Tài liệu Đề tài " A shape theorem for the spread of an infection " pdf

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704 HARRY KESTEN AND VLADAS SIDORAVICIUS
growth of the distances of reached half-spaces to the full asymptotic shape
result. We will give more heuristics before some of our lemmas.
Remark 2. Our proof in [KSb] shows that the right-hand inclusion in (1.1)
remains valid for arbitrary jump rates of the A and the B-particles. However,
it is still not known whether the left-hand inclusion holds in general. The lower
bound for B(t) is known only when D
A
= D
B
, or when D
A
= 0, that is, when
the A and B-particles move according to the same random walk (see [KSb]),
or in the frog model, when the A-particles stand still (see [AMP], [RS]).
Here is some general notation which will be used throughout the paper:
x without subscript denotes the 
∞
-norm of a vector x =(x(1), ,x(d)) ∈
R
d
, i.e.,
x = max
1≤i≤d
|x(i)|.
We will also use the Euclidean norm of x; this will be denoted by the usual x
2
.
x, u denotes the (Euclidean) inner product of two vectors x, u ∈ R
d
, and 0
denotes the origin (in Z
d
or R
d
).ForaneventE, E
c
denotes its complement.
K
1
,K
2
, will denote various strictly positive, finite constants whose
precise value is of no importance to us. The same symbol K
i
may have different
values in different formulae. Further, C
i
denotes a strictly positive constant
whose value remains the same throughout this paper (a.s. is an abbreviation
of almost surely).
Acknowledgement. The research for this paper was started during a stay
by H. Kesten at the Mittag-Leffler Institute in 2001–2002. H. Kesten thanks
the Swedish Research Council for awarding him a Tage Erlander Professorship
for 2002. Further support for HK came from the NSF under Grant DMS-
9970943 and from Eurandom. HK thanks Eurandom for appointing him as
Eurandom Professor in the fall of 2002. He also thanks the Mittag-Leffler
Institute and Eurandom for providing him with excellent facilities and for
their hospitality during his visits.
V. Sidoravicius thanks Cornell University and the Mittag-Leffler Insti-
tute for their hospitality and travel support. His research was supported by
FAPERJ Grant E-26/151.905/2001, CNPq (Pronex).
2. Results from [KSb]
Throughout the rest of this paper we assume that
D
A
= D
B
(2.1)
and we abbreviate their common value to D. We begin this section with some
further facts about the setup. More details can be found in Section 2 of [KSb]
which deals with the construction of our particle system. {S
t
}
t≥0
will be a
continuous-time simple random walk on Z
d
with jump rate D and starting at 0.
SHAPE THEOREM FOR SPREAD OF AN INFECTION
705
To each initial particle ρ is assigned a path {π
A
(t, ρ)}
t≥0
which is distributed
like {S
t
}
t≥0
. The paths π
A
(·,ρ) for different ρ’s are independent and they are
all independent of the initial N
A
(x, 0−),x ∈ Z
d
. The position of ρ at time t
equals π(0,ρ)+π
A
(t, ρ), and this can be assigned to ρ without knowing the
paths of any of the other particles. The type of ρ at time s is denoted by
η(s, ρ). This equals A for 0 ≤ s<θ(ρ) and equals B for s ≥ θ(ρ), where θ(ρ),
the so-called switching time of ρ, is the first time at which ρ coincides with a
B-particle. Note that this is simpler than in the construction of [KSb] for the
general case which may have D
A
= D
B
. In that case we had simple random
walks {S
η
}
t≥0
with jump rate D
η
for η ∈{A, B}, and there were two paths
associated with each initial particle ρ : π
η
(·,ρ),η ∈{A, B}, with {π
η
(t, ρ)}
having the same distribution as {S
η
t
}.Ifρ had initial position z, its position
was then equal to z + π
A
(t, ρ) until ρ first coincided with a B-particle at time
θ(ρ); for t ≥ θ(ρ) the position of ρ was z +π
A
(θ(ρ),ρ)+[π
B
(t, ρ)−π
B
(θ(ρ),ρ)].
This depends on θ(ρ) and therefore on the movement of all the other particles.
In the present case we can take π
B
= π
A
, which has the great advantage
that the path of ρ does not depend on the paths of the other particles. This
is the reason why the case D
A
= D
B
is special. We proved in [KSb] that on
a certain state space Σ
0
(which we shall not describe here), the collection of
positions and types of all particles at time t, with t running from 0 to ∞,is
well defined and forms a strong Markov process with respect to the σ-fields
F
t
= ∩
h>0
F
0
t+h
,t≥ 0, where F
0
t
is the σ-field generated by the positions and
types of all particles during [0,t]. The elements of these σ-fields are subsets
of Σ
[0,∞)
, where Σ =

