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Large deviation principles for telegraph processes
Statistics and Probability Letters 82 (2012) 1874–1882
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Large deviation principles for telegraph processes Alessandro De Gregorio a , Claudio Macci b,∗ a
Dipartimento di Scienze Statistiche, Sapienza Università di Roma, Piazzale Aldo Moro 5, I-00185 Rome, Italy
b
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Rome, Italy
article
info
Article history: Received 7 February 2012 Received in revised form 20 June 2012 Accepted 22 June 2012 Available online 28 June 2012 MSC: 60F10 60J27
abstract The aim of this paper is to present large deviation results for some telegraph random motions. We are not aware of any other results of this kind except the ones for the classical telegraph process (with drift). We start with the large deviation principle of the conditional laws given the number of changes of direction for the classical case; moreover, we compare the rate function with the one obtained for the non-conditional distributions. Finally, we study an inhomogeneous model and a planar telegraph motion. © 2012 Elsevier B.V. All rights reserved.
Keywords: Conditional laws Inhomogeneous Poisson processes Markov additive processes Random motions
1. Introduction The stochastic processes are often used to describe the real motions. An important example is given by the Brownian motion, which was introduced in order to describe the movement of the particles in a liquid. However the Brownian motion is not free from defects such as the unbounded first variation. Therefore, over the years, many authors have introduced random models having the main characteristics of the real movements: finite velocity and persistence. A typical example in this direction is the telegraph process, which is represented by a particle moving on the real line, alternatively forward and backward, with finite speed. Furthermore, one suppose that the changes of direction are governed by a homogeneous Poisson process. The probabilistic properties of this random model have been analyzed, for example, in Beghin et al. (2001), Fong and Kanno (1994) and Orsingher (1990, 1995). Other authors have also studied generalizations of the telegraph process, concerning the waiting times between two consecutive changes of direction or the randomization of the velocity (see Di Crescenzo (2001), Stadje and Zacks (2004), Zacks (2004), Di Crescenzo and Martinucci (2010) and De Gregorio (2010)). It is interesting to observe that applications of the telegraph process emerge in different fields. For instance, in physics the propagation of a damped wave along a wire is described by the telegraph equation. In order to model the dynamics of the price of risky assets, Di Crescenzo and Pellerey (2002) have considered the geometric telegraph process. A model of financial market based on a telegraph process with drift, that is with two different velocities and switching rates, has been introduced by Ratanov (2007). In this paper, we present large deviation results for some telegraph random motions. We are not aware of any other results of this kind except Proposition 2.1 recalled below; this result concerns the non-conditional laws for the classical case (with drift). Both Proposition 2.1 and the results in this paper, concern the large-time asymptotic behavior.
∗
Corresponding author. E-mail addresses: [email protected] (A. De Gregorio), [email protected] (C. Macci).
0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.06.023
A. De Gregorio, C. Macci / Statistics and Probability Letters 82 (2012) 1874–1882
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The theory of large deviations gives an asymptotic computation of small probabilities on exponential scale; estimates based on large deviations play a crucial role in resolving a variety of questions in statistics, engineering, statistical mechanics and applied probability. We recall the basic definitions in Dembo and Zeitouni (1998, pp. 4–5). Let Z be a Hausdorff topological space with Borel σ -algebra BZ . A lower semi-continuous function I : Z → [0, ∞] is called the rate function. A family of probability measures {νt : t > 0} on (Z, BZ ) satisfies the large deviation principle (LDP), as t → ∞, with rate function I if lim sup
1
log νt (F ) ≤ − inf I (z ) for all closed sets F
t
t →∞
z ∈F
and lim inf t →∞
1 t
log νt (G) ≥ − inf I (z ) z ∈G
for all open sets G.
A rate function I is said to be good if all the level sets {{z ∈ Z : I (z ) ≤ γ } : γ ≥ 0} are compact. In view of the proof of Proposition 2.2 below, we recall that the lower bound for open sets is equivalent to condition (b) with Eq. (1.2.8) in Dembo and Zeitouni (1998), i.e.: lim inf t →∞
1 t
log νt (G) ≥ −I (z )
for all z ∈ Z such that I (z ) < ∞ and for all open sets G such that z ∈ G.
(1.1)
For completeness we say that a family of Z-valued random variables {Zt : t > 0} satisfies the LDP if the family of probability measures {P (Zt ∈ ·) : t > 0} does (in the sense of the above definition). We conclude with the outline of the paper. In Section 2, we present two LDPs for the classical telegraph process (with drift): Proposition 2.1 recalls a known LDP for non-conditional distributions; Proposition 2.2 provides a new LDP for the conditional laws given the number of changes of direction. In the proof of Proposition 2.2 we handle the conditional densities in a standard way. Furthermore, we present inequalities between the rate functions in Propositions 2.1–2.2 which allow to compare the convergence of non-conditional and conditional laws. We conclude with two other LDPs in Sections 3–4, respectively: the first one concerns an inhomogeneous model in Iacus (2001), the second result concerns a planar telegraph motion which is a slight generalization of the random processes in Orsingher (2000) and Orsingher and Ratanov (2002). We remark that we do not need to know explicitly the expression of the laws (as happens in the proof of Proposition 2.2) because we can combine the LDP in Proposition 2.1 with other large deviation tools: Theorem 4.2.13 in Dembo and Zeitouni (1998) concerning exponentially equivalent families of laws in Section 3, and the contraction principle (see e.g. Theorem 4.2.1 in Dembo and Zeitouni (1998)) in Section 4. 2. The classical telegraph process with drift The telegraph process with drift is a random motion {X (t ) : t ≥ 0} on the real line which starts at the origin of R and moves with a two-valued integrated telegraph signal: for some λ1 , λ2 , c1 , c2 > 0, we have a rightward velocity c1 , a leftward velocity −c2 , and the rates of the occurrences of velocity switches are λ1 and λ2 , respectively. More precisely, we have X (t ) =
t 0
c1 − c2 2
+
c1 + c2 1{V (0)=c1 } − 1{V (0)=−c2 } (−1)N (s) ds 2
where the random variable V (0) is such that P (V (0) ∈ {−c2 , c1 }) = 1; the process {N (t ) : t ≥ 0} which counts the number of changes of direction is an inhomogeneous Poisson process such that N (t ) = n≥1 1{S1 +···+Sn ≤t } , the random variables {Sn : n ≥ 1} are conditionally independent given V (0), and the conditional law is the following:
{S2k−1 : k ≥ 1}
are exponentially distributed with mean
1
λ1 1 {S2k : k ≥ 1} are exponentially distributed with mean ; λ2 1 {S2k−1 : k ≥ 1} are exponentially distributed with mean λ 2 if V (0) = −c2 , then 1 {S2k : k ≥ 1} are exponentially distributed with mean . λ1 In the literature the case c1 = c2 and λ1 = λ2 is said to be the case without drift. We remark that, if we have λ1 = λ2 = λ for some λ > 0, {N (t ) : t ≥ 0} is a Poisson process with intensity λ and, moreover, V (0) and {N (t ) : t ≥ 0} are independent (because V (0) and {Sn : n ≥ 1} are independent). Finally, we allow a general initial distribution of the initial velocity V (0), if V (0) = c1 ,
then
i.e. we set
(P (V (0) = c1 ), P (V (0) = −c2 )) = (q, 1 − q) for some q ∈ [0, 1], which has no influence in the expression of the rate functions below.
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We start with a known large deviation result which can be proved by referring to more general results for Markov additive processes (see e.g. Iscoe et al. (1985) and Ney and Nummelin (1987a,b,c)); see also Macci (2009, Section 3). We remark that there is a link between the tail behavior of the rare events studied in this paper and the sample paths of the process, and in particular the support of the paths. Actually the rate function IλX1 ,λ2 ,c1 ,c2 is not finite outside the closed interval [−c2 , c1 ] because we have P
X (t ) t
∈ [−c2 , c1 ] = 1, and we can consider the lower bound for the open set G = R \ [−c2 , c1 ] in the
definition of LDP.
Proposition 2.1. The family P
IλX1 ,λ2 ,c1 ,c2 (x) =
X (t ) t
∈ · : t > 0 satisfies the LDP with good rate function IλX1 ,λ2 ,c1 ,c2 defined by
2 c1 − x − λ2 λ1 c1 + c2 c1 + c2 x + c2
∞
if x ∈ [−c2 , c1 ] otherwise.
Remark 2.1. Note that, if λ1 = λ2 = λ and c1 = c2 = c for some λ, c > 0, i.e. {X (t ) : t ≥ 0} is a standard telegraph process without drift, the rate function in Proposition 2.1 becomes
x2 λ 1− 1− 2 X Iλ,λ, c c ,c (x) = ∞
if x ∈ [−c , c ] otherwise.
Now we prove the LDP for the conditional laws assuming that λ1 = λ2 = λ for some λ > 0. We remark that, since X |N the conditional densities {fn (·, t ) : n ≥ 1, t > 0} below do not depend on λ, the rate function Iλ,λ,c1 ,c2 (·; w) in the next proposition has the same feature. Proposition 2.2. Assume that λ1 = λ 2 = λ > 0. Let λ for some t → wt be a deterministic function such that limt →∞ wt = w N (t ) X |N X (t ) for some w ∈ (0, ∞). Then the family P ∈ · t = wt : t > 0 satisfies the LDP with good rate function Iλ,λ,c1 ,c2 (·; w) t defined by X |N
Iλ,λ,c1 ,c2 (x; w) =
w log
∞
c1 + c2 √ 2 (c2 + x)(c1 − x)
if − c2 < x < c1 otherwise.
