Power and exponential moments of the number of visits and related quantities for perturbed random walks

0.35 MB·English

by  Gerold Alsmeyer

Power and exponential moments of the number
of visits and related quantities for perturbed
random walks
Gerold Alsmeyer∗, Alexander Iksanov†and Matthias Meiners‡
December 13, 2011
1
1
0
2

c
e Abstract
D
Let (ξ1,η1),(ξ2,η2),... be a sequence of i.i.d. copies of a random
2
vector (ξ,η) taking values in R2, and let S := ξ + ... + ξ . The
1 n 1 n
sequence (S +η ) is then called perturbed random walk.
n−1 n n≥1
]
We study random quantities defined in terms of the perturbed ran-
R
P dom walk: τ(x), the first time the perturbed random walk exits the
. interval (−∞,x], N(x), the number of visits to the interval (−∞,x],
h
t and ρ(x), the last time the perturbed random walk visits the interval
a
m (−∞,x]. We provide criteria for the a.s. finiteness and for the finiteness
of exponential moments of thesequantities. Further, weprovidecriteria
[
for the finiteness of power moments of N(x) and ρ(x).
2
2010 Mathematics Subject Classification: Primary: 60G50
v
9 Secondary: 60G40
5 Keywords: firstpassagetime, lastexittime, numberofvisits, perturbed
1
random walk, random walk, renewal theory, shot-noise process
4
.
1
1
1 Introduction
1
1
:
v The purpose of this article is to study the moments of certain basic renewal-
i
X theoretic quantities for a class of perturbed random walks formally defined
r below. Such random sequences arise as derived processes in various areas of
a
Applied Probability and we refer to Subsection 1.2 for a number of examples.
∗Institutfu¨rMathematischeStatistik,Westfa¨lischeWilhelms-Universit¨atMu¨nster,48149
Mu¨nster, Germany, e-mail: [email protected]
†FacultyofCybernetics,NationalT.ShevchenkoUniversityofKiev,01033Kiev,Ukraine,
e-mail: [email protected]
A part of this work was done while A. Iksanov was visiting Mu¨nster in January/February
and May 2011. Grateful acknowledgment is made for financial support and hospitality.
The research of A. Iksanov was partly supported by a grant from Utrecht University, the
Netherlands.
‡Institutfu¨rMathematischeStatistik,Westfa¨lischeWilhelms-Universit¨atMu¨nster,48149
Mu¨nster,Germany,e-mail: [email protected]. ResearchsupportedbyDFG-
grant Me 3625/1-1
1
It is an interesting question and in fact the main motivation behind this work
to what extent classical moment results for ordinary random walks must be
adjusted in the presence of a perturbating sequence.
1.1 Setup
Let (ξ ,η ),(ξ ,η ),... be a sequence of i.i.d. two-dimensional random vectors
1 1 2 2
with generic copy (ξ,η). For notational convenience, we assume that (ξ,η) is
defined on the same probability space as the (ξ ,η ), k ≥ 1 and independent of
k k
this sequence. No condition is imposed on the dependence structure between ξ
and η. Let (S ) be the zero-delayed ordinary random walk with increments
n n≥0
ξ for n ∈ N, i.e., S = 0 and S = ξ + ... + ξ , n ∈ N. Then define its
n 0 n 1 n
perturbed variant (T ) , called perturbed random walk (PRW), by
n n≥1
T := S +η , n ∈ N. (1.1)
n n−1 n
It has appeared in a number of recent publications, see for instance [4, 11, 21,
34]. Here we should mention that, motivated by certain problems in sequen-
tial statistics, a very different class of perturbations, which may roughly be
characterized by having slowly varying paths in a stochastic sense, has been
considered under the label “nonlinear renewal theory”, see [28, 29, 38] and [15,
Section 6].
For x ∈ R, define the level x first passage time
τ(x) := inf{n ∈ N : T > x}, (1.2)
n
the number of visits to (−∞,x]
N(x) := #{n ∈ N : T ≤ x}, (1.3)
n
and the associated last exit time
ρ(x) := sup{n ∈ N : T ≤ x} (1.4)
n
with the usual conventions that sup∅ := 0 and inf∅ := ∞.Our aim is to find
criteria for the a.s. finiteness of these quantities and for the finiteness of their
power and exponential moments.