k≥1

(Z
d
∪ ∂
k
) ×{A, B}

.Σ
[0,∞)
is the pathspace for
the positions and types of all particles. More explicit definitions are given in
[KSb] but are probably not needed for this paper. It was also shown in [KSb]
that if one chooses the number of initial A-particles at z, with z varying over
Z
d
, as i.i.d. mean μ
A
Poisson variables, then the process starts off in Σ
0
and
stays in Σ
0
forever, almost surely.
We write N
η
(z,t) for the number of particles of type η at the space-
time point (z, t),z∈ Z
d
,η ∈{A, B}, while N
A
(z,0−) denotes the number of
A-particles to be put at z ‘just before’ the system starts evolving. Note that
our model always has only particles of one type at each given site, because an
A-particle which meets a B-particle changes instantaneously to a B-particle.
Thus, if N
A
(z,0−)=N for some site z and we add M(> 0) B-particles at z at
time 0, then we have to say that N
A
(z,0)=0,N
B
(z,0) = N + M. We call a
site x occupied at time t by a particle of type η if there is at least one particle
of type η at x at time t; in this case all particles at (x, t) have type η. Also,
x is occupied at time t if there is at least one particle at (x, t), irrespective of
the type of that particle.
We shall rely heavily on basic upper and lower bounds for the growth of
B(t) which come from Theorems 1 and 2 in [KSb].
706 HARRY KESTEN AND VLADAS SIDORAVICIUS
Theorem A. If D
A
= D
B
, then there exist constants 0 <C
2
≤ C
1
< ∞
such that for every fixed K
P

C(C
2
t) ⊂ B(t) ⊂C(2C
1
t)

≥ 1 −
1
t
K
(2.2)
for all sufficiently large t.
We also have some information about the presence of A-particles in the
regions which have already been visited by B-particles. The following is Propo-
sition 3 of [KSb].
Proposition B. If D
A
= D
B
, then for all K there exists a constant
C
3
= C
3
(K) such that
P {there are a vertex z and an A-particle at the space-time point (z, t)(2.3)
while there also was a B-particle at z at some time ≤ t −C
3
[t log t]
1/2
}
≤
1
t
K
for all sufficiently large t.
Consequently, for large t
P {at time t there is a site in C

C
2
t/2

which(2.4)
is occupied by an A-particle}≤
2
t
K
.
Finally we reproduce here Lemma 15 of [KSb] which gives an impor-
tant monotonicity property. We repeat that in the present setup, with the
N
A
(x, 0−) i.i.d. Poisson variables, our process a.s. has values in Σ
0
at all
times (see Proposition 5 of [KSb]).
Lemma C. Assume D
A
= D
B
and let σ
(2)
∈ Σ
0
. Assume further that
σ
(1)
lies below σ
(2)
in the following sense:
For any site z ∈ Z
d
, all particles present in(2.5)
σ
(1)
at z are also present in σ
(2)
at z,
and
At any site z at which the particles in σ
(2)
have type A,(2.6)
the particles also have type A in σ
(1)
.
Let π
A
(·,ρ) be the random-walk paths associated to the various particles and
assume that the Markov processes {Y
(1)
t
} and {Y
(2)
t
} are constructed by means
of the same set of paths π
A
(·,ρ) starting with state σ
(1)
and σ
(2)
, respectively
(as defined in Section 2 of [KSb], but with π
A
(s, ρ)=π
B
(s, ρ) for all s,ρ; see
(2.6), (2.7) there). Then, almost surely, {Y
(1)
t
} and {Y
(2)
t
} satisfy (2.5) and
(2.6) for all t, with σ
(i)
replaced by Y
(i)
t
,i=1, 2. In particular, σ
(1)
∈ Σ
0
.
SHAPE THEOREM FOR SPREAD OF AN INFECTION
707
In particular, this monotonicity property says that if σ
(1)
is obtained
from σ
(2)
by removal of some particles and/or changing some B-particles to
A-particles, then the process starting from σ
(1)
has no more B-particles at
each space-time point than the process starting from σ
(2)
. We note that this
monotonicity property holds only under our basic assumption that D
A
= D
B
.
3. A subadditivity relation
In this section we shall prove the basic subadditivity relation of Proposi-
tion 3 and deduce from it, in Corollary 5, that the B-particles spread in each
fixed direction over a distance which grows asymptotically linearly with time.
This statement is ambiguous because we haven’t made precise what ‘spread in
a fixed direction’ means. Here this will be measured by
max{x, u : x ∈

