.
0
0
2
0
1 Introduction
6
0
/
h
at Let Xi be symmetric i.i.d. random vectors taking values in Z2 with mean 0
m andfinitecovariancematrixΓ,setSn = ni=1Xi, andsupposethatnoproper
subgroup of Z2 supports the random walk S . For any random variable Y
: n
v P
we will use the notation
i
X Y = Y EY.
r −
a
Let
(1.1) R = # S ,...,S
n 1 n
{ }
be the range of the random walk up to time n. The purpose of this paper is
to obtain moderate deviation results for R and R .
n n
−
∗Research partially supported by NSF grant #DMS-0244737
†Research partially supported by NSF grant #DMS-0405188.
‡Research partially supported by grants from the NSF and from PSC-CUNY.
1
For moderate deviations of R we have the following. Let
n
n
(1.2) (n) = P0(S = 0).
k
H
k=0
X
Since the X have two moments, then by [23], Section 2,
i
n
logn
(n) = P0(S = 0)
k
H ∼ 2π√detΓ
k=0
X
and
n
logb
(n) ([n/b ]) = P0(S = 0) n .
n k
H −H ∼ 2π√detΓ
k=[nX/bn]+1
Theorem 1.1 Let b be apositive sequencesatisfyingb andlogb =
n n n
{ } → ∞
o((logn)1/2) as n . There are two constants C ,C > 0 independent of
1 2
→ ∞
the choice of the sequence b such that
n
{ }
n
C liminfb 1logP R ( (n) ([n/b ]))
− 1 ≤ n −n n ≥ (n)2 H −H n
→∞ n H n o
(1.3) limsupb 1logP R ( (n) ([n/b ]) C .
≤ −n n ≥ (n)2 H −H n ≤ − 2
n
→∞ n H o
Remark 1.2 The proof will show that C in the statement of Theorem 1.1
2
is equal to the constant L given in Theorem 1.3 in [2]. We believe that C is
1
also equal to L, but we do not have a proof of this fact.
A more precise statement than Theorem 1.1 is possible when the X have
i
slightly more than two moments.
Corollary 1.3 Suppose E[ X 2(log+( X ))1+δ] < for some δ > 0. Let
i i 2
| | | | ∞
b be a positive sequence satisfying b and logb = o((logn)1/2) as
n n n
{ } → ∞
n . There are two constants C ,C > 0 independent of the choice of the
1 2
→ ∞
sequence b such that
n
{ }
n
C liminfb θlogP R 2θπ√detΓ logb
− 1 ≤ n −n n ≥ (logn)2 n
→∞
n n o
(1.4) limsupb θlogP R 2θπ√detΓ logb C
≤ −n n ≥ (logn)2 n ≤ − 2
n
→∞ n o
for any θ > 0.
2
Remark 1.4 The constants C , C are the same as in the statement of
1 2
Theorem 1.1. See Remark 1.2.
For b tending to infinity faster than the rate given in Theorem 1.1, e.g.,
n
logb = (logn)2, then we are in the realm of large deviations. For results on
n
large deviations of the range, see [14], [18], [19].
For the moderate deviations of R = ER R we have the following.
n n n
− −
Let κ(2,2) be the smallest A such that
1/2 1/2
f A f f
k k4 ≤ k∇ k2 k k2
for all f C1 with compact support. (This constant appeared in [2].)
∈
Theorem 1.5 Suppose b and b = o((logn)1/5) as n . For
n n
→ ∞ → ∞
λ > 0
1 nb
lim logP R > λ n = (2π) 2(detΓ) 1/2κ(2,2)4λ.
n→∞ bn − n log2n − − −
(cid:16) (cid:17)
Comparing Theorems 1.1 and 1.5, we see that the upper and lower tails of
R are quite different. This is similar to the behavior of the distribution of
n
the self-intersection local time of planar Brownian motion. This is not sur-
prising, since LeGall, [21, Theorem 6.1], shows that R , properly normalized,
n
converges in distribution to the self-intersection local time.
The moderatedeviations ofR arequite similar innatureto those of L ,
n n
−
where L is the number of self-intersections of the random walk S ; see [4].
n n
Again, [21, Theorem 6.1] gives a partial explanation of this. However the
case of the range is much more difficult than the corresponding results for
intersection local times. The latter case can be represented as a quadratic
functional of the path, which is amenable to the techniques of large deviation
theory, while the range cannot be so represented. This has necessitated the
development of several new tools, see in particular Sections 5 and 6, which
we expect will have further applications in the study of the range of random
walks.