Let us further denote by τ∗(x), N∗(x) and ρ∗(x) the corresponding quan-
tities for the ordinary random walk (S ) which is obtained in the special
n n≥0
case η = 0 a.s. after a time shift. If ξ = 0, then (T ) reduces to a sequence
n n≥1
of i.i.d. r.v.’s. In this case N(x) = ρ(x) = ∞ a.s. and τ(x) has a geometric
distribution whenever 0 < P{η ≤ x} < 1. Neither of the two afore-mentioned
cases will be subject of our analysis and therefore be ruled out by making the
Standing Assumption: P{ξ = 0} < 1 and P{η = 0} < 1.
1.2 Examples and applications
Functionals of PRW’s appear in several areas ofApplied Probability as demon-
strated by the following examples.
2
Example 1.1 (Perpetuities). Provided that eTn is a.s. convergent, this
n≥1
sum is called perpetuity due to its interpretation as a sum of discounted pay-
P
mentstreamsininsuranceandfinance. Perpetuitieshavereceivedanenormous
amount of attention which by now has led to a more or less complete theory.
A partial survey of the relevant literature may be found in [3], for more recent
contributions see [9, 18, 19, 20]. Presumably one of the most challenging open
problems in the area is to provide sufficient (and close to necessary) conditions
for the absolute continuity of the law of a perpetuity. In the light of serious
complications that already arise in the “simple” case ξ = const < 0 (see [3] for
more information), there is only little hope for the issue being settled in the
near future.
Example 1.2 (The Bernoulli sieve). The Bernoulli sieve is an infinite occu-
pancy scheme in a random environment (P ) , where
k k≥1
P := W W ···W (1−W ), k ∈ N, (1.5)
k 1 2 k−1 k
and (W ) are independent copies of a random variable W taking values in
k k≥1
(0,1). One may think of balls that, given (P ) , are independently placed
k k≥1
into one of infinitely many boxes 1,2,3,..., the probability for picking box
k being P . Assuming that the number of balls equals n, denote by K the
k n
number of nonempty boxes. If the law of |logW| is non-lattice, it was shown
in [11] that the weak convergence of K , properly centered and normalized, is
n
completely determined by the weak convergence of
N(x) := #{k ∈ N : P ≥ e−x}
k
= #{k ∈ N : W ···W (1−W ) ≥ e−x}, x > 0,
1 k−1 k
again properly centered and normalized. Notice that N(x) is the number of
visits to (−∞,x] by the PRW generated by the couples (|logW |,|log(1 −
1
W )|),(|logW |,|log(1 − W )|),.... A summary of known results including
1 2 2
relevant literature for the Bernoulli sieve can be found in the recent survey
[10].
Example 1.3 (Regenerative processes). Let (W(t)) be a c`adla`g process
t≥0
starting at W(0) = 0 and drifting to −∞ a.s. Suppose there exists a zero-
delayed renewal sequence of random epochs (τ ) such that the segments
n n≥0
(also called cycles)
(W(t)) , (W(τ +t)−W(τ )) ,...
0≤t≤τ1 1 1 0≤t≤τ2−τ1
are i.i.d. Then (W(t)) is a (strong-sense) regenerative process, see [5]. For
t≥0
n ∈ N, put
ξ := W(τ )−W(τ ) and η := sup W(t)−W(τ ).
n n n−1 n n−1
τn−1≤t<τn
Then (ξ ,η ),(ξ ,η ),... are i.i.d., and
1 1 2 2
supW(t) = sup(ξ +...+ξ +η ),
1 n−1 n
t≥0 n≥1
3
i.e., the supremum of the regenerative process can be represented as the supre-
mum of an appropriate PRW. The supremum M, say, of a PRW is a relatively
simple functional that has received considerable attention in the literature.
For instance, the tail behavior of M was investigated in [4, 12, 16, 21, 33, 34].
Some moment results on M can be found in [2, 3].
Example 1.4 (Queues and branching processes). Suppose that ξ and η are
both non-negative and define for t ≥ 0
∞
R(t) := = τ∗(t)−N(t), t ≥ 0.
