Asymptotic results for random walks in
continuous time with alternating rates
Antonio Di Crescenzo
∗
Claudio Macci
†
Barbara Martinucci
‡
Abstract
We investigate some large deviation problems for a random walk in continuous time {N(t); t ≥
0} with spatially inhomogeneous rates of alternating type. We first deal with the large devia-
tion principle for the convergence of N(t)/t to a suitable constant. Then, the case of moderate
deviations is also discussed. Motivated by possible applications in chemical physics context,
we finally obtain an asymptotic lower bound for level crossing probabilities both in the case
of finite and infinite horizon.
AMS Subject Classification: 60F10; 60J27
Keywords: Large deviations, Moderate deviations, Probability generating function.
1 Introduction
Random walks in continuous time are largely employed in population dynamics, queueing the-
ory, epidemiology, and many other areas of both theoretical and applied interest. In this paper
we investigate certain features of skip-free random walks in continuous time over the whole set
of integers, and having alternating rates, which extend some birth-death processes previously
introduced in chemical context. Markov chains with alternating rates are useful in the study
of chain molecular diffusion. We recall the paper [Stockmayer et al. (1971)], where a molecule
is modeled as a freely-joined chain of two regularly alternating kinds of atoms, which have al-
ternating jump rates. Moreover, a simple birth-death process with alternating rates has been
studied in [Conolly et al. (1997)] as a model for an infinitely long chain of atoms joined by
∗
Dipartimento di Matematica, Universit`a di Salerno, Via Giovanni Paolo II n. 132, 84084 Fisciano (SA), Italy.
e-mail: adicrescenzo@unisa.it
†
Dipartimento di Matematica, Universit`a di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Rome, Italy.
e-mail: macci@mat.uniroma2.it
‡
Dipartimento di Matematica, Universit`a di Salerno, Via Giovanni Paolo II n. 132, 84084 Fisciano (SA), Italy.
e-mail: bmartinucci@unisa.it
1
links which are subject to random alternating shocks. Recent results on the transient proba-
bilities of such model, also in the presence of suitable reflecting or absorbing states, are pro-
vided in [Parthasarathy and Dharmaraja (1998)], [Tarabia (2009)], [Tarabia and El-Baz (2007)]
and [Tarabia et al. (2009)]. An investigation on the two-periodic discrete-time random walk
which is obtained by observing the previous process at its jump times has been performed in
[B¨ohm and Hornik (2010)].
A random walk in continuous time characterized by alternating rates, in which the death
rates are shifted with respect to those of [Conolly et al. (1997)], has been studied more recently
in [Di Crescenzo et al. (2012)]. An extension of both such models is proposed and studied in the
present paper. In Section 2 we describe the process and obtain the closed form of the probability
generating function. The expressions of mean and variance are also provided. Section 3 is devoted
to the study of large and moderate deviations for the proposed model. In particular, Section 3.2
deals with an asymptotic lower bound for level crossing probabilities of interest in chemical physics.
We remark that several results in the literature are concerning sample path large deviations
for skip-free random walks in continuous time; see, for instance, the cases of scaled birth-death
processes studied in [Chan (1998)] and in [Pakdaman et al. (2010)]. Sample path large deviations
for general Markov processes are widely studied by Feng and Kurtz in [Feng and Kurtz (2006)];
a formulation of the main result in [Feng and Kurtz (2006)] for d-variate pure jump Markov
processes is Theorem 5.1 in [Shwartz and Weiss (1995)], where the jump rates have to be regular
functions defined on all R
d
. We also recall [Redig and Wang (2012)] where the connection of
Feng-Kurtz theory with several examples of Gibbs-non-Gibbs transition of mean-field type is
illustrated. The results in the present paper are concerning one dimensional random variables.
The interest of this model relies on the fact that our proofs involve only the jump rates on Z.
Moreover, our results give more insight on the role of the parameters in the asymptotic behavior
of the model (see e.g. Remark 3.1 on the limit value of the normalized process).