B(t)},(3.1)
where u is a given unit vector (in the Euclidean norm) in R
d
(see the abstract
for

B). In addition we will not prove subadditivity (which is an almost sure
relation), but only superconvolutivity, in the terminology of [Ha] (which is
a relation between distribution functions). The tool of superconvolutivity in
other models with no obvious subadditivity in the strict sense goes back to
[Ri], and was also used in [BG] and [W].
Actually we prove superconvolutivity only for half-space processes, which
we shall introduce now. We define the closed half-space
S(u, c)={x ∈ R
d
: x, u≥c}.
Given a u ∈ S
d−1
and r ≥ 0 we consider the half-space process corresponding to
(u, −r) (also called (u, −r) half-space-process). We define this to be the process
whose initial state is obtained by replacing N
A
(x, 0−) by 0 for all x ∈S(u, −r).
Thus the initial state of the (u, −r)-half-space-process is
N
A
(x, 0−)

=0ifx/∈S(u, −r)
= original N
A
(x, 0−)ifx ∈S(u, −r),
where the N(A, x, −0) are i.i.d., mean μ
A
Poisson variables. In addition
the particles at w
−r
are turned into B-particles at time 0, where w
−r
is
the site in S(u, −r) nearest to the origin (in 
∞
-norm) with N
A
(w
−r
, 0−)
> 0. If there are several possible choices for w
−r
, the tie is broken in the
following manner. All vertices of Z
d
are first ordered in some deterministic
manner, say lexicographically. Then among all occupied vertices in S(u, −r)
which are nearest to the origin we take w
−r
to be the first one in this order.
There will be many other occasions where ties may occur. These will be broken
in the same way as here, but we shall not mention ties or the breaking of them
anymore. Note that no extra B-particles are introduced at time 0, but that
708 HARRY KESTEN AND VLADAS SIDORAVICIUS
only the type of the particles at w
−r
is changed. Thus,
N
A
(x, 0) + N
B
(x, 0) = N
A
(x, 0−) for all x.(3.2)
From time 0 on the particles move and change type as described in the abstract.
Note that only the initial state is restricted to S(u, −r). Once the particles
start to move they are free to leave S(u, −r). The (u, −r) half-space process
will often be denoted by P
h
(u, −r).
We further define the (u, −r) half-space process starting at (x, t). This
process is defined for times t

≥ t only. We define it as follows: at time t let
w
−r
(x, t) be the nearest site to x which is occupied in the (u, −r) half-space
process. We then reset the types of the particles at w
−r
(x, t)toB and the
types of all other particles present in the (u, −r) half-space process at time t
to A. The particles then move along the same path in the (u, −r) half-space
process starting at (x, t)asinP
h
(u, −r) (which starts at (0, 0)). However,
the types of the particles in the (u, −r) half-space process starting at (x, t)
are determined on the basis of the reset types at time t. Thus the half-space
process starting at (x, t) has at any time only particles which were in S(u, −r)
at time 0.
Moreover, at any site y and time t