Proof. We refer to the conditional laws P (X (t ) ∈ dy|N (t ) = n) = fn (y, t )dy (for n ≥ 1), where f2k+1 (y, t ) =
(2k + 1)! (c2 t + y)k (c1 t − y)k 1{−c2 t
and f2k (y, t ) =
(2k)! (c2 t + y)k−1 (c1 t − y)k−1 {q(c2 t + y) + (1 − q)(c1 t − y)}1{−c2 t
for k ≥ 1. The conditional densities {fn (·, t ) : n ≥ 1, t > 0} are the extension of Eqs. (2.17)–(2.18) in De Gregorio et al. (2005) which concern the case q = 21 and c1 = c2 = c for some c > 0; see Appendix for details. The proof is divided into two parts. (1) Proof of the lower bound for open sets. We want to check the equivalent condition (1.1), i.e. lim inf t →∞
1 t
log P
X (t ) t
N (t ) ∈ G = wt ≥ −w log √
c1 + c2
2 (c2 + x)(c1 − x)
t
for all x such that −c2 < x < c1 and for all open sets G such that x ∈ G. First we can take ε > 0 small enough to have (x − ε, x + ε) ⊂ G ∩ (−c2 , c1 ). Then, for some yε,t ∈ (x − ε, x + ε), we have N (t ) N (t ) X (t ) X (t ) ∈ G = wt ≥ P ∈ (x − ε, x + ε) = wt P t
t
t
(x+ε)t
= (x−ε)t
t
ft wt (y, t )dy =
x+ε
ft wt (yt , t )tdy = 2ε ft wt (yε,t t , t )t . x−ε
A. De Gregorio, C. Macci / Statistics and Probability Letters 82 (2012) 1874–1882
Hence, since limn→∞ lim inf
1
t →∞
t
log(n!) n log(n)
log P
≥ lim inf t →∞
= 1, we obtain N (t ) X (t ) ∈ G = wt t
1
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t
t
t wt log(t wt ) − 2
t wt 2
w
= w log 2 − w log(c1 + c2 ) +
log
t wt 2
− t wt log(c1 + c2 ) +
t wt 2
log{(c2 + x − ε)(c1 − (x + ε))}
log{(c2 + x − ε)(c1 − x − ε)} = −w log
2
c1 + c2 , √ 2 (c2 + x − ε)(c1 − x − ε)
and we complete the proof of the lower bound letting ε go to zero. (2) Proof of the upper bound for closed sets. We have to check lim sup
1 t
t →∞
log P
X (t ) t
N (t ) X |N ∈ F = wt ≤ − inf Iλ,λ,c1 ,c2 (x; w) for all closed sets F . x∈F t X |N
c1 − c2 ∈ F (we would have infx∈F Iλ,λ,c1 ,c2 (x; w) = 0) and if F ∩ (−c2 , c1 ) = ∅ 2 N (t ) ∈ F t = wt = 0). Thus, from now on, we assume that c1 −2 c2 ̸∈ F and F ∩ (−c1 , c2 ) ̸= ∅. For the
First note that this condition trivially holds if (we would have P
X (t ) t
moment we also assume that
F∩
c1 − c2 2
, c1 ,
F∩
c1 − c2 −c2 , ̸= ∅. 2
X |N
X |N
We remark that Iλ,λ,c1 ,c2 (·; w) is decreasing in (−c2 , 1 2 2 ] and Iλ,λ,c1 ,c2 (·; w) is increasing in [ 1 2 2 , c1 ); then, if we set c −c c −c c −c xF := sup(F ∩ (−c2 , 1 2 2 )) and xF := inf(F ∩ ( 1 2 2 , c1 )), we have F ⊂ (−∞, xF ] ∪ [xF , ∞), where xF ∈ (−c2 , 1 2 2 ) and xF ∈ (
c −c
c −c
c t , c1 ). Then, since we have P X (tt ) ≥ xF N (tt ) = wt = xF1t ft wt (y, t )dy, we get N (t ) X (t ) ≥ xF = wt ≤ ft wt (xF t , t )(c1 − xF )t (if t wt ∈ {1, 3, 5, . . .}) P c1 − c2 2
t
t
and, since q(c2 t + y) + (1 − q)(c1 t − y) ≤ (c1 + c2 )t for y ∈ (−c2 t , c1 t ),
P
N (t ) (t wt )! ≥ xF = wt ≤ t wt t wt t t wt t wt t ! − 1 !(( c + c ) t ) 1 2 2 2
X (t ) t
· {(c2 + xF )(c1 − xF )}
t wt 2
−1
(c1 + c2 )(c1 − xF ) (if t wt ∈ {2, 4, 6, . . .}).
log(n!) n log(n)
= 1 as in the first part of the proof, we obtain N (t ) X (t ) 1 ≥ xF = wt lim sup log P t t t t →∞ 1 t wt t wt t wt ≤ lim sup t wt log(t wt ) − 2 log − t wt log(c1 + c2 ) + log{(c2 + xF )(c1 − xF )}
Thus, by taking into account that limn→∞
t →∞
t
= w log 2 − w log(c1 + c2 ) +
w 2
2
2
2
X |N
log{(c2 + xF )(c1 − xF )} = −Iλ,λ,c1 ,c2 (xF ; w).
Similarly, we can also check the inequality lim sup t →∞
1 t
log P
X (t ) t
N (t ) X |N ≤ xF = wt ≤ −Iλ,λ,c1 ,c2 (xF ; w). t
Then, by the inclusion F ⊂ (−∞, xF ] ∪ [xF , ∞), by Lemma 1.2.15 in Dembo and Zeitouni (1998) and the inequalities above, we get lim sup t →∞
1 t
log P
X (t ) t
N (t ) X |N X |N ∈F = wt ≤ max{−Iλ,λ,c1 ,c2 (xF ; w), −Iλ,λ,c1 ,c2 (xF ; w)} t
X |N
X |N
X |N
= − min{Iλ,λ,c1 ,c2 (xF ; w), Iλ,λ,c1 ,c2 (xF ; w)} = − inf Iλ,λ,c1 ,c2 (x; w). x∈F
Finally we remark that if F ∩ (
c1 − c2 2
, c1 ) is empty, we can adapt the above procedure by considering the inclusion c1 − c2 ) is empty, we can adapt the above procedure by considering the inclusion 2
F ⊂ (−∞, xF ]; similarly, if F ∩ (−c2 , F ⊂ [xF , ∞).
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X |N
Fig. 1. The functions I1X,1,1,2 (x) and I1,1,1,2 (x; w), where x ∈ (−2, 1). Two choices for w : w = 1.5 ≥ 1; w = 0.6 ∈ (0, 1).
Remark 2.2. A close inspection ofthe proof (and of the content of the Appendix) reveals that we can also prove the LDP for P
X (t ) t
∈ · N (tt ) = wt , V (0) = v : t > 0 for v ∈ {−c2 , c1 }, and we have the same rate function. X |N
We remark that Iλ,λ,c1 ,c2 (·; w) in Proposition 2.2 uniquely vanishes at 1 2 2 and, if λ1 = λ2 , this also happens for IλX1 ,λ2 ,c1 ,c2 in Proposition 2.1. So, in order to compare the convergence of non-conditional and conditional laws, it is interesting to find c −c strict inequalities between these two rate functions in a neighborhood of 1 2 2 (except this point) when λ1 = λ2 . c −c
Proposition 2.3. We have two cases. X |N
c1 − c2 X (i) For w ≥ λ, we have Iλ,λ, c1 ,c2 (x) ≤ Iλ,λ,c1 ,c2 (x; w) for all x ∈ R; moreover, the inequality is strict for x ∈ [−c2 , c1 ]\{ 2 }. 2 (ii) For w ∈ (0, λ) set α± = 21 c1 − c2 ± 1 − wλ2 (c1 + c2 ) ; then there exist β+ ∈ (α+ , c1 ) and β− ∈ (−c2 , α− ) such X |N
X |N
X X that Iλ,λ,c1 ,c2 (x; w) > Iλ,λ, c1 ,c2 (x) for x ∈ [−c2 , β− ) ∪ (β+ , c1 ], Iλ,λ,c1 ,c2 (x; w) < Iλ,λ,c1 ,c2 (x) for all x ∈ (β− , β+ ) \ { X |N X Iλ,λ,c1 ,c2 (x; w) = Iλ,λ, c1 ,c2 (x) otherwise. X |N
Proof. First note that, for all w > 0, we have Iλ,λ,c1 ,c2 (
c1 − c2 2
} and
X |N
c1 −c2 X X ; w) = Iλ,λ, c1 ,c2 ( 2 ) = 0; Iλ,λ,c1 ,c2 (x; w) = Iλ,λ,c1 ,c2 (x) = ∞ X for x ̸∈ [−c2 , c1 ]; λ = Iλ,λ, c1 ,c2 (x) < Iλ,λ,c1 ,c2 (x; w) = ∞ for x ∈ {−c2 , c1 }. Then (i) and (ii) in the statement can be checked by considering the difference function (for x ∈ (−c2 , c1 )) defined by √ c1 + c2 2 (c2 + x)(c1 − x) X |N X ∆(x) := Iλ,λ,c1 ,c2 (x; w) − Iλ,λ, − λ 1 − , ( x ) = w log √ c1 , c2 c1 + c2 2 (c2 + x)(c1 − x) c −c √ x− 1 2 2λ (c2 +x)(c1 −x) and studying its derivative ∆′ (x) = (c +x)(c2 −x) w − . c1 + c2 2 1 c1 − c2 2
X |N
Thus, as depicted in Fig. 1 below, for x in a neighborhood of X |N
c1 − c2 2
X |N
X (except this point) we have Iλ,λ,c1 ,c2 (x; w) > Iλ,λ, c1 , c2 ( x )
if w ≥ λ and Iλ,λ,c1 ,c2 (x; w) < Iλ,λ,c1 ,c2 (x) if w ∈ (0, λ). Then, if λ1 = λ2 = λ, in some sense the convergence of the c −c conditional laws to the limit 1 2 2 is faster than the convergence of the non-conditional laws if and only if w ≥ λ. We remark that it is not surprising to have a faster convergence of the conditional laws as w increases; indeed the more w is large, the more the changes of direction are frequent. We conclude this section by pointing out a connection between Proposition 2.2 and a LDP in Macci (2008). Let T0 = 0 and {Tn : n ≥ 1} are the epochs of the occurrences of the transition number process, i.e. N (t ) = n≥1 1{Tn ≤t } ; moreover, let {Jn : n ≥ 0} be the random sequence defined by Jn = 1 (resp. Jn = 2) if X (t ) is increasing (resp. decreasing) for t ∈ [Tn , Tn+1 ). Here we are interested in the sequence {(πˆ n(1) , πˆ n(2) ) : n ≥ 1} defined by n−1
πˆ n(i) =
X
(Tk+1 − Tk )1{Jk =i}
k=0
Tn
.