1{Sk≤t<Sk+ηk+1}
k=0
X
In a GI/G/∞-queuing system, where customers arrive at times S = 0 < S <
0 1
S < ... and are immediately served by one of infinitely many idle servers, the
2
service time of the kth customer being η , R(t) gives the number of busy
k+1
servers at time t ≥ 0. Another interpretation of R(t) emerges in the context
of a degenerate pure immigration Bellman-Harris branching process in which
each individual is sterile, immigration occurs at the epochs S , S etc., and the
1 2
lifetimes of the ancestor and the subsequent immigrants are η ,η ,.... Then
1 2
R(t) gives the number of particles alive at time t ≥ 0. The process (R(t))
t≥0
was also used to model the number of active sessions in a computer network
[27, 32].
2 Main results
2.1 Almost sure finiteness
It is well-known that a non-trivial zero-delayed random walk (S ) (i.e. a
n n≥0
random walk starting at the origin with increment distribution not degenerate
at 0) exhibits one of the following three regimes:
1) drift to +∞ (positive divergence): lim S = ∞ a.s.;
n→∞ n
2) drift to −∞ (negative divergence): lim S = −∞ a.s.;
n→∞ n
3) oscillation: liminf S = −∞ and limsup S = ∞ a.s.
n→∞ n n→∞ n
PRW’s exhibit the same trichotomy. In order to state the result precisely some
further notation is needed. As usual, let ξ+ = max(ξ,0) and ξ− = max(−ξ,0).
Then, for x > 0, define
x x
A (x) := P{±ξ > y}dy = Emin(ξ±,x) and J (x) := ,
± ±
A (x)
Z0 ±
whenever the denominators are non-zero. Notice that J (x) for x > 0 is well-
±
defined iff P{±ξ > 0} > 0. In this case, we define J (0) := P{±ξ > 0}−1. The
±
following theorem, though not stated explicitly in [13], can be read off from
the results obtained there.
4
Theorem 2.1. Any PRW (T ) satisfying the standing assumption is either
n n≥1
positively divergent, negatively divergent or oscillating. Positive divergence
takes place iff
lim S = ∞ and EJ (η−) < ∞, (2.1)
n +
n→∞
while negative divergence takes place iff
lim S = −∞ a.s. and EJ (η+) < ∞. (2.2)
n −
n→∞
Oscillation occurs in the remaining cases, thus iff either
−∞ = liminfS < limsupS = ∞ a.s., (2.3)
n n
n→∞ n→∞
or
lim S = ∞ a.s. and EJ (η−) = ∞, (2.4)
n +
n→∞
or
lim S = −∞ a.s. and EJ (η+) = ∞. (2.5)
n −
n→∞
Remark 2.2. As a consequence of Theorem 2.1 it should be observed that a
PRW (T ) may oscillate even if the corresponding ordinary random walk
n n≥1
(S ) drifts to ±∞.
n n≥0
Inviewoftheprevious resultitisnaturaltotakealookatthea.s.finiteness
of the first passage times τ(x). Plainly, if limsup T = ∞ a.s., then
n→∞ n
τ(x) < ∞ a.s. for all x ∈ R. On the other hand, one might expect in the
opposite case, viz. lim T = −∞ a.s., that P{τ(x) = ∞} > 0 for all x ≥ 0,
n→∞ n
for this holds true for ordinary random walks. Namely, if lim S = −∞
n→∞ n
a.s., then P{sup S ≤ 0} = P{τ∗ = ∞} > 0. The following result shows
n≥1 n
that this conclusion may fail for a PRW. It further provides a criterion for the
a.s. finiteness of τ(x) formulated in terms of (ξ,η).
Theorem 2.3. Let (T ) be negatively divergent and x ∈ R. Then τ(x) < ∞
n n≥1
a.s. iff P{ξ < 0,η ≤ x} = 0. Furthermore, P{η ≤ x} < 1 holds true in this
case.
In order to establish a criterion for the a.s. finiteness of the r.v.’s N(x) and
ρ(x), it only takes to observe that, if one of those is a.s. finite for some x, then
liminf T > −∞ a.s. Hence, by Theorem 2.1, (T ) must be positively
n→∞ n n n≥1
divergent. Since the converse holds trivially true, we can state the following
result analogous to the case of ordinary random walks.
Theorem 2.4. The following assertions are equivalent:
(i) (T ) is positively divergent.
n n≥1
(ii) N(x) < ∞ a.s. for some/all x ∈ R.