≥ t, P
h
(u, −r) and the (u, −r) half-
space process started at (x, t) contain exactly the same particles. We see from
this that the paths of the particles in the (u, −r) half-space processes starting
at (x, t) and at (0, 0) are coupled so that they coincide from time t on, but
the types of a particle in these two processes may differ. Lemma C shows that
if there is a B-particle in P
h
(u, −r)atx at time t, then in this coupling any
B-particle in the (u, −r) half-space process starting at (x, t) also has to have
type B in P
h
(u, −r).
The coupling between the two half-space processes clearly relies heavily
on the assumption D
A
= D
B
, so that we can assign the same path to a particle
in the two processes, even though the types of the particle in the two processes
may be different.
It is somewhat unnatural to start the (u, −r) half-space process with
B-particles at w
−r
in case r<0, so that the origin does not lie in the half-space
S(u, −r). We shall avoid that situation. We can, however, use the (u, −r) half-
space process starting at (x, t). This is well defined for all r. We merely need
to find the site nearest to x which has at time t a particle which started in
S(u, −r) at time 0. We can then reset the type of the particles at this site to
B at time t. We shall consider the (u, −r) half-space process starting at (x, t)
mostly in cases where we already know that x itself is occupied at time t in
the (u, −r) half-space process.
Finally we shall occasionally talk about the full-space process and the
full-space process starting at (x, t). These are defined just as the half-space
processes, but with r = ∞. In particular, the full-space process starts with
SHAPE THEOREM FOR SPREAD OF AN INFECTION
709
B-particles only at the nearest occupied site to the origin and (3.2) applies.
The full-space process starting at (x, t) has B-particles at time t only at the
nearest occupied site to x. The type of all particles at other sites are reset
to A at time t. Being stationary in time, the full-space process started at
(x, t) has the same distribution at the space-time point (x + y, t + s)asthe
full-space process (started at (0, 0)) at the point (y, s). Again we shall use the
same random walk paths π
A
for all the full-state processes and the half-space
processes, so that these processes are automatically coupled. We shall denote
the full-space process by P
f
.
We point out that if 0 ≤ r
1
≤ r
2
, and if w
−r
2
≤r
1
/
√
d, then w
−r
2
∈
S(u, −r
1
) ⊂S(u, −r
2
) and w
−r
1
= w
−r
2
. In this case, both P
h
(u, −r
1
) and
P
h
(u, −r
2
) start with changing the type to B at the site w
−r
1
only and all
other particles are given by type A. In this situation, by Lemma C, at any
time,
any B-particle in P
h
(u, −r
1
) is also a B-particle in P
h
(u, −r
2
).(3.3)
This comment also applies if P
h
(u, −r
2
) is replaced by P
f
(which is the case
r
2
= ∞).
Rather than introduce formal notation for the probability measures gov-
erning the many processes here, we shall abuse notation and write P {A in
the process P} for the probability of the event A according to the probability
measure governing the process P. Neither shall we describe the probability
space on which P lives.
It seems worthwhile to discuss more explicitly the relation of the full-
space process to our process as described in the abstract. The latter has some
B-particles introduced at time 0 at one or more sites, in addition to the Poisson
numbers of particles, N
A
(x, 0−),x∈ Z
d
. If exactly one B-particle is added at
time 0, and this particle is placed at 0, then we shall call the resulting process
the original process.
Suppose we want to estimate P {A(x
0
)} in the full-space process, where
x
0
:= the nearest occupied site to the origin at time 0 in P
f
,(3.4)
A is some event and A(x) is the translation by x of this event (which takes
N
A
(0,s)toN
A
(x, s)). Then, for C a subset of Z
d
,
P {x
0
∈ C, A(x
0
)inP
f
} =

x∈C
P {x
0
= x, A(x)}(3.5)
≤

x∈C
P {x is occupied at time 0, A(x)inP
f
}
=

x∈C
∞

k=1
e
−μ
A
[μ
A
]
k
k!
P {A|there are kB-particles at 0 at time 0}.
710 HARRY KESTEN AND VLADAS SIDORAVICIUS
(The probability in the last sum is the same in P
f
as in the original process.)
On the other hand, in the original process we have
(3.6) P {A in the original process}
=
∞