A. De Gregorio, C. Macci / Statistics and Probability Letters 82 (2012) 1874–1882
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(1)
(2)
Then, by Proposition 3.2 in Macci (2008), for any fixed integer w ≥ 1 the sequence of random variables {(πˆ nw , πˆ nw ) : n ≥ 1} satisfies the LDP with good rate function Hw defined by Hw (π1 , π2 ) := w H1 (π1 , π2 ), where H1 (π1 , π2 ) =
1
log
2
λ1 π1 + λ2 π2
λ2 π2 λ1 π1
if π1 , π2 ∈ (0, 1) and π1 + π2 = 1 otherwise.
∞
Thus, if we follow the lines of the proof of Proposition 2.1 illustrated in Macci 3),we get the LDP of (2009, Section 1) 2) {c1 πˆ n(w − c2 πˆ n(w : n ≥ 1} (or, equivalently, the LDP of the conditional laws P X (nn) ∈ · N (nn) = w : n ≥ 1 ), and the rate function ˜IλX1 ,λ2 ,c1 ,c2 (·; w) is defined by
˜IλX ,λ
2 , c1 , c2
1
(x; w) := inf{Hw (π1 , π2 ) : c1 π1 − c2 π2 = x}.
X Finally, it is easy to check that ˜Iλ,λ, c1 ,c2 (·; w) coincides with the rate function in Proposition 2.2.
3. An inhomogeneous telegraph process Iacus (2001) has shown that the finite dimensional law of an inhomogeneous telegraph process driven by a Poisson process with intensity {λ(t ) : t ≥ 0} is a solution of the telegraph equation with nonconstant coefficients. A rare example for which the finite dimensional law can be provided explicitly is the process {Yθ (t ) : t ≥ 0} defined by Yθ (t ) = V (0)
t
(−1)Nθ (s) ds,
(3.1)
0
where the random variable V (0) is such that P (V (0) = c ) = P (V (0) = −c ) = 21 ; the process {Nθ (t ) : t ≥ 0} is an inhomogeneous Poisson process with intensity {λθ (t ) : t ≥ 0} defined by λθ (t ) := θ tanh(θ t ); V (0) and {Nθ (t ) : t ≥ 0} are independent. The main goal of this section is to prove the LDP for the non-conditional laws of
Yθ ( t ) t
: t > 0 . The method used
for from the one in Proposition 2.1; indeed we take into account Proposition 2.1 and we show that the proofis different X (t ) t
: t > 0 and
Yθ (t ) t
: t > 0 are exponentially equivalent as t → ∞ in the sense of Definition 4.2.10 in Dembo and
Zeitouni (1998). Proposition 3.1. The family
P
Yθ (t ) t
∈ · : t > 0 satisfies the LDP with good rate function IθX,θ ,c ,c as in Proposition 2.1
(see Remark 2.1). Proof. We shall show that
Xθ (t ) Yθ (t ) >δ lim sup log P − = −∞ for all δ > 0 (3.2) t t t t →∞ Yθ (t ) for a suitable version of the processes : t > 0 and Xθt(t ) : t > 0 defined on the same probability space, where t {Xθ (t ) : t ≥ 0} is the process {X (t ) : t ≥ 0} in Proposition 2.1 with λ1 = λ2 = θ and c1 = c2 = c. This gives the proof because, if condition (3.2)holds, the LDP is an immediate consequence of Theorem 4.2.13 in Dembo and Zeitouni (1998) and X (t ) the LDP for θt : t > 0 (i.e. Proposition 2.1 with λ1 = λ2 = θ and c1 = c2 = c). In what follows, we check (3.2) assuming that both the processes {Yθ (t ) : t ≥ 0} and {Xθ (t ) : t ≥ 0} have the same random initial velocity V (0) such that P (V (0) = c ) = P (V (0) = −c ) = 21 ; thus we have (3.1) and t ∗ Xθ (t ) = V (0) (−1)Nθ (s) ds 1
0
for a homogeneous Poisson process {Nθ∗ (t ) : t ≥ 0} with intensity θ . Moreover, for a homogeneous Poisson process
{M (t ) : t ≥ 0} with intensity 1, we set Nθ∗ (t ) := M (θ t ) and Nθ (t ) := M (Λθ (t )), where Λθ (t ) :=
Then we have Xθ (t ) t
:=
V (0)
t
t
(−1)M (θ s) ds =
0
V (0)
θt
θt
(−1)M (r ) dr
0
and Yθ (t ) t
:=
V (0) t
0
t
M (Λθ (s))
(−1)
ds =
V (0)
θt
0
Λθ (t )
(−1)
er
M (r )
er
√
√
e2r − 1 + e2r
e2r − 1 + e2r − 1
dr ;
t 0
λθ (u)du = log cosh(θ t ).
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A. De Gregorio, C. Macci / Statistics and Probability Letters 82 (2012) 1874–1882
1 the latter equality can be derived by considering the change of variable r = Λθ (s), whence we obtain s = Λ− θ (r ) := 1
θ
log(er +
√
e2r − 1). Thus we get
√ Λθ (t ) θt er e2r − 1 + e2r M (r ) M (r ) dr (−1) (−1) dr − √ 0 er e2r − 1 + e2r − 1 0 Λθ (t ) 1 c dr + θ t − Λθ (t ) =: ac ,θ (t ) ≤ √ θt er e2r − 1 + e2r − 1 0
Xθ (t ) Yθ (t ) = c − t t θt
(where ac ,θ (t ) is deterministic) and, by the L’Hopital Theorem, we have lim ac ,θ (t ) =
t →∞
=
θ
lim
t →∞
eΛθ (t )
√
e2Λθ (t ) − 1 + e2Λθ (t ) − 1
c
θ
Λ′θ (t )
c
lim θ
t →∞
+ θ − Λ′θ (t )
1 + tanh(θ t )
cosh(θ t ) cosh2 (θ t ) − 1 + cosh2 (θ t ) − 1
because Λ′θ (t ) = θ tanh(θ t ). In conclusion (3.2) holds because
{ac ,θ (t ) > δ} is empty for t large enough.
1
Xθ (t ) t −
Yθ (t ) t
−1
=0
> δ ⊂ {ac ,θ (t ) > δ} and the event
Proposition 3.1 can be proved with the same techniques used above for the proof of Proposition 2.2 and by considering the density provided by Theorem 2.1 in Iacus (2001). This density can be expressed in terms of the modified Bessel function of the first kind I1 (r ) := are omitted.
(r /2)1+2k k=0 k!(k+1)!
∞
and we also have to take into account the limit
log I1 (x) x
→ 1 as x → ∞. The details
4. A planar telegraph motion In this section we consider a planar telegraph motion {(Y1 (t ), Y2 (t )) : t ≥ 0} with four directions and four speeds, switching at exponential times, which is a slight generalization of the random motion considered in Orsingher and Ratanov (2002). This process is introduced by means of two independent telegraph motions with drift {X1 (t ) : t ≥ 0} and {X2 (t ) : t ≥ 0}. More precisely, for λ0 , λ1 , λ2 , λ3 > 0 and c0 , c1 , c2 , c3 > 0, we consider the process {X1 (t ) : t ≥ 0} with switching rates (λ0 /2, λ2 /2) and velocities (c0 , −c2 ), the process {X2 (t ) : t ≥ 0} with switching rates (λ1 /2, λ3 /2) and velocities (c1 , −c3 ), and the process {(Y1 (t ), Y2 (t )) : t ≥ 0} defined by
c0 c1 X1 (t ) + c2 X2 (t ) Y ( t ) = 1 c c +c 1
0
2
Y2 (t ) = c0 X2 (t ) − c1 X1 (t ) . c0 + c2
Note that, by Proposition 2.1, P
X1 (t ) t
∈ · : t > 0 and P X2t(t ) ∈ · : t > 0 satisfy the LDP with good rate functions
I1 = IλX0 /2,λ2 /2,c0 ,c2 and I2 = IλX1 /2,λ3 /2,c1 ,c3 , respectively.