Theorem 1.1 gives rise to the following LIL for R .
n
3
Theorem 1.6
R
(1.5) limsup n = 2π√detΓ, a.s.
nlogloglogn/log2n
n
→∞
This result is an improvement of that in [6]; there it was required that
the X be bounded random variables and the constant was not identified.
i
Theorem 1.1 is a more precise estimate than is needed for Theorem 1.6; this
is why Theorem 1.1 needs to be stated in terms of (n) while Theorem 1.6
H
does not.
For an LIL for R we have a different rate.
n
−
Theorem 1.7 We have
R
limsup − n = (2π) 2√detΓ κ(2,2)4, a.s.
−
nloglogn/log2n
n
→∞
The study of the range of a lattice-valued (or Zd-valued) random walk
has a long history in probability and the results show a strong dependence
on the dimension d. See [15], [20], [21], [23], [18], [19], [14], and [6] and
the references in these papers, to cite only a few. The two dimensional case
seems to be the most difficult; in one dimension no renormalization is needed
(see [9]), while for d 3 the tails are sub-Gaussian and have asymptotically
≥
symmetric behavior. In two dimensions, renormalization is needed and the
tails have non-symmetric behavior. In this case, the central limit theorem
was proved in 1986 in [21], while the first law of the iterated logarithm was
not proved until a few years ago in [6].
Acknowledgment: We would like to thank Greg Lawler and Takashi Ku-
magai for helpful discussions and their interest in this paper.
2 Moments of the range
In this section we first give an estimate for the expectation of the range.
By [23], Theorem 6.9, we have
n 1 n
(2.1) ER = + (1+o(1)),
n (n) 2π√detΓ (n)2
H H
4
where is defined in (1.2). By [23], Section 2,
H
logn
(2.2) (n)
H ∼ 2π√detΓ
and
log(n/m)
(2.3) (n) (m)
H −H ∼ 2π√detΓ
as n and m tend to infinity.
Throughout this paper we will mostly be concerned with random walks
that have only second moments. The exception is the following proposition,
which supposes slightly more than two moments, and Corollary 1.3.
Proposition 2.1 Suppose X is a sequence of i.i.d. mean zero random
i
vectors taking values in Z2 w{ith}
(2.4) E X 2(log+ X )21+δ <
| | | | ∞
(cid:16) (cid:17)
for some δ > 0 and nondegenerate covariance matrix Γ. Let S = n X
n i=1 i
and suppose S is strongly aperiodic. Then
n
P
1 1
(2.5) P(S = 0) = +O .
n
2πn√detΓ n(logn)(1+δ)/2
(cid:16) (cid:17)
Proof. Let ϕ be the characteristic function of X , let x y denote the inner
i
product in R2, let Q(u) = u Γu, and let C = [ π,π]2. W· e observe that
· −
(2.6) 1 ϕ(u) Q(u)
| − − |
= E 1 eiuX +iu X +(1/2)(iu X)2
·
| − · · |
c u 3E 1 X 3 +c u 2E 1 X 2
1 (cid:0) X 1/u 1 (cid:1)X>1/u
≤ | | {| |≤ | |}| | | | {| | | |}| |
(cid:0) (cid:1) (cid:0) (cid:1)
5
and consequently for any fixed M > 0
(2.7)
1 ϕ(u/√n) Q(u/√n)
| − − |
1 1
c E 1 ( u X )3 +c E 1 ( u X )2
≤ 2 n3/2 {|u||X|≤√n} | || | 2 n {|u||X|>√n} | || |
(cid:18) (cid:19) (cid:18) (cid:19)
1 (cid:0) 1 (cid:1) (cid:0) (cid:1)
c +c E 1 ( u X )3
≤ 3n3/2 3 n3/2 {M<|u||X|≤√n} | || |
(cid:18) (cid:19)
1 (cid:0) (cid:1)
+c E 1 ( u X )2 .
3 n {|u||X|>√n} | || |
(cid:18) (cid:19)
(cid:0) (cid:1)
Choose M so that x/log1/2+δ(x) is monotone increasing on x M, and
≥
therefore
(2.8) E(1 ( u X )3)
M<u X √n
{ | || |≤ } | || |
u X
E 1 ( u X )2log1/2+δ( u X ) | || |
≤ {M<|u||X|≤√n} | || | | || | log1/2+δ( u X )
(cid:18) | || | (cid:19)
√n
E 1 ( u X )2log1/2+δ( u X ) .