k=1
e
−μ
A
[μ
A
]
k−1
(k −1)!
P {A|there are kB-particles at 0 at time 0}.
Comparison of the right-hand sides in (3.5) and (3.6) yields the crude bound
(3.7) P {x
0
∈ C, A(x
0
) in the full-space process}
≤ (cardinality of C)μ
A
P {A in original process}.
We shall repeatedly use a somewhat more general version of this inequality
(see for instance (3.25), (3.77), (3.78), (5.35)). Suppose s ≥ 0 is fixed and X
is a random vertex in Z
d
, and suppose further that
(3.8) P {A(X) but (X, s) is not occupied
in the full-space process starting at (X, s)} =0.
(Note that this is satisfied if (X, s) is occupied almost surely in P
f
.) Let C ⊂ Z
d
as before. Now, given that there are k ≥ 1 particles at the (nonrandom) space-
time point (x, s), the full-space process starting at (x, s) is simply a translation
by (x, s) in space-time of the original process, conditioned to start with k −
1 points at the origin and one B-particle added at the origin. Therefore,
essentially for the same reasons as for (3.7),
(3.9) P {X ∈ C,A(X) in the full-space process starting at (X, s)}
≤ (cardinality of C)μ
A
P {A in original process}.
For a rather trivial comparison in the other direction we note that if
P {A in P
f
} = 0 for the full-space process, then we certainly have for each
k ≥ 1 that
0=P {A in P
f
,x
0
= 0,k particles at x
0
}(3.10)
= P {A in P
f
,k particles at 0}
= e
−μ
A
[μ
A
]
k
k!
P {A|there are kB-particles at 0 at time 0}.
This implies, via (3.6), that also P {A in original process} =0.
It is somewhat more complicated to compare P
f
with the process described
in the abstract if more than one B-particle is introduced at time 0. Rather
than develop general results in this direction we merely show in our first lemma
that it suffices to prove (1.3) for the full-space process.
Lemma 1. If (1.3) holds in P
f
, then it also holds in the original process
of the abstract with any fixed finite number of B-particles added at time 0.
SHAPE THEOREM FOR SPREAD OF AN INFECTION
711
Proof. The preceding discussion shows that if (1.3) has probability 1 in
P
f
, then it has probability 1 in the original process (with one particle added at
the origin at time 0). By translation invariance (1.3) will then have probability
1 in the process of the abstract with one particle added at any fixed site at
time 0.
Lemma C implies that one can couple two processes as in the abstract,
with collections of B-particles A
(1)
⊂ A
(2)
added at time 0, respectively, in such
a way that the process corresponding to A
(1)
always has no more B-particles
than the one corresponding to A
(2)
. Therefore, if the left-hand inclusion in
(1.3) holds when only one B-particle is added at time 0, then it certainly holds
if more than one B-particles are added.
It follows that we only have to prove the right-hand inclusion in (1.3) for
the process from the abstract with more than one particle added, if we already
know it when exactly one particle is added. Assume first that we run this last
process with one B-particle ρ
0
added at z
0
. We now have to refer the reader to
the genealogical paths introduced in the proof of Proposition 5 of [KSb]. The
right-hand inclusion in (1.3) then says that for all ε>0
(3.11) P {there exist genealogical paths from z
0
to some point
outside (1 + ε)tB
0
for arbitrarly large t} =0.
From the construction of the genealogical paths in Proposition 5 of [KSb] and
the fact that a.s. there are only finitely many B-particles at finite times (see
(2.18) in [KSb]) it is not hard to deduce that
{

B(t) ⊂ (1 + ε)tB
0
at time t if one adds a B-particle ρ
i
(3.12)
at z
i
, 1 ≤ i ≤ k, at time 0}
= {there is a genealogical path from some z
i
, 1 ≤ i ≤ k,
to the complement of (1 + ε)tB
0
at time t if one
adds a B-particle ρ
i
at z
i
, 1 ≤ i ≤ k, at time 0}
⊂
k

i=1
{there is a genealogical path from z
i
to the complement of
(1 + ε)tB
0
at time t if one adds a B-particle ρ
i
at z
i
at time 0}
(the z
i
do not have to be distinct here). It follows that
P {

B(t) ⊂ (1 + ε)tB
0
for arbitrarily large times t if one(3.13)
adds a B-particle ρ
i
at z
i
, 1 ≤ i ≤ k, at time 0}
≤
k