, Y2t(t ) ∈ · : t > 0 . We remark that we can obtain explicitly such non-conditional laws when the random initial velocities of {X1 (t ) : t ≥ 0} and {X2 (t ) : t ≥ 0}, denoted by V1 (0) and V2 (0), are such that The aim is to prove the LDP for the non-conditional laws P
Y1 (t ) t
P (V1 (0) = c0 ) = P (V1 (0) = −c2 ) = P (V2 (0) = c1 ) = P (V2 (0) = −c3 ) =
1 2
;
see Orsingher and Ratanov (2002) for the case λ0 = λ1 = λ2 = λ3 , and see Orsingher (2000) if we also have c0 = c1 = c2 = c3 . We also remark that, if λ0 = λ1 = λ2 = λ3 = λ for some λ, the number of changes of directions are governed by a Poisson process with intensity λ; more precisely, when a Poisson event occurs with rate λ, the motion takes – with probability 21 – one of the two directions orthogonal to the previous one.
It is interesting that we can avoid to handle the non-conditional laws P
Y1 (t ) t
, Y2t(t ) ∈ · : t > 0 . Actually the LDP
will be proved by a simple application of the contraction principle (see e.g. Theorem 4.2.1 in Dembo and Zeitouni (1998)): roughly speaking, if a family of probability measures {νt : t > 0} on (Z, BZ ) satisfies the LDP with a good rate function I and f : Z → Y is a continuous function, then the family of probability measures {νt ◦ f −1 : t > 0} on (Y, BY ) satisfies the LDP with a good rate function J defined by J (y) = inf{I (z ) : f (z ) = y}.
A. De Gregorio, C. Macci / Statistics and Probability Letters 82 (2012) 1874–1882
Proposition 4.1. The family P
Y1 (t ) t
, Y2t(t )
1881
∈ · : t > 0 satisfies the LDP with good rate function IY1 ,Y2 defined by
2 c 2 c λ0 y1 − c12 y2 + c2 λ2 c0 − y1 − c1 y2 − 2 c0 + c2 2 c0 + c2 2 IY1 ,Y2 (y1 , y2 ) = c 1 c λ1 y2 + c10 y1 + c3 λ3 c1 − y2 + c0 y1 + − 2 c1 + c3 2 c1 + c3 ∞ c c where S = (y1 , y2 ) : −c2 ≤ y1 − c2 y2 ≤ c0 , −c3 ≤ y2 + c1 y1 ≤ c1 . 1 0
if (y1 , y2 ) ∈ S otherwise,
Proof. Firstly, by Proposition 2.1 (with suitable choices of the parameters) and by Corollary 2.9 in Lynch and Sethuraman
(1987), the family of laws P
X1 (t ) t
, X2t(t ) ∈ · : t > 0 satisfies the LDP with good rate function JX1 ,X2 defined by
JX1 ,X2 (x1 , x2 ) := I1 (x1 ) + I2 (x2 )
2 λ0 x1 + c2 λ2 c0 − x1 − 2 c0 + c2 2 c0 + c2 2 = λ1 x2 + c3 λ3 c1 − x2 − + 2 c1 + c3 2 c1 + c3 ∞
if (x1 , x2 ) ∈ [−c2 , c0 ] × [−c3 , c1 ] otherwise.
Then, since
(x1 , x2 ) → (y1 , y2 ) = f (x1 , x2 ) =
c0
c1 x1 + c2 x2
,
c0 + c2
c1
is a continuous function, the family of laws P
Y1 (t ) t
c0 x2 − c1 x1
c0 + c2
, Y2t(t ) ∈ · : t > 0 satisfies the LDP with good rate function IY1 ,Y2
defined by IY1 ,Y2 (y1 , y2 ) := inf JX1 ,X2 (x1 , x2 ) : f (x1 , x2 ) = (y1 , y2 )
by the contraction principle. Finally, it is easy to check that IY1 ,Y2 (y1 , y2 ) = JX1 ,X2 y1 − c2 y2 , y2 + 1 Orsingher and Ratanov (2002)) and this meets the expression of the rate function in the statement. c
c1 y c0 1
(see Eq. (3.13) in
Acknowledgments The authors thank the referees for the careful reading of the paper. One of them suggested some extensions of the results in an earlier version. Appendix. The extension of Eqs. (2.17)–(2.18) in De Gregorio et al. (2005) Here we consider the telegraph process {X (t ) : t ≥ 0}, with λ1 = λ2 = λ; actually, if λ1 ̸= λ2 , we cannot adapt the technique used in this Appendix. The aim is to study the conditional law of X (t ) given N (t ) = n, for t > 0 and n ≥ 1. In this case the changes of direction are governed by a homogeneous Poisson process {N (t ) : t ≥ 0}. If N (t ) = n for n ≥ 1, then the displacements of the telegraph process are of two types, i.e. c1 t (Tk − Tk−1 ) or −c2 t (Tk − Tk−1 ), where tTk is the instant in which the k-th Poisson event happens. In other words, Tk is the k-th order statistics from the uniform law in [0, 1]. By means of the exchangeability of Tk − Tk−1 , we can put together the n1 forward steps as well as the n + 1 − n1 backward ones. Then, when n changes of direction happen, the random variables X (t ) and t [c1 Tn1 − c2 (1 − Tn1 )] (for t > 0) are equally distributed; thus, for v ∈ {−c2 , c1 }, we obtain
N (t ) = n, V (0) = v , (c1 + c2 )t and, being the order statistic Tn1 a Beta(n1 , n + 1 − n1 ) distributed random variable, we have P (X (t ) ≤ x|N (t ) = n, V (0) = v) = P
P (X (t ) ∈ dx|N (t ) = n, V (0) = v) = dx
Tn1 ≤
c2 t + x
(c2 t + x)n1 −1 (c1 t − x)n−n1 1{−c2 t
1882
A. De Gregorio, C. Macci / Statistics and Probability Letters 82 (2012) 1874–1882
Clearly, there is a strict relationship between n and n1 , indeed we have
if n = 2k + 1 (for some k ≥ 0), if n = 2k (for some k ≥ 1) and V (0) = c1 , if n = 2k (for some k ≥ 1) and V (0) = −c2 ,
then n1 = k + 1; then n1 = k + 1; then n1 = k.
In conclusion, we can provide the following formulas: for all k ≥ 0, P (X (t ) ∈ dx|N (t ) = 2k + 1, V (0) = v) = P (X (t ) ∈ dx|N (t ) = 2k + 1) (2k + 1)! = dx [(c2 t + x)(c1 t − x)]k 1{−c2 t
(2k)! (c2 t + x)k (c1 t − x)k−1 1{−c2 t
whence we obtain
P (X (t ) ∈ dx|N (t ) = 2k) =
P (X (t ) ∈ dx|N (t ) = 2k, V (0) = v)P (V (0) = v)
v∈{−c2 ,c1 }
(2k)! (c2 t + x)k−1 (c1 t − x)k−1 {(c2 t + x)p k!(k − 1)!((c1 + c2 )t )2k + (c1 t − x)(1 − p)}1{−c2 t
= dx
References Beghin, L., Nieddu, L., Orsingher, E., 2001. Probabilistic analysis of the telegrapher’s process with drift by means of relativistic transformations. J. Appl. Math. Stoch. Anal. 14, 11–25. De Gregorio, A., 2010. Stochastic velocity motions and processes with random time. Adv. Appl. Probab. 42, 1028–1056. De Gregorio, A., Orsingher, E., Sakhno, L., 2005. Motions with finite velocity analyzed with order statistics and differential equations. Theory Probab. Math. Statist. 71, 63–79. Dembo, A., Zeitouni, O., 1998. Large Deviations Techniques and Applications, second ed. Springer, New York. Di Crescenzo, A., 2001. On random motions with velocities alternating at Erlang-distributed random times. Adv. Appl. Probab. 33, 690–701. Di Crescenzo, A., Martinucci, B., 2010. A damped telegraph random process with logistic stationary distribution. J. Appl. Probab. 47, 84–96. Di Crescenzo, A., Pellerey, F., 2002. On prices’ evolutions based on geometric telegrapher’s process. Appl. Stoch. Models Bus. Ind. 18, 171–184. Fong, S.K., Kanno, S., 1994. Properties of the telegrapher’s random process with or without a trap. Stochastic Process. Appl. 53, 147–173. Iacus, S.M., 2001. Statistical analysis of the inhomogeneous telegrapher’s process. Statist. Probab. Lett. 55, 83–88. Iscoe, I., Ney, P., Nummelin, E., 1985. Large deviations of uniformly recurrent Markov additive processes. Adv. Appl. Math. 6, 373–412. Lynch, J., Sethuraman, J., 1987. Large deviations for processes with independent increments. Ann. Probab. 15, 610–627. Macci, C., 2009. Convergence of large deviation rates based on a link between wave governed random motions and ruin processes. Statist. Probab. Lett. 79, 255–263. Macci, C., 2008. Large deviations for empirical estimators of the stationary distribution of a semi-Markov process with finite state space. Comm. Statist. Theory Methods 37, 3077–3089. Ney, P., Nummelin, E., 1987a. Markov additive processes I, eigenvalue properties and limit theorems. Ann. Probab. 15, 561–592. Ney, P., Nummelin, E., 1987b. Markov additive processes II, large deviations. Ann. Probab. 15, 593–609. Ney, P., Nummelin, E., 1987c. Markov additive processes: large deviations for the continuous time case. In: Prohorov, Y.V., Statulevicius, V.A., Sazonov, V.V., Grigelionis, B. (Eds.), Probability Theory and Mathematical Statistics (Vol. II). VNU Sci. Press, Utrecht, pp. 377–389. Orsingher, E., 1990. Probability law, flow function, maximum distribution of wave governed random motions and their connections with Kirchoff’s laws. Stochastic Process. Appl. 34, 49–66. Orsingher, E., 1995. Motions with reflecting and absorbing barriers driven by the telegraph equation. Random Oper. Stoch. Equ. 1, 9–21. Orsingher, E., 2000. Exact joint distribution in a model of planar random motion. Stoch. Stoch. Rep. 69, 1–10. Orsingher, E., Ratanov, N., 2002. Planar random motions with drift. J. Appl. Math. Stoch. Anal. 15, 205–221. Ratanov, N., 2007. A jump telegraph model for option pricing. Quant. Finance 7, 575–583. Stadje, W., Zacks, S., 2004. Telegraph processes with random velocities. J. Appl. Probab. 41, 665–678. Zacks, S., 2004. Generalized integrated telegraph processes and the distribution of related random times. J. Appl. Probab. 41, 497–507.