(iii) ρ(x) < ∞ a.s. for some/all x ∈ R.
5
2.2 Finiteness of exponential moments
The following theorems are on finiteness of exponential moments of τ(x), N(x)
and ρ(x).
Theorem 2.5. Let a > 0 and x ∈ R.
(a) If P{ξ < 0,η ≤ x} = 0, then Eexp(aτ(x)) < ∞ iff
ea P{ξ = 0,η ≤ x} < 1. (2.6)
(b) If P{ξ < 0,η ≤ x} > 0, then
Eexp(aτ(x)) < ∞, (2.7)
Eexp(aτ(y)) < ∞ for all y ∈ R, (2.8)
Eexp(aτ∗) < ∞, (2.9)
R := −loginf Ee−tξ ≥ a (2.10)
t≥0
are equivalent assertions.
Turning toexponential moments ofN(x), thenumber ofvisits of(T ) to
n n≥1
(−∞,x], for x ∈ R, let us point out before-hand that these random variables
are a.s. finite iff (T ) is positively divergent which in turn holds true iff
n n≥1
(S ) is positively divergent and
n n≥0
EJ (η−) < ∞ (2.11)
+
(see Theorem 2.1) which will therefore be assumed hereafter.
Theorem 2.6. Let (T ) be a positively divergent PRW.
n n≥1
(a) If ξ ≥ 0 a.s., then the assertions
Eexp(aN(x)) < ∞, (2.12)
ea P{ξ = 0,η ≤ x}+P{ξ = 0,η > x} < 1 (2.13)
are equivalent for each a > 0 and x ∈ R. As a consequence,
a > 0 : EeaN(x) < ∞ = (0,a(x)) (2.14)
for any x ∈ R, wher(cid:8)e a(x) ∈ (0,∞] equal(cid:9)s the supremum of all positive a
satisfying (2.13). As a function of x, a(x) is nonincreasing with lower bound
−logP{ξ = 0}.
(b) If ξ > 0 a.s., then a(x) = ∞ for all x ∈ R, thus EeaN(x) < ∞ for any
a > 0 and x ∈ R.
(c) If P{ξ < 0} > 0, then the following assertions are equivalent:
Eexp(aN(x)) < ∞ for some/all x ∈ R, (2.15)
Eexp(aN∗(x)) < ∞ for some/all x ∈ R, (2.16)
R = −loginf Ee−tξ ≥ a. (2.17)
t≥0
6
Theorem 2.7. Let (T ) be a positively divergent PRW, a > 0 and R =
n n≥1
−loginf Ee−tξ.
t≥0
(a) Assume that P{ξ ≥ 0} = 1. Let x ∈ R and assume that P{η ≤ x} > 0.
Then the following assertions are equivalent:
Eexp(aρ(x)) < ∞; (2.18)
V (y) := eanP{T ≤ y} < ∞ for some/all y ≥ x; (2.19)
a n
n≥1
X
a < −logP{ξ = 0} and Ee−γη < ∞, (2.20)
where γ is the unique positive number satisfying Ee−γξ = e−a.
(b) If P{ξ < 0} > 0, then the following assertions are equivalent:
Eexp(aρ(x)) < ∞ for some/all x ∈ R; (2.21)
V (x) = eanP{T ≤ x} < ∞ for some/all x ∈ R; (2.22)
a n
n≥1
X
a < R and Ee−γη < ∞ or a = R, Eξe−γξ < 0 and Ee−γη < ∞
(2.23)
where γ is the minimal positive number satisfying Ee−γξ = e−a.
Remark 2.8. Notice that in Theorem 2.7 the case P{ξ ≥ 0} = 1, P{η ≤ x} = 0
is not treated. But this case is trivial since then ρ(x) = 0 a.s., cf. Lemma 4.3.