≤ log1/2+δ(√n) {M<|u||X|≤√n} | || | | || |
(cid:18) (cid:19)
(cid:16) (cid:17)
Also
(2.9) E(1 ( u X )2)
u X>√n
{| || | } | || |
1
E 1 ( u X )2log1/2+δ( u X ) .
≤ log1/2+δ(√n) {|u||X|>√n} | || | | || |
(cid:18) (cid:19)
(cid:16) (cid:17)
(2.7) then implies that
u 2 log1/2+δ( u )
(2.10) 1 ϕ(u/√n) Q(u/√n) c| | | | | |.
| − − | ≤ nlog1/2+δ(n)
Following the proof in Spitzer [34], pp. 76–77,
2πnP(S = 0) = (2π) 1 ϕ(u/√n)ndu
n −
Z√nC
= I +I (n,A )+I (n,A )+I (n,A ,r)+I (n,r),
0 1 n 2 n 3 n 4
6
where
I = (2π) 1 e Q(u)/2du = (detQ) 1/2,
0 − − −
R2
Z
I (n,A ) = (2π) 1 [ϕ(u/√n)n e Q(u)/2]du,
1 n − −
−
Zu An
| |≤
I (n,A ) = (2π) 1 e Q(u)/2du,
2 n − −
−
Zu>An
| |
I (n,A ,r) = (2π) 1 ϕ(u/√n)ndu,
3 n −
ZAn<u<r√n
| |
I (n,r) = (2π) 1 ϕ(u/√n)ndu.
4 −
Zu r√n,u √nC
| |≥ ∈
Still following [34], we can choose r such that ϕ(u/√n)n e Q(u)/4 if u
−
| | ≤ | | ≤
r√n and by the strong aperiodicity there exists γ > 0 such that ϕ(u/√n)
| | ≤
1 γ if u > r√n and u √nC. Set A = c √loglogn. We have
n 4
− | | ∈
I (n,r) (2π) 1 (1 γ)ndu = O(n p)
4 − −
| | ≤ −
Zu √nC
∈
for every positive integer p. Next
I (n,A ,r) e Q(u)/4du = O((logn) 2)
3 n − −
| | ≤
Zu>c4√loglogn
| |
forc largeandsimilarly we have thesame bound for I (n,A ) . To estimate
4 2 n
| |
I (n,A ) we use the inequality an bn n a b if a , b 1 with a =
1 n
| − | ≤ | − | | | | | ≤
ϕ(u/√n) and b = e Q(u)/2n. Using (2.10) and the analogous expansion for
−
e Q(u)/2n we have
−
ϕ(u/√n)n e Q(u)/2 n ϕ(u/√n) e Q(u)/2n
− −
| − | ≤ | − |
u 2 log1/2+δ( u ) u 2 log1/2+δ( u )
c n| | | | | | = c | | | | | |.
≤ 5 nlog1/2+δ(n) 5 log1/2+δ(n)
Integrating this over the set u A , we see
n
{| | ≤ }
I (n,A ) = O((loglogn)2+δ/2/(logn)1/2+δ) = O(1/(logn)(1+δ)/2).
1 n
| |
Summing I through I , we obtain
0 4
2πnP(S = 0) = (detΓ) 1/2 +O(1/(logn)(1+δ)/2).
n −
7
Next we establish some sharp exponential estimates for the range and
intersection of ranges. Aside from their intrinsic interest, they will be used
to estimate the tail probabilities in our first main theorem.
We write S(I) for S : k I . Let S(i), i = 1,...,p be p independent
k
{ ∈ }
copies of S. First, by Corollary 1 of [8], for any integers a 1, n , n 1,
1 a
≥ ··· ≥
m!
(2.11) EJm 1/p EJk1 1/p EJka 1/p,
n1+ +na ≤ k ! k ! n1 ··· na
··· 1 a
(cid:0) (cid:1) k1k+1,·X··+,kkaa=0m ··· (cid:0) (cid:1) (cid:0) (cid:1)
··· ≥
where
J = # S(1)[1,n] S(p)[1,n] n = 1,2, .
n
∩···∩ ···
(cid:8) (cid:9)
In the next Theorem we deduce from this the exponential integrability
of J , which was established in [6] in the special case p = 2 and under the
n
condition that S had bounded increments.
Theorem 2.2 Assume that the planar random walk S has finite second mo-
ments and zero mean. There exists θ > 0 such that
(logn)p 1/(p 1)
(2.12) sup sup E(y1, ,yp)exp θ − J1/(p 1) < .