2 The model and preliminary results
Let {N(t); t ≥ 0} be a skip-free random walk in continuous time with state-space Z, also known
as ‘bilateral birth-death process’. We denote its transition probabilities by
p
k,n
(t) = P {N(t) = n |N(0) = k}, t ≥ 0, n ∈ Z,
where k ∈ Z is the initial state. We assume that, for all n ∈ Z, the upward transition rates are
given by
λ
n
= lim
h→0
1
h
P {N(t + h) = n + 1 |N(t) = n} =
(
λη + µ(1 − η), n even,
µη + λ(1 − η), n odd,
(1)
2
whereas the downward transition rates are
µ
n
= lim
h→0
1
h
P {N(t + h) = n − 1 |N(t) = n} =
(
λθ + µ(1 − θ), n even,
µθ + λ(1 − θ), n odd,
(2)
where λ, µ > 0, and θ, η ∈ [0, 1].
According to the criteria developed in [Pruitt (1963)], from assumptions (1) and (2) it follows
that both boundaries −∞ and +∞ are natural. We remark that the transition rates (1) and
(2) have been defined as suitable extension of periodic rates treated in other models. Indeed,
we notice that when (θ, η) = (0, 1) the above model identifies with the model investigated by
[Conolly et al. (1997)] and [Tarabia et al. (2009)], whereas when (θ, η) = (1, 1) the rates (1) and
(2) refer to the bilateral birth-death process studied in [Di Crescenzo et al. (2012)]. Moreover, we
notice that in the special case λ = µ, process {N(t); t ≥ 0} identifies with the time-continuous
nearest neighbor random walk with rate λ, studied in [Conolly (1971)].
Hereafter and throughout the paper we adopt the notation E
k
[ ·] = E
k
[ · |N(0) = k], and
similarly Var
k
[ ·] = Var[ · |N(0) = k].
Let us now obtain the probability generating function of N(t), namely
F
k
(z, t) := E
k
z
N(t)
z > 0, t ≥ 0, (3)
for any fixed initial state k ∈ Z. To this purpose, for all z, α, β > 0 we define the following
function:
C(z; α, β) := [ηα + (1 − η)β]z
2
+ (α − β)(1 − θ − η)z + θα + (1 − θ)β. (4)
Note that C(z; α, β) > 0 for all z ≥ 0. Indeed, if (α − β)(1 − θ − η) ≥ 0 then C(0; α, β) > 0 and
C(z; α, β) is increasing for z > 0, whereas if (α − β)(1 − θ − η) < 0 then the discriminant of the
quadratic polynomial in the right-hand-side of (4) is negative.
Proposition 2.1 For z > 0, t ≥ 0 and k ∈ Z, the explicit expression of the probability generating
function of N(t) is:
F
k
(z, t) = e
−(λ+µ)t
z
k
cosh
t
h(z)
z
+
c
k
(z)
h(z)
sinh
t
h(z)
z
, (5)
where
c
k
(z) =
C(z; λ, µ) if k is even
C(z; µ, λ) if k is odd,
(6)
with C(z; ·, ·) defined in (4), and
h(z) =
p
(µz
2
+ λ)(λz
2
+ µ) + (λ − µ)
2
(1 − z
2
)[θ(1 − θ) − z
2
η(1 − η)]. (7)
3
Proof. For any t ≥ 0, n ∈ Z and for any initial state k ∈ Z, the transition probabilities of N(t)
satisfy the following system of differential-difference equations:
d
dt
p
k,2n
(t) = [µη + λ(1 − η)] p
k,2n−1
(t) + [µθ + λ(1 − θ)] p
k,2n+1
(t)
−[λη + µ(1 − η) + λθ + µ(1 − θ)] p
k,2n
(t),
d
dt
p
k,2n+1
(t) = [λη + µ(1 − η)] p
k,2n
(t) + [λθ + µ(1 − θ)] p
k,2n+2
(t)
−[µη + λ(1 − η) + µθ + λ(1 − θ)] p
k,2n+1
(t),
(8)
with initial condition p
k,n
(0) = δ
n,k
, where δ
n,k
is the Kronecker’s delta. For z > 0, let
G
k
(z, t) :=
+∞
X
j=−∞
z
2j
p
k,2j
(t), H
k
(z, t) :=
+∞
X
j=−∞
z
2j+1
p
k,2j+1
(t), (9)
be the probability generating functions of the sets of even and odd states of N(t), respectively.
From system (8) it follows that the generating functions (9) satisfy the differential system
∂
∂t
"
G
k
(z, t)
H
k
(z, t)
#
= A ·
"
G
k
(z, t)
H
k
(z, t)
#
, (10)
where
A :=
−[λη + µ(1 − η) + λθ + µ(1 − θ)]
µθ + λ(1 − θ) + z
2
[µη + λ(1 − η)]
z
λθ + µ(1 − θ) + z
2
[λη + µ(1 − η)]
z
−[µη + λ(1 − η) + µθ + λ(1 − θ)]
,
with initial conditions
G
k
(z, 0) =
(
z
k
, k even
0, k odd,
H
k
(z, 0) =
(
0, k even
z
k
, k odd.