2.3 Finiteness of power moments
Theorem 2.9. Let (T ) be a positively divergent PRW and p > 0. The
n n≥0
following conditions are equivalent:
EN(x)p < ∞ for some/all x ∈ R; (2.24)
EN∗(x)p < ∞ for some/all x ≥ 0 and EJ (η−) < ∞; (2.25)
+
EJ (ξ−)p+1 < ∞ and EJ (η−) < ∞. (2.26)
+ +
Theorem 2.10. Let (T ) be a positively divergent PRW and p > 0. Then
n n≥0
the following assertions are equivalent:
Eρ(x)p < ∞ for some/all x ∈ R; (2.27)
Eρ∗(y)p < ∞ for some/all y ≥ 0 and EJ (η−)p+1 < ∞; (2.28)
+
EJ (ξ−)p+1 < ∞ and EJ (η−)p+1 < ∞. (2.29)
+ +
7
Remark 2.11. According to Theorem 2.7, for fixed a > 0,
Eeaρ(x) < ∞ for some/all x ∈ R iff
eanP{T ≤ x} < ∞ for some/all x ∈ R.
n
n≥1
X
According to [26, Theorem 2.1], for fixed p > 0,
Eρ∗(x)p < ∞ for some/all x ≥ 0 iff
np−1P{S ≤ x} < ∞ for some/all x ≥ 0.
n
n≥1
X
In the light of these results it may appear to be unexpected that, in gen-
eral, the finiteness of Eρ(x)p is not equivalent to the convergence of the series
np−1P{T ≤ x}. Indeed, it can be checked (but we omit the details)
n≥1 n
that a criterion for the convergence of the latter series is as follows:
P
Eρ∗(x)p < ∞ for some/all y ≥ 0 and EJ (η−)p < ∞.
+
2.4 Notation and overview
At thispoint, we introduce some notationwhich is used throughout the article.
First of all, whenever it is convenient, we write τ, N and ρ for τ(0), N(0) and
ρ(0), respectively. Analogously, we write τ∗, N∗ and ρ∗ for τ∗(0), N∗(0) and
ρ∗(0), respectively.
As usual, f(t) ∼ g(t) as t → ∞ for functions f and g, means that
f(t)/g(t) → 1 as t → ∞. Similarly, f(t) ≍ g(t) as t → ∞ means that
0 < liminf f(t)/g(t) ≤ limsup f(t)/g(t) < ∞.
t→∞ t→∞
We finish this section with an overview over the further organization of
the article. The proofs of the main results are given in Section 4. The proofs
concerning finiteness of moments of N(x), Theorems 2.6 and 2.9, are based on
general results on finiteness of exponential moments of shot-noise processes.
These results and their proofs can be found in Section 3. The appendix con-
tains auxiliary results from random walk theory (Subsection A.1) and some
elementary facts (Subsection A.2).
3 Shot-noise processes
Let ξ be a real-valued random variable with P{ξ = 0} < 1 and (X(t))t∈R
a doubly infinite non-negative stochastic process with non-decreasing paths
such that limt→−∞X(t) = 0 a.s. Any dependence between (X(t))t∈R and ξ is
allowed. Further, given a sequence ((Xn(t))t∈R,ξn))n≥1 of independent copies
of ((X(t))t∈R,ξ), define
S := 0, S := ξ +...+ξ , n ∈ N,
0 n 1 n
and then the renewal shot-noise process Z(·) with random response functions
X (·) by
n
Z(t) := X (t−S ), t ∈ R.
n n−1
n≥1
X
8
3.1 Examples of shot-noise processes
In this subsection, we give some examples of shot-noise processes.
Example 3.1. The current at time t induced by an electron that arrives at
time s at the anode of a vacuum tube equals f(t − s) for some appropriate
deterministic response function f vanishing on thenegative halfline. Assuming
that X(t) = f(t) and the S are the arrival times in a homogeneous Poisson
n
process, the total current at time t equals
Z(t) = f(t−S ), t ≥ 0.
n−1
n≥1
X
This is the classical shot-noise process [36].
Example 3.2. A very popular model in the literature has X(·) in multi-
plicative form X(t) = ηf(t) for a non-negative random variable η and some
deterministic f (see [7, 24, 30, 31, 35, 37] and the references therein). In the
particular case f(t) = eat for some a 6= 0, the corresponding shot-noise process
is a perpetuity, namely
Z(t) = eat e−aSn−1η , t ∈ R.
n
n≥1
X
Themomentresultsforshot-noiseprocesses wearegoingtoderivehereafter
will be a key in the analysis of the moments of N(t), the number of visits to
(−∞,t] of a PRW (T ) . The link between N(t) and shot-noise processes is
n n≥1
disclosed in the following example.