··· n n − ∞
n y1, ,yp
··· n (cid:16) (cid:17) o
Proof. We recall the fact (see Remarks, p. 664, in [23]) that
(2.13) EJk (k!)p(EJ )k, k = 0,1, ,
n ≤ n ···
and for some C <
∞
Cn
(2.14) EJ , n = 1, .
n ≤ (logn)p ···
The proof of (2.12) is a modification of the approach used in Lemma 1 of [8].
We begin by showing that there is a constant C > 0 such that
n m
(2.15) supEJm Cm(m!)p 1 , m,n = 1,2, .
n ≤ − (logn)p ···
n
(cid:16) (cid:17)
8
We first consider the case m (logn)(p 1)/p. Write l(n,m) = [n/m] + 1.
−
≤
Then by (2.11) and (2.14),
m!
EJm 1/p EJk1 1/p EJkm 1/p
n ≤ k ! k ! l(n,m) ··· l(n,m)
1 m
(cid:0) (cid:1) k1k+1,··X·+,kkmm=0m ··· (cid:0) (cid:1) (cid:0) (cid:1)
··· ≥
m!
k ! k ! EJ k1/p EJ km/p
1 m l(n,m) l(n,m)
≤ k ! k ! ··· ···
1 m
k1k+1,··X·+,kkmm=0m ··· (cid:0) (cid:1) (cid:0) (cid:1)
··· ≥
2m 1 m/p 2m 1 (n/m) m/p
= − m! EJ − m!Cm
m l(n,m) ≤ m (logn)p
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:16) (cid:17) (cid:16) (cid:17)
2m (m!)p−p1Cm n m/p,
≤ m (logn)p
(cid:18) (cid:19)
(cid:16) (cid:17)
where the second inequality follows from (2.13) and the third from (2.14)
using the fact that m = O(logn) so that logn = O(log(n/m)). Hence,
taking p-th powers we obtain
p
2m n m
EJm Cpm(m!)p 1 ,
n ≤ m − (logn)p
(cid:18) (cid:19)
(cid:16) (cid:17)
and (2.15) for the case of m (logn)(p 1)/p follows from the fact
−
≤
2m
4m.
m ≤
(cid:18) (cid:19)
For the case m > (logn)(p 1)/p, notice from the definition of J that
− n
J n. So we have
n
≤
n m n m
EJm nm = (logn)pm m(p 1)m
n ≤ (logn)p ≤ − (logn)p
(cid:16) n (cid:17) m (cid:16) (cid:17)
(m!)p 1Cm ,
−
≤ (logn)p
(cid:16) (cid:17)
where the last step follows from Stirling’s formula. This completes the proof
of (2.15).
9
By H¨older’s inequality this shows that
(logn)p m/(p 1)
(2.16) − sup E(y1, ,yp) Jm/(p 1)
n ··· n −
y1, ,yp
(cid:16) (cid:17) ··· (cid:16) (cid:17)
(logn)p m/(p 1) 1/(p 1)
− sup E(y1, ,yp) Jm −
≤ n ··· n
y1, ,yp
(cid:16) (cid:17) ··· n (cid:16) (cid:17)o
(logn)p m/(p 1) 1/(p 1)
− E Jm − Cmm!
≤ n n ≤
(cid:16) (cid:17) n (cid:16) (cid:17)o
where the second inequality used [8], p.1053. Our theorem then follows from
a Taylor expansion.
Remark. Theorem 2.2 is sharp in the sense that (2.12) does not hold if
θ is too large. Indeed, by [21], for any m = 1,2, ,
···
(logn)pm
EJm (2π)pmdet(Γ)m/2Eα [0,1]p m
nm n −→
(cid:0) (cid:1)
as n , where α [0,1]p is the Brownian intersection local time formally
→ ∞
defined by
(cid:0) (cid:1) p 1
α [0,1]p = δ W (s) ds dx,
x j
ZRd (cid:20)j=1Z0 (cid:21)
(cid:0) (cid:1) Y (cid:0) (cid:1)
and by Theorem 2.1 in [7]
Eexp θα [0,1]p (p−1)−1 =
∞
n o
(cid:0) (cid:1)
for large θ. The following theorem is sharp in the same sense.
Theorem 2.3 Assume that the planar random walk S has finite second mo-
ments and zero mean. Then there exists θ > 0 such that
(logn)2
(2.17) supEexp θ R < .
n
n | | ∞
n
n o
Proof. We first consider the case where n is replaced by 2n. Let
N = [2(log2) 1logn]
−
10