Hence, by solving system (10) and recalling relation F
k
(z, t) = G
k
(z, t) + H
k
(z, t) we immediately
come to Eq. (5).
We are now able to obtain the mean of N(t) conditional on the initial state. This is expressed
in terms of
M(α, β) :=
(α − β)(η − θ)[(α − β)(η + θ − 1) + (α + β)]
2(α + β)
2
. (11)
Proposition 2.2 For any initial state k ∈ Z and for all t ≥ 0 the explicit expression of the mean
of N(t) is:
E
k
[N(t)] = k + m
k
1 − e
−2(λ+µ)t
+
(λ − µ)
2
[η(1 − η) −θ(1 − θ)]
λ + µ
t, (12)
where
m
k
=
M(λ, µ) if k is even
M(µ, λ) if k is odd,
with M(·, ·) defined in (11).
4
Remark 2.1 From Proposition 2.2 we have
lim
t→+∞
1
t
E
k
[N(t)] =
(λ − µ)
2
[η(1 − η) −θ(1 − θ)]
λ + µ
=: `
1
,
where `
1
= 0 if λ = µ, or θ = η, or θ = 1 − η. We also note that, if λ 6= µ, we have:
• `
1
> 0 if and only if η(1 − η) > θ(1 − θ), i.e. if and only if |η −
1
2
| < |θ −
1
2
|;
• `
1
< 0 if and only if η(1 − η) < θ(1 − θ), i.e. if and only if |η −
1
2
| > |θ −
1
2
|.
Let us now obtain the variance of N(t) conditional on the initial state.
Proposition 2.3 For all t ≥ 0, if k is even the conditional variance of N (t) is:
Var
k
[N(t)] = a
1
t + a
2
t
1 − 2e
−2(λ+µ)t
− a
3
1 − e
−2(λ+µ)t
2
− a
4
1 − e
−2(λ+µ)t
, (13)
where
a
1
=
1
(λ + µ)
3
n
4(1 − η)η(λ −µ)
2
(λ + µ)
2
+ 4λµ(λ + µ)
2
+ (λ − µ)
2
(η − θ)(−1 + η + θ)
×[2(λ + µ)
2
− 2(λ − µ)
2
(η − θ)(−1 + η + θ) − (λ − µ)(λ + µ)(η − θ)]
o
,
a
2
=
1
(λ + µ)
3
(λ − µ)
3
(η − θ)
2
(−1 + η + θ)[(λ − µ)(−1 + η + θ) + λ + µ],
a
3
=
(λ − µ)
2
(η − θ)
2
[(λ − µ)(−1 + η + θ) + λ + µ]
2
4(λ + µ)
4
,
a
4
=
(λ − µ)
2(λ + µ)
4
n
(2η − 1)(λ − µ)(2θ − 1)
h
2(λ − µ)
2
(−1 + η + θ)
2
+ [(−1 + η + θ)(λ − µ) + λ + µ]
2
i
− 3(λ − µ)
3
(−1 + η + θ)
4
− (λ + µ)(−1 + η + θ)
×
h
(λ − µ)
2
(−1 + η + θ)
2
+ [(−1 + η + θ)(λ − µ) + λ + µ]
2
io
.
If k is odd, Var
k
[N(t)] is obtained by Eq. (13) by interchanging λ with µ.
The proof of Propositions 2.2 and 2.3 is omitted, since it follows from Eq. (5) via straightfor-
ward calculations.
Remark 2.2 From Proposition 2.3 we have
lim
t→+∞
1
t
Var
k
[N(t)] =
1
(λ + µ)
3
−(η − θ)
2
(1 − η − θ)
2
(λ − µ)
4
− 2(η − θ)(1 −η − θ)(λ
2
− µ
2
)
2
+4η(1 − η)(λ
2
− µ
2
)
2
+ 4λµ(λ + µ)
2
:= `
2
.
Note that `
2
> 0 for all λ, µ > 0, 0 ≤ η ≤ 1 and 0 ≤ θ ≤ 1. Clearly, it is `
2
= a
1
+ a
2
.
5