Contents lists available at SciVerse ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Large deviation principles for telegraph processes Alessandro De Gregorio a , Claudio Macci b,∗ a
Dipartimento di Scienze Statistiche, Sapienza Università di Roma, Piazzale Aldo Moro 5, I-00185 Rome, Italy
b
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Rome, Italy
article
info
Article history: Received 7 February 2012 Received in revised form 20 June 2012 Accepted 22 June 2012 Available online 28 June 2012 MSC: 60F10 60J27
abstract The aim of this paper is to present large deviation results for some telegraph random motions. We are not aware of any other results of this kind except the ones for the classical telegraph process (with drift). We start with the large deviation principle of the conditional laws given the number of changes of direction for the classical case; moreover, we compare the rate function with the one obtained for the non-conditional distributions. Finally, we study an inhomogeneous model and a planar telegraph motion. © 2012 Elsevier B.V. All rights reserved.
Keywords: Conditional laws Inhomogeneous Poisson processes Markov additive processes Random motions
1. Introduction The stochastic processes are often used to describe the real motions. An important example is given by the Brownian motion, which was introduced in order to describe the movement of the particles in a liquid. However the Brownian motion is not free from defects such as the unbounded first variation. Therefore, over the years, many authors have introduced random models having the main characteristics of the real movements: finite velocity and persistence. A typical example in this direction is the telegraph process, which is represented by a particle moving on the real line, alternatively forward and backward, with finite speed. Furthermore, one suppose that the changes of direction are governed by a homogeneous Poisson process. The probabilistic properties of this random model have been analyzed, for example, in Beghin et al. (2001), Fong and Kanno (1994) and Orsingher (1990, 1995). Other authors have also studied generalizations of the telegraph process, concerning the waiting times between two consecutive changes of direction or the randomization of the velocity (see Di Crescenzo (2001), Stadje and Zacks (2004), Zacks (2004), Di Crescenzo and Martinucci (2010) and De Gregorio (2010)). It is interesting to observe that applications of the telegraph process emerge in different fields. For instance, in physics the propagation of a damped wave along a wire is described by the telegraph equation. In order to model the dynamics of the price of risky assets, Di Crescenzo and Pellerey (2002) have considered the geometric telegraph process. A model of financial market based on a telegraph process with drift, that is with two different velocities and switching rates, has been introduced by Ratanov (2007). In this paper, we present large deviation results for some telegraph random motions. We are not aware of any other results of this kind except Proposition 2.1 recalled below; this result concerns the non-conditional laws for the classical case (with drift). Both Proposition 2.1 and the results in this paper, concern the large-time asymptotic behavior.
∗
Corresponding author. E-mail addresses: [email protected] (A. De Gregorio), [email protected] (C. Macci).
0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.06.023
A. De Gregorio, C. Macci / Statistics and Probability Letters 82 (2012) 1874–1882
1875
The theory of large deviations gives an asymptotic computation of small probabilities on exponential scale; estimates based on large deviations play a crucial role in resolving a variety of questions in statistics, engineering, statistical mechanics and applied probability. We recall the basic definitions in Dembo and Zeitouni (1998, pp. 4–5). Let Z be a Hausdorff topological space with Borel σ -algebra BZ . A lower semi-continuous function I : Z → [0, ∞] is called the rate function. A family of probability measures {νt : t > 0} on (Z, BZ ) satisfies the large deviation principle (LDP), as t → ∞, with rate function I if lim sup
1
log νt (F ) ≤ − inf I (z ) for all closed sets F
t
t →∞
z ∈F
and lim inf t →∞
1 t
log νt (G) ≥ − inf I (z ) z ∈G
for all open sets G.
A rate function I is said to be good if all the level sets {{z ∈ Z : I (z ) ≤ γ } : γ ≥ 0} are compact. In view of the proof of Proposition 2.2 below, we recall that the lower bound for open sets is equivalent to condition (b) with Eq. (1.2.8) in Dembo and Zeitouni (1998), i.e.: lim inf t →∞
1 t
log νt (G) ≥ −I (z )
for all z ∈ Z such that I (z ) < ∞ and for all open sets G such that z ∈ G.
(1.1)
For completeness we say that a family of Z-valued random variables {Zt : t > 0} satisfies the LDP if the family of probability measures {P (Zt ∈ ·) : t > 0} does (in the sense of the above definition). We conclude with the outline of the paper. In Section 2, we present two LDPs for the classical telegraph process (with drift): Proposition 2.1 recalls a known LDP for non-conditional distributions; Proposition 2.2 provides a new LDP for the conditional laws given the number of changes of direction. In the proof of Proposition 2.2 we handle the conditional densities in a standard way. Furthermore, we present inequalities between the rate functions in Propositions 2.1–2.2 which allow to compare the convergence of non-conditional and conditional laws. We conclude with two other LDPs in Sections 3–4, respectively: the first one concerns an inhomogeneous model in Iacus (2001), the second result concerns a planar telegraph motion which is a slight generalization of the random processes in Orsingher (2000) and Orsingher and Ratanov (2002). We remark that we do not need to know explicitly the expression of the laws (as happens in the proof of Proposition 2.2) because we can combine the LDP in Proposition 2.1 with other large deviation tools: Theorem 4.2.13 in Dembo and Zeitouni (1998) concerning exponentially equivalent families of laws in Section 3, and the contraction principle (see e.g. Theorem 4.2.1 in Dembo and Zeitouni (1998)) in Section 4. 2. The classical telegraph process with drift The telegraph process with drift is a random motion {X (t ) : t ≥ 0} on the real line which starts at the origin of R and moves with a two-valued integrated telegraph signal: for some λ1 , λ2 , c1 , c2 > 0, we have a rightward velocity c1 , a leftward velocity −c2 , and the rates of the occurrences of velocity switches are λ1 and λ2 , respectively. More precisely, we have X (t ) =
t 0
c1 − c2 2
+
c1 + c2 1{V (0)=c1 } − 1{V (0)=−c2 } (−1)N (s) ds 2
where the random variable V (0) is such that P (V (0) ∈ {−c2 , c1 }) = 1; the process {N (t ) : t ≥ 0} which counts the number of changes of direction is an inhomogeneous Poisson process such that N (t ) = n≥1 1{S1 +···+Sn ≤t } , the random variables {Sn : n ≥ 1} are conditionally independent given V (0), and the conditional law is the following:
{S2k−1 : k ≥ 1}
are exponentially distributed with mean
1
λ1 1 {S2k : k ≥ 1} are exponentially distributed with mean ; λ2 1 {S2k−1 : k ≥ 1} are exponentially distributed with mean λ 2 if V (0) = −c2 , then 1 {S2k : k ≥ 1} are exponentially distributed with mean . λ1 In the literature the case c1 = c2 and λ1 = λ2 is said to be the case without drift. We remark that, if we have λ1 = λ2 = λ for some λ > 0, {N (t ) : t ≥ 0} is a Poisson process with intensity λ and, moreover, V (0) and {N (t ) : t ≥ 0} are independent (because V (0) and {Sn : n ≥ 1} are independent). Finally, we allow a general initial distribution of the initial velocity V (0), if V (0) = c1 ,
then
i.e. we set
(P (V (0) = c1 ), P (V (0) = −c2 )) = (q, 1 − q) for some q ∈ [0, 1], which has no influence in the expression of the rate functions below.