Example 3.3. If X (t) = for a real-valued random variable η , n ≥ 1,
n 1{ηn≤t} n
then Z(t) equals the number of visits to (−∞,t] of the PRW (S +η ) ,
n−1 n n≥1
thus Z(t) = N(t).
3.2 Finiteness of exponential moments of shot-noise pro-
cesses
Our first moment result for shot-noise processes, assuming ξ ≥ 0 a.s., provides
two conditions which combined are necessary and sufficient for the finiteness of
EeaZ(t) for fixed a > 0 and t ∈ R. As before, let τ∗(x) = inf{n ≥ 1 : S > x}.
n
Moreover, we denote by U := P{S ∈ ·} the renewal measure associated
n≥0 n
with (S ) .
n n≥0
P
Theorem 3.4. Let ξ ≥ 0 a.s. Then, for any a > 0 and t ∈ R,
EeaZ(t) < ∞ (3.1)
holds if and only if
r(t) := EeaX(t−y) −1 U(dy) < ∞ (3.2)
Z (cid:18) (cid:19)
τ∗
and l(t) := E eaXn(t−Sn−1) < ∞. (3.3)
!
n=1
Y
Moreover, (3.2) alone implies EeaZ(t0) < ∞ for some t ≤ t.
0
9
Remark 3.5. It is easily seen from the proof given next that we may replace τ∗
in (3.3) by any other (S ) -stopping time τ ≥ τ∗. Note also that, unlike the
n n≥0
case when P{ξ < 0} > 0 to be discussed later, τ∗ coincides with τ := inf{n ≥
1 : ξ > 0} and thus has a geometric distribution with parameter P{ξ > 0}.
n
Finally, (3.3) is a trivial consequence of (3.2) if ξ > 0 a.s.
b
Proof. Observe that
eaZ(t) −1 = eaXn(t−Sn−1) −1 eaXk(t−Sk−1)
n≥1(cid:18) (cid:19)k≥n+1
X Y (3.4)
≥ eaXn(t−Sn−1) −1
n≥1(cid:18) (cid:19)
X
τ∗
and eaZ(t) ≥ eaXn(t−Sn−1) (3.5)
n=1
Y
hold whenever Z(t) < ∞. Taking expectations in the above inequalities there-
fore gives the implications “(3.1)⇒(3.2)” and “(3.1)⇒(3.3)”.
In turn, assume that (3.2) and (3.3) hold. Let (τ∗) be the zero-delayed
n n≥0
renewal sequence of strictly ascending ladder epochs of (S ) , thus τ∗ = τ∗
n n≥0 1
and define
τ∗
L(s) := eaXn(s−Sn−1)
n=1
Y
for s ∈ R. Then EL(s) ≤ EL(t) < ∞ for all s ≤ t, for L(·) is non-decreasing
and (3.3) holds. Pick ε > 0 so small that
EL(s) ≤ β := EL(t) < 1
1{Sτ∗≤ε} 1{Sτ∗≤ε}
for all s ≤ t. Next define Z (·) = Z′(·) = 0 and
0 0
n τ∗+n
Z (·) := X (·−S ), Z′(·) := X (·−(S −S ))
n k k−1 n k k−1 τ∗
k=1 k=τ∗+1
X X
for n ∈ N. Plainly, Z (·) ↑ Z(·) and similarly
n
Z′(·) ↑ Z′(·) := X (·−(S −S ))
n n n−1 τ∗
n≥τ∗+1
X
as n → ∞. Note that each Z′(·) is a copy of Z (·) and further independent of
n n
(L(·),S ). Now observe that
τ∗
Z (t) ≤ Z (t)+Z′ (t)
n τ∗ n−τ∗ 1{τ∗≤n,Sτ∗≤ε}
+Z′ (t−ε)
n−τ∗ 1{τ∗≤n,Sτ∗>ε}
≤ Z (t)+Z′(t) +Z′(t−ε)
τ∗ n 1{Sτ∗≤ε} n 1{Sτ∗>ε}
and therefore, using the stated independence properties,
EeaZn(t) ≤ E L(t) eaZn′(t) +L(t) eaZn′(t−ε)
1{Sτ∗≤ε} 1{Sτ∗>ε}
≤ β E(cid:16)eaZn(t) +EL(t) EeaZn(t−ε) (cid:17) (3.6)
10