i=1
P {there are genealogical paths from z
i
to the complement
of (1 + ε)tB
0
at arbitrarily large times t
if one adds a B-particle ρ
i
at z
i
at time 0}
= 0 (by (3.11)).
712 HARRY KESTEN AND VLADAS SIDORAVICIUS
Thus the right-hand inclusion in (1.3) holds a.s., even if one adds kB-particles
at time 0.
We recall that
P
h
(u, −r) is short for the (u, −r) half-space process,
P
f
is short for the full-space process,
and we further introduce
B
h
(y, s; u, −r):={there is a B-particle at (y, s)inP
h
(u, −r)},(3.14)
h(t, u, −r) = max{x, u : B
h
(x, t; u, −r) occurs}.(3.15)
P
or
will denote the probability measure for the original process (with one
B-particle added at the origin at time 0); E
or
is expectation with respect to
P
or
. (The superscripts h, f and or are added to various symbols which refer to a
half-space process, the full-space process, or the original process, respectively).
We use P without superscript if it is clear from the context with which process
we are dealing or when we are discussing the probability of an event which is
described entirely in terms of the N
A
(x, 0−) and the paths π
A
.
The following technical lemma will be useful. It tells us that, with high
probability, P
h
(u, −r) moves out in the direction of u at least at the speed C
4
,
provided r is large enough (see (3.15) and (3.16)). Its proof would be nicer
if we made use of the fact that even the (u, 0) half-space-process has, with a
probability at least 1 − t
−K
,B-particles at time t at sites x with x, u≥Ct,
for some constant C>0. However, it takes some work to prove this fact and
we decided to do without it.
Lemma 2. Let C
1
,C
2
be as in Theorem A and let
C
4
=
2
√
dC
1
C
2
32
√
dC
1
+ C
2
.
For all constants K ≥ 0, there exists a constant r
0
= r
0
(K) ≥ 0 such that for
r ≥ r
0
P

h(t, u, −r) ≤ C
4
t for some t ≥ t
1
:=
1
4
√
dC
1

1+
C
2
32
√
dC
1

r

≤ r
−K
.
(3.16)
Proof. The lemma is proven in three steps. In the first step we intro-
duce exponentially growing sequences of times {t
k
} and distances {d
k
}, and
prove that we only need a good bound on the probability that there are no
B-particles in P
h
(u, −r) at time t
k
in S(u, d
k
) ∩{x : x≤2C
1
d
k
}. In Step 2
we recursively define further events E
k,1
−E
k,5
and reduce the lemma to provid-
ing a good estimate for the probability that at least one E
k,i
,k ≥ 1, 1 ≤ i ≤ 5,
SHAPE THEOREM FOR SPREAD OF AN INFECTION
713
fails. The required estimates for these probabilities are derived in Step 3. This
last step relies on the left-hand inclusion in (2.2) and on (2.4). Once we know
that there is a B-particle far out in the direction u at time t
k−1
, or more pre-
cisely a B-particle at some point x
k−1
with x
k−1
,u≥d
k−1
, (2.2) and (2.4)
allow us to conclude that with high probability there is a B-particle at time t
k
at some x
k
with x
k
,u≥d
k
.
Step 1. For k ≥ 1 define the times
t
k
=
1
4
√
dC
1

1+
C
2
32
√
dC
1

k
r,
and the real numbers
d
k
=
C
2
32
√
dC
1

1+
C
2
32
√
dC
1

k
r.
Also define for each k ≥ 1 the event
(3.17) D
k
:=

B
h
(x
k
,t
k
; u, −r) occurs for some x
k
which
satisfies x
k
,u≥d
k
and x
k
≤2C
1
t
k

.
In this step we shall reduce the lemma to an estimate for the probability
that D
k
fails for some k ≥ 1. Indeed, assume that D
k
occurs for all k ≥ 1.
By definition, there is then a B-particle at (x
k
,t
k
) in the (u, −r) half-space
process (starting at (0, 0)), so that
h(t
k
,u,−r) ≥x
k
,u≥d
k
=
C
2
32
√
dC
1

1+
C
2
32
√
dC
1

k
r, k ≥ 1.(3.18)
Recall that F
t
is defined in the beginning of Section 2. In addition to (3.18),
we have on the event {x
k
,u≥d
k
}, for k ≥ 1,
P {h(t, u, −r) ≤
1
2
d
k
for some t ∈ [t
k
,t
k+1
)|F
t
k
}(3.19)
≤ P {each B-particle in P
h
(u, −r)at(x
k
,t
k
) moves during
[t
k
,t
k+1
] to some site x with x, u≤
1
2
d
k
}
≤ P { min
q≤t
k+1
−t
k
S
q
,u≤−
1
2
d
k
= −C
4
t
k+1
}
≤ K
1
exp[−K
2
t
k+1
]
for some constants K
1
,K
2
depending on d, D
A
only; see (2.42) in [KSa] for the
last inequality. It follows that the left-hand side of (3.16) is bounded by
P {D
k
fails for some k ≥ 1}+
∞

k=1
K
1
exp[−K
2
t
k
].(3.20)