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A. De Gregorio, C. Macci / Statistics and Probability Letters 82 (2012) 1874–1882
We start with a known large deviation result which can be proved by referring to more general results for Markov additive processes (see e.g. Iscoe et al. (1985) and Ney and Nummelin (1987a,b,c)); see also Macci (2009, Section 3). We remark that there is a link between the tail behavior of the rare events studied in this paper and the sample paths of the process, and in particular the support of the paths. Actually the rate function IλX1 ,λ2 ,c1 ,c2 is not finite outside the closed interval [−c2 , c1 ] because we have P
X (t ) t
∈ [−c2 , c1 ] = 1, and we can consider the lower bound for the open set G = R \ [−c2 , c1 ] in the
definition of LDP.
Proposition 2.1. The family P
IλX1 ,λ2 ,c1 ,c2 (x) =
X (t ) t
∈ · : t > 0 satisfies the LDP with good rate function IλX1 ,λ2 ,c1 ,c2 defined by
2 c1 − x − λ2 λ1 c1 + c2 c1 + c2 x + c2
∞
if x ∈ [−c2 , c1 ] otherwise.
Remark 2.1. Note that, if λ1 = λ2 = λ and c1 = c2 = c for some λ, c > 0, i.e. {X (t ) : t ≥ 0} is a standard telegraph process without drift, the rate function in Proposition 2.1 becomes
x2 λ 1− 1− 2 X Iλ,λ, c c ,c (x) = ∞
if x ∈ [−c , c ] otherwise.
Now we prove the LDP for the conditional laws assuming that λ1 = λ2 = λ for some λ > 0. We remark that, since X |N the conditional densities {fn (·, t ) : n ≥ 1, t > 0} below do not depend on λ, the rate function Iλ,λ,c1 ,c2 (·; w) in the next proposition has the same feature. Proposition 2.2. Assume that λ1 = λ 2 = λ > 0. Let λ for some t → wt be a deterministic function such that limt →∞ wt = w N (t ) X |N X (t ) for some w ∈ (0, ∞). Then the family P ∈ · t = wt : t > 0 satisfies the LDP with good rate function Iλ,λ,c1 ,c2 (·; w) t defined by X |N
Iλ,λ,c1 ,c2 (x; w) =
w log
∞
c1 + c2 √ 2 (c2 + x)(c1 − x)
if − c2 < x < c1 otherwise.
Proof. We refer to the conditional laws P (X (t ) ∈ dy|N (t ) = n) = fn (y, t )dy (for n ≥ 1), where f2k+1 (y, t ) =
(2k + 1)! (c2 t + y)k (c1 t − y)k 1{−c2 t
and f2k (y, t ) =
(2k)! (c2 t + y)k−1 (c1 t − y)k−1 {q(c2 t + y) + (1 − q)(c1 t − y)}1{−c2 t
for k ≥ 1. The conditional densities {fn (·, t ) : n ≥ 1, t > 0} are the extension of Eqs. (2.17)–(2.18) in De Gregorio et al. (2005) which concern the case q = 21 and c1 = c2 = c for some c > 0; see Appendix for details. The proof is divided into two parts. (1) Proof of the lower bound for open sets. We want to check the equivalent condition (1.1), i.e. lim inf t →∞
1 t
log P
X (t ) t
N (t ) ∈ G = wt ≥ −w log √
c1 + c2
2 (c2 + x)(c1 − x)
t
for all x such that −c2 < x < c1 and for all open sets G such that x ∈ G. First we can take ε > 0 small enough to have (x − ε, x + ε) ⊂ G ∩ (−c2 , c1 ). Then, for some yε,t ∈ (x − ε, x + ε), we have N (t ) N (t ) X (t ) X (t ) ∈ G = wt ≥ P ∈ (x − ε, x + ε) = wt P t
t
t
(x+ε)t
= (x−ε)t
t
ft wt (y, t )dy =
x+ε
ft wt (yt , t )tdy = 2ε ft wt (yε,t t , t )t . x−ε
A. De Gregorio, C. Macci / Statistics and Probability Letters 82 (2012) 1874–1882
Hence, since limn→∞ lim inf
1
t →∞
t
log(n!) n log(n)
log P
≥ lim inf t →∞
= 1, we obtain N (t ) X (t ) ∈ G = wt t
1
1877
t
t
t wt log(t wt ) − 2
t wt 2
w
= w log 2 − w log(c1 + c2 ) +
log
t wt 2
− t wt log(c1 + c2 ) +
t wt 2
log{(c2 + x − ε)(c1 − (x + ε))}
log{(c2 + x − ε)(c1 − x − ε)} = −w log
2
c1 + c2 , √ 2 (c2 + x − ε)(c1 − x − ε)
and we complete the proof of the lower bound letting ε go to zero. (2) Proof of the upper bound for closed sets. We have to check lim sup
1 t
t →∞
log P
X (t ) t
N (t ) X |N ∈ F = wt ≤ − inf Iλ,λ,c1 ,c2 (x; w) for all closed sets F . x∈F t X |N
c1 − c2 ∈ F (we would have infx∈F Iλ,λ,c1 ,c2 (x; w) = 0) and if F ∩ (−c2 , c1 ) = ∅ 2 N (t ) ∈ F t = wt = 0). Thus, from now on, we assume that c1 −2 c2 ̸∈ F and F ∩ (−c1 , c2 ) ̸= ∅. For the
First note that this condition trivially holds if (we would have P
X (t ) t
moment we also assume that
F∩
c1 − c2 2
, c1 ,
F∩
c1 − c2 −c2 , ̸= ∅. 2
X |N
X |N
We remark that Iλ,λ,c1 ,c2 (·; w) is decreasing in (−c2 , 1 2 2 ] and Iλ,λ,c1 ,c2 (·; w) is increasing in [ 1 2 2 , c1 ); then, if we set c −c c −c c −c xF := sup(F ∩ (−c2 , 1 2 2 )) and xF := inf(F ∩ ( 1 2 2 , c1 )), we have F ⊂ (−∞, xF ] ∪ [xF , ∞), where xF ∈ (−c2 , 1 2 2 ) and xF ∈ (
c −c
c −c
c t , c1 ). Then, since we have P X (tt ) ≥ xF N (tt ) = wt = xF1t ft wt (y, t )dy, we get N (t ) X (t ) ≥ xF = wt ≤ ft wt (xF t , t )(c1 − xF )t (if t wt ∈ {1, 3, 5, . . .}) P c1 − c2 2
t
t
and, since q(c2 t + y) + (1 − q)(c1 t − y) ≤ (c1 + c2 )t for y ∈ (−c2 t , c1 t ),
P
N (t ) (t wt )! ≥ xF = wt ≤ t wt t wt t t wt t wt t ! − 1 !(( c + c ) t ) 1 2 2 2
X (t ) t
· {(c2 + xF )(c1 − xF )}
t wt 2
−1
(c1 + c2 )(c1 − xF ) (if t wt ∈ {2, 4, 6, . . .}).
log(n!) n log(n)
= 1 as in the first part of the proof, we obtain N (t ) X (t ) 1 ≥ xF = wt lim sup log P t t t t →∞ 1 t wt t wt t wt ≤ lim sup t wt log(t wt ) − 2 log − t wt log(c1 + c2 ) + log{(c2 + xF )(c1 − xF )}
Thus, by taking into account that limn→∞
t →∞
t
= w log 2 − w log(c1 + c2 ) +
w 2
2
2
2
X |N
log{(c2 + xF )(c1 − xF )} = −Iλ,λ,c1 ,c2 (xF ; w).
Similarly, we can also check the inequality lim sup t →∞
1 t
log P
X (t ) t
N (t ) X |N ≤ xF = wt ≤ −Iλ,λ,c1 ,c2 (xF ; w). t
Then, by the inclusion F ⊂ (−∞, xF ] ∪ [xF , ∞), by Lemma 1.2.15 in Dembo and Zeitouni (1998) and the inequalities above, we get lim sup t →∞
1 t
log P
X (t ) t
N (t ) X |N X |N ∈F = wt ≤ max{−Iλ,λ,c1 ,c2 (xF ; w), −Iλ,λ,c1 ,c2 (xF ; w)} t
X |N
X |N
X |N
= − min{Iλ,λ,c1 ,c2 (xF ; w), Iλ,λ,c1 ,c2 (xF ; w)} = − inf Iλ,λ,c1 ,c2 (x; w). x∈F
Finally we remark that if F ∩ (
c1 − c2 2
, c1 ) is empty, we can adapt the above procedure by considering the inclusion c1 − c2 ) is empty, we can adapt the above procedure by considering the inclusion 2
F ⊂ (−∞, xF ]; similarly, if F ∩ (−c2 , F ⊂ [xF , ∞).
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A. De Gregorio, C. Macci / Statistics and Probability Letters 82 (2012) 1874–1882
X |N
Fig. 1. The functions I1X,1,1,2 (x) and I1,1,1,2 (x; w), where x ∈ (−2, 1). Two choices for w : w = 1.5 ≥ 1; w = 0.6 ∈ (0, 1).
Remark 2.2. A close inspection ofthe proof (and of the content of the Appendix) reveals that we can also prove the LDP for P
X (t ) t
∈ · N (tt ) = wt , V (0) = v : t > 0 for v ∈ {−c2 , c1 }, and we have the same rate function. X |N
We remark that Iλ,λ,c1 ,c2 (·; w) in Proposition 2.2 uniquely vanishes at 1 2 2 and, if λ1 = λ2 , this also happens for IλX1 ,λ2 ,c1 ,c2 in Proposition 2.1. So, in order to compare the convergence of non-conditional and conditional laws, it is interesting to find c −c strict inequalities between these two rate functions in a neighborhood of 1 2 2 (except this point) when λ1 = λ2 . c −c
Proposition 2.3. We have two cases. X |N
c1 − c2 X (i) For w ≥ λ, we have Iλ,λ, c1 ,c2 (x) ≤ Iλ,λ,c1 ,c2 (x; w) for all x ∈ R; moreover, the inequality is strict for x ∈ [−c2 , c1 ]\{ 2 }. 2 (ii) For w ∈ (0, λ) set α± = 21 c1 − c2 ± 1 − wλ2 (c1 + c2 ) ; then there exist β+ ∈ (α+ , c1 ) and β− ∈ (−c2 , α− ) such X |N
X |N
X X that Iλ,λ,c1 ,c2 (x; w) > Iλ,λ, c1 ,c2 (x) for x ∈ [−c2 , β− ) ∪ (β+ , c1 ], Iλ,λ,c1 ,c2 (x; w) < Iλ,λ,c1 ,c2 (x) for all x ∈ (β− , β+ ) \ { X |N X Iλ,λ,c1 ,c2 (x; w) = Iλ,λ, c1 ,c2 (x) otherwise. X |N
Proof. First note that, for all w > 0, we have Iλ,λ,c1 ,c2 (
c1 − c2 2
} and
X |N
c1 −c2 X X ; w) = Iλ,λ, c1 ,c2 ( 2 ) = 0; Iλ,λ,c1 ,c2 (x; w) = Iλ,λ,c1 ,c2 (x) = ∞ X for x ̸∈ [−c2 , c1 ]; λ = Iλ,λ, c1 ,c2 (x) < Iλ,λ,c1 ,c2 (x; w) = ∞ for x ∈ {−c2 , c1 }. Then (i) and (ii) in the statement can be checked by considering the difference function (for x ∈ (−c2 , c1 )) defined by √ c1 + c2 2 (c2 + x)(c1 − x) X |N X ∆(x) := Iλ,λ,c1 ,c2 (x; w) − Iλ,λ, − λ 1 − , ( x ) = w log √ c1 , c2 c1 + c2 2 (c2 + x)(c1 − x) c −c √ x− 1 2 2λ (c2 +x)(c1 −x) and studying its derivative ∆′ (x) = (c +x)(c2 −x) w − . c1 + c2 2 1 c1 − c2 2
X |N
Thus, as depicted in Fig. 1 below, for x in a neighborhood of X |N
c1 − c2 2
X |N
X (except this point) we have Iλ,λ,c1 ,c2 (x; w) > Iλ,λ, c1 , c2 ( x )
if w ≥ λ and Iλ,λ,c1 ,c2 (x; w) < Iλ,λ,c1 ,c2 (x) if w ∈ (0, λ). Then, if λ1 = λ2 = λ, in some sense the convergence of the c −c conditional laws to the limit 1 2 2 is faster than the convergence of the non-conditional laws if and only if w ≥ λ. We remark that it is not surprising to have a faster convergence of the conditional laws as w increases; indeed the more w is large, the more the changes of direction are frequent. We conclude this section by pointing out a connection between Proposition 2.2 and a LDP in Macci (2008). Let T0 = 0 and {Tn : n ≥ 1} are the epochs of the occurrences of the transition number process, i.e. N (t ) = n≥1 1{Tn ≤t } ; moreover, let {Jn : n ≥ 0} be the random sequence defined by Jn = 1 (resp. Jn = 2) if X (t ) is increasing (resp. decreasing) for t ∈ [Tn , Tn+1 ). Here we are interested in the sequence {(πˆ n(1) , πˆ n(2) ) : n ≥ 1} defined by n−1
πˆ n(i) =
X
(Tk+1 − Tk )1{Jk =i}
k=0
Tn
.
A. De Gregorio, C. Macci / Statistics and Probability Letters 82 (2012) 1874–1882
1879
(1)
(2)
Then, by Proposition 3.2 in Macci (2008), for any fixed integer w ≥ 1 the sequence of random variables {(πˆ nw , πˆ nw ) : n ≥ 1} satisfies the LDP with good rate function Hw defined by Hw (π1 , π2 ) := w H1 (π1 , π2 ), where H1 (π1 , π2 ) =
1
log
2
λ1 π1 + λ2 π2
λ2 π2 λ1 π1
if π1 , π2 ∈ (0, 1) and π1 + π2 = 1 otherwise.
∞
Thus, if we follow the lines of the proof of Proposition 2.1 illustrated in Macci 3),we get the LDP of (2009, Section 1) 2) {c1 πˆ n(w − c2 πˆ n(w : n ≥ 1} (or, equivalently, the LDP of the conditional laws P X (nn) ∈ · N (nn) = w : n ≥ 1 ), and the rate function ˜IλX1 ,λ2 ,c1 ,c2 (·; w) is defined by
˜IλX ,λ
2 , c1 , c2
1
(x; w) := inf{Hw (π1 , π2 ) : c1 π1 − c2 π2 = x}.
X Finally, it is easy to check that ˜Iλ,λ, c1 ,c2 (·; w) coincides with the rate function in Proposition 2.2.
3. An inhomogeneous telegraph process Iacus (2001) has shown that the finite dimensional law of an inhomogeneous telegraph process driven by a Poisson process with intensity {λ(t ) : t ≥ 0} is a solution of the telegraph equation with nonconstant coefficients. A rare example for which the finite dimensional law can be provided explicitly is the process {Yθ (t ) : t ≥ 0} defined by Yθ (t ) = V (0)
t
(−1)Nθ (s) ds,
(3.1)
0
where the random variable V (0) is such that P (V (0) = c ) = P (V (0) = −c ) = 21 ; the process {Nθ (t ) : t ≥ 0} is an inhomogeneous Poisson process with intensity {λθ (t ) : t ≥ 0} defined by λθ (t ) := θ tanh(θ t ); V (0) and {Nθ (t ) : t ≥ 0} are independent. The main goal of this section is to prove the LDP for the non-conditional laws of
Yθ ( t ) t
: t > 0 . The method used
for from the one in Proposition 2.1; indeed we take into account Proposition 2.1 and we show that the proofis different X (t ) t
: t > 0 and
Yθ (t ) t
: t > 0 are exponentially equivalent as t → ∞ in the sense of Definition 4.2.10 in Dembo and
Zeitouni (1998). Proposition 3.1. The family
P
Yθ (t ) t
∈ · : t > 0 satisfies the LDP with good rate function IθX,θ ,c ,c as in Proposition 2.1
(see Remark 2.1). Proof. We shall show that
Xθ (t ) Yθ (t ) >δ lim sup log P − = −∞ for all δ > 0 (3.2) t t t t →∞ Yθ (t ) for a suitable version of the processes : t > 0 and Xθt(t ) : t > 0 defined on the same probability space, where t {Xθ (t ) : t ≥ 0} is the process {X (t ) : t ≥ 0} in Proposition 2.1 with λ1 = λ2 = θ and c1 = c2 = c. This gives the proof because, if condition (3.2)holds, the LDP is an immediate consequence of Theorem 4.2.13 in Dembo and Zeitouni (1998) and X (t ) the LDP for θt : t > 0 (i.e. Proposition 2.1 with λ1 = λ2 = θ and c1 = c2 = c). In what follows, we check (3.2) assuming that both the processes {Yθ (t ) : t ≥ 0} and {Xθ (t ) : t ≥ 0} have the same random initial velocity V (0) such that P (V (0) = c ) = P (V (0) = −c ) = 21 ; thus we have (3.1) and t ∗ Xθ (t ) = V (0) (−1)Nθ (s) ds 1
0
for a homogeneous Poisson process {Nθ∗ (t ) : t ≥ 0} with intensity θ . Moreover, for a homogeneous Poisson process
{M (t ) : t ≥ 0} with intensity 1, we set Nθ∗ (t ) := M (θ t ) and Nθ (t ) := M (Λθ (t )), where Λθ (t ) :=
Then we have Xθ (t ) t
:=
V (0)
t
t
(−1)M (θ s) ds =
0
V (0)
θt
θt
(−1)M (r ) dr
0
and Yθ (t ) t
:=
V (0) t
0
t
M (Λθ (s))
(−1)
ds =
V (0)
θt
0
Λθ (t )
(−1)
er
M (r )
er
√
√
e2r − 1 + e2r
e2r − 1 + e2r − 1
dr ;
t 0
λθ (u)du = log cosh(θ t ).
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A. De Gregorio, C. Macci / Statistics and Probability Letters 82 (2012) 1874–1882
1 the latter equality can be derived by considering the change of variable r = Λθ (s), whence we obtain s = Λ− θ (r ) := 1
θ
log(er +
√
e2r − 1). Thus we get
√ Λθ (t ) θt er e2r − 1 + e2r M (r ) M (r ) dr (−1) (−1) dr − √ 0 er e2r − 1 + e2r − 1 0 Λθ (t ) 1 c dr + θ t − Λθ (t ) =: ac ,θ (t ) ≤ √ θt er e2r − 1 + e2r − 1 0
Xθ (t ) Yθ (t ) = c − t t θt
(where ac ,θ (t ) is deterministic) and, by the L’Hopital Theorem, we have lim ac ,θ (t ) =
t →∞
=
θ
lim
t →∞
eΛθ (t )
√
e2Λθ (t ) − 1 + e2Λθ (t ) − 1
c
θ
Λ′θ (t )
c
lim θ
t →∞
+ θ − Λ′θ (t )
1 + tanh(θ t )
cosh(θ t ) cosh2 (θ t ) − 1 + cosh2 (θ t ) − 1
because Λ′θ (t ) = θ tanh(θ t ). In conclusion (3.2) holds because
{ac ,θ (t ) > δ} is empty for t large enough.
1
Xθ (t ) t −
Yθ (t ) t
−1
=0
> δ ⊂ {ac ,θ (t ) > δ} and the event
Proposition 3.1 can be proved with the same techniques used above for the proof of Proposition 2.2 and by considering the density provided by Theorem 2.1 in Iacus (2001). This density can be expressed in terms of the modified Bessel function of the first kind I1 (r ) := are omitted.
(r /2)1+2k k=0 k!(k+1)!
∞
and we also have to take into account the limit
log I1 (x) x
→ 1 as x → ∞. The details
4. A planar telegraph motion In this section we consider a planar telegraph motion {(Y1 (t ), Y2 (t )) : t ≥ 0} with four directions and four speeds, switching at exponential times, which is a slight generalization of the random motion considered in Orsingher and Ratanov (2002). This process is introduced by means of two independent telegraph motions with drift {X1 (t ) : t ≥ 0} and {X2 (t ) : t ≥ 0}. More precisely, for λ0 , λ1 , λ2 , λ3 > 0 and c0 , c1 , c2 , c3 > 0, we consider the process {X1 (t ) : t ≥ 0} with switching rates (λ0 /2, λ2 /2) and velocities (c0 , −c2 ), the process {X2 (t ) : t ≥ 0} with switching rates (λ1 /2, λ3 /2) and velocities (c1 , −c3 ), and the process {(Y1 (t ), Y2 (t )) : t ≥ 0} defined by
c0 c1 X1 (t ) + c2 X2 (t ) Y ( t ) = 1 c c +c 1
0
2
Y2 (t ) = c0 X2 (t ) − c1 X1 (t ) . c0 + c2
Note that, by Proposition 2.1, P
X1 (t ) t
∈ · : t > 0 and P X2t(t ) ∈ · : t > 0 satisfy the LDP with good rate functions
I1 = IλX0 /2,λ2 /2,c0 ,c2 and I2 = IλX1 /2,λ3 /2,c1 ,c3 , respectively.
, Y2t(t ) ∈ · : t > 0 . We remark that we can obtain explicitly such non-conditional laws when the random initial velocities of {X1 (t ) : t ≥ 0} and {X2 (t ) : t ≥ 0}, denoted by V1 (0) and V2 (0), are such that The aim is to prove the LDP for the non-conditional laws P
Y1 (t ) t
P (V1 (0) = c0 ) = P (V1 (0) = −c2 ) = P (V2 (0) = c1 ) = P (V2 (0) = −c3 ) =
1 2
;
see Orsingher and Ratanov (2002) for the case λ0 = λ1 = λ2 = λ3 , and see Orsingher (2000) if we also have c0 = c1 = c2 = c3 . We also remark that, if λ0 = λ1 = λ2 = λ3 = λ for some λ, the number of changes of directions are governed by a Poisson process with intensity λ; more precisely, when a Poisson event occurs with rate λ, the motion takes – with probability 21 – one of the two directions orthogonal to the previous one.
It is interesting that we can avoid to handle the non-conditional laws P
Y1 (t ) t
, Y2t(t ) ∈ · : t > 0 . Actually the LDP
will be proved by a simple application of the contraction principle (see e.g. Theorem 4.2.1 in Dembo and Zeitouni (1998)): roughly speaking, if a family of probability measures {νt : t > 0} on (Z, BZ ) satisfies the LDP with a good rate function I and f : Z → Y is a continuous function, then the family of probability measures {νt ◦ f −1 : t > 0} on (Y, BY ) satisfies the LDP with a good rate function J defined by J (y) = inf{I (z ) : f (z ) = y}.
A. De Gregorio, C. Macci / Statistics and Probability Letters 82 (2012) 1874–1882
Proposition 4.1. The family P
Y1 (t ) t
, Y2t(t )
1881
∈ · : t > 0 satisfies the LDP with good rate function IY1 ,Y2 defined by
2 c 2 c λ0 y1 − c12 y2 + c2 λ2 c0 − y1 − c1 y2 − 2 c0 + c2 2 c0 + c2 2 IY1 ,Y2 (y1 , y2 ) = c 1 c λ1 y2 + c10 y1 + c3 λ3 c1 − y2 + c0 y1 + − 2 c1 + c3 2 c1 + c3 ∞ c c where S = (y1 , y2 ) : −c2 ≤ y1 − c2 y2 ≤ c0 , −c3 ≤ y2 + c1 y1 ≤ c1 . 1 0
if (y1 , y2 ) ∈ S otherwise,
Proof. Firstly, by Proposition 2.1 (with suitable choices of the parameters) and by Corollary 2.9 in Lynch and Sethuraman
(1987), the family of laws P
X1 (t ) t
, X2t(t ) ∈ · : t > 0 satisfies the LDP with good rate function JX1 ,X2 defined by
JX1 ,X2 (x1 , x2 ) := I1 (x1 ) + I2 (x2 )
2 λ0 x1 + c2 λ2 c0 − x1 − 2 c0 + c2 2 c0 + c2 2 = λ1 x2 + c3 λ3 c1 − x2 − + 2 c1 + c3 2 c1 + c3 ∞
if (x1 , x2 ) ∈ [−c2 , c0 ] × [−c3 , c1 ] otherwise.
Then, since
(x1 , x2 ) → (y1 , y2 ) = f (x1 , x2 ) =
c0
c1 x1 + c2 x2
,
c0 + c2
c1
is a continuous function, the family of laws P
Y1 (t ) t
c0 x2 − c1 x1
c0 + c2
, Y2t(t ) ∈ · : t > 0 satisfies the LDP with good rate function IY1 ,Y2
defined by IY1 ,Y2 (y1 , y2 ) := inf JX1 ,X2 (x1 , x2 ) : f (x1 , x2 ) = (y1 , y2 )
by the contraction principle. Finally, it is easy to check that IY1 ,Y2 (y1 , y2 ) = JX1 ,X2 y1 − c2 y2 , y2 + 1 Orsingher and Ratanov (2002)) and this meets the expression of the rate function in the statement. c
c1 y c0 1
(see Eq. (3.13) in
Acknowledgments The authors thank the referees for the careful reading of the paper. One of them suggested some extensions of the results in an earlier version. Appendix. The extension of Eqs. (2.17)–(2.18) in De Gregorio et al. (2005) Here we consider the telegraph process {X (t ) : t ≥ 0}, with λ1 = λ2 = λ; actually, if λ1 ̸= λ2 , we cannot adapt the technique used in this Appendix. The aim is to study the conditional law of X (t ) given N (t ) = n, for t > 0 and n ≥ 1. In this case the changes of direction are governed by a homogeneous Poisson process {N (t ) : t ≥ 0}. If N (t ) = n for n ≥ 1, then the displacements of the telegraph process are of two types, i.e. c1 t (Tk − Tk−1 ) or −c2 t (Tk − Tk−1 ), where tTk is the instant in which the k-th Poisson event happens. In other words, Tk is the k-th order statistics from the uniform law in [0, 1]. By means of the exchangeability of Tk − Tk−1 , we can put together the n1 forward steps as well as the n + 1 − n1 backward ones. Then, when n changes of direction happen, the random variables X (t ) and t [c1 Tn1 − c2 (1 − Tn1 )] (for t > 0) are equally distributed; thus, for v ∈ {−c2 , c1 }, we obtain
N (t ) = n, V (0) = v , (c1 + c2 )t and, being the order statistic Tn1 a Beta(n1 , n + 1 − n1 ) distributed random variable, we have P (X (t ) ≤ x|N (t ) = n, V (0) = v) = P
P (X (t ) ∈ dx|N (t ) = n, V (0) = v) = dx
Tn1 ≤
c2 t + x
(c2 t + x)n1 −1 (c1 t − x)n−n1 1{−c2 t
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A. De Gregorio, C. Macci / Statistics and Probability Letters 82 (2012) 1874–1882
Clearly, there is a strict relationship between n and n1 , indeed we have
if n = 2k + 1 (for some k ≥ 0), if n = 2k (for some k ≥ 1) and V (0) = c1 , if n = 2k (for some k ≥ 1) and V (0) = −c2 ,
then n1 = k + 1; then n1 = k + 1; then n1 = k.
In conclusion, we can provide the following formulas: for all k ≥ 0, P (X (t ) ∈ dx|N (t ) = 2k + 1, V (0) = v) = P (X (t ) ∈ dx|N (t ) = 2k + 1) (2k + 1)! = dx [(c2 t + x)(c1 t − x)]k 1{−c2 t
(2k)! (c2 t + x)k (c1 t − x)k−1 1{−c2 t
whence we obtain
P (X (t ) ∈ dx|N (t ) = 2k) =
P (X (t ) ∈ dx|N (t ) = 2k, V (0) = v)P (V (0) = v)
v∈{−c2 ,c1 }
(2k)! (c2 t + x)k−1 (c1 t − x)k−1 {(c2 t + x)p k!(k − 1)!((c1 + c2 )t )2k + (c1 t − x)(1 − p)}1{−c2 t
= dx
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