On asymptotic behaviour of probabilities of small deviations for compound Cox processes
We derive logarithmic asymtotics for probabilities of small deviations for compound Cox processes in the space of trajectories. We find conditions under which these asymptotics are the same as those for sums of independent identically distributed random variables and homogeneous processes with indep...
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Інститут математики НАН України
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| Цитувати: | On asymptotic behaviour of probabilities of small deviations for compound Cox processes / A.N. Frolov // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 19–27. — Бібліогр.: 10 назв.— англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860059819779555328 |
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| author | Frolov, A.N. |
| author_facet | Frolov, A.N. |
| citation_txt | On asymptotic behaviour of probabilities of small deviations for compound Cox processes / A.N. Frolov // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 19–27. — Бібліогр.: 10 назв.— англ. |
| collection | DSpace DC |
| description | We derive logarithmic asymtotics for probabilities of small deviations for compound Cox processes in the space of trajectories. We find conditions under which these asymptotics are the same as those for sums of independent identically distributed random variables and homogeneous processes with independent increments. We show that if these conditions do not hold, the asymptotics of small deviations for compound Cox processes are quite different.
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Theory of Stochastic Processes
Vol. 14 (30), no. 2, 2008, pp. 19–27
UDC 519.21
ANDREI N. FROLOV
ON ASYMPTOTIC BEHAVIOUR OF PROBABILITIES OF
SMALL DEVIATIONS FOR COMPOUND COX PROCESSES
We derive logarithmic asymtotics for probabilities of small deviations for compound
Cox processes in the space of trajectories. We find conditions under which these
asymptotics are the same as those for sums of independent identically distributed
random variables and homogeneous processes with independent increments. We show
that if these conditions do not hold, the asymptotics of small deviations for compound
Cox processes are quite different.
Introduction
The asymptotic behaviour of probabilities of small deviations has been investigated for
various classes of stochastic processes. Asymptotics of probabilities of small deviations
for sums of independent random variables and homogeneous processes with independent
increments were found in Mogul’skii [1], Borovkov and Mogul’skii [2] and references
therein. Note that the last class of processes includes stable, Poisson and compound
Poisson processes. Various results for Gaussian processes and references may be found
in surveys by Ledoux [3], Li and Shao [4] and Lifshits [5]. Further results were obtained
for various stochastic processes, generated by sums of random numbers of independent
random variables. In this case, the number of summands is a stochastic process which
is usually independent with the summands. Increments of such the processes may be
dependent. These processes are called compound processes. Now we give definitions for
some of them.
Let X,X1, X2, . . . be a sequence of independent, identically distributed random vari-
ables. Put Sn = X1 +X2 + · · · +Xn for n � 1, S0 = 0.
Let ν(t) be a standard Poisson process, independent with the sequence {Xk}. Then
the stochastic process η(t) = Sν(λt), λ > 0, is called a compound Poisson process.
If δ(t) is a renewal process, independent with {Xk}, then ζ(t) = Sδ(t) is called a
compound renewal process. When the renewal times have an exponential distribution,
the compound renewal process coincides with the compound Poisson process.
Small deviations of the renewal and compound renewal processes has been studied in
Frolov, Martikainen, Steinebach [6].
Now we turn to the definition of the compound Cox process. We start with the defi-
nition of a Cox process which we borrow from the paper of Embrechts and Klüppelberg
[7].
Let Λ(t), t > 0, be a random measure, i.e. a.s. (almost surely) Λ(0) = 0, Λ(t) < ∞
for all t > 0 and Λ(t) has non-decreasing trajectories. Assume that Λ(t) does not depend
on the standard Poisson process ν(t). The point process N(t) = ν(Λ(t)) is called a
Cox process. If, in particular, the trajectories of Λ(t) are continuous a.s., then for every
2000 AMS Mathematics Subject Classification. Primary 60F15; Secondary 60K05.
Key words and phrases. Small deviation, limit theorems, compound Cox processes.
This article was partially supported by RFBR, grant 05–01–00486.
19
20 ANDREI N. FROLOV
realization λ(t) of the measure Λ(t) the process N(t) is a non-homogeneous Poisson
process with the intensity measure λ(t).
Now we define the compound Cox process in the same way as we introduce the pro-
cesses η(t) and ζ(t) above.
Let N(t) be a Cox process, independent with the sequence {Xk}. The stochastic
process S(t) = SN(t) is called a compound Cox process.
Compound Cox processes play an important role in actuarial and financial mathemat-
ics. They describe, for example, the processes of total claims of an insurance company
in a collective risk model (cf. [7]).
Note that under additional conditions (cf.,e.g. [7]), the Cox processes are renewal
processes. Nevertheless, in the sequel, we assume that Λ(t) is such that our Cox process
will be the renewal process only if it is a Poisson process. So, we do not consider renewal
processes here.
The logarithmic asymptotics of small deviations of the compound Cox processes has
been investigated by Frolov [8]. In there, we have described the behaviour of
PT = P
(
sup
0�t�T
∣∣∣S(t) − cΛ(t)
∣∣∣ � xT
)
,
where {xT } is a positive real function such that xT → ∞ and x2
T = o(f(T )) as T → ∞,
f(T ) is a function, depending on properties of the measure Λ(t), c = EX for EX < ∞
and c = 0 otherwise.
Note that if EX <∞, then S(t) is centered by a random function, generally speaking.
Nevertheless, in the case of the homogeneous Poisson process, Λ(t) = λt and our center-
ing coincides with ES(t). The same holds true when N(t) is a non-homogeneous Poisson
process. Moreover, centering functions of this type are used in appropriate models of the
risk process. It turns out that fluctuations of risk in such models have to be compensate
by insurance premiums of random amounts since, otherwise, the ruin probability of in-
surance company may be separated from zero and may be non-decreasing with increasing
of initial capital of insurance company (cf. [7]). Therefore we consider this centering.
Here, we present generalizations of the results in Frolov [8]. PT is the probability that
trajectories of the compound Cox process, centered at cΛ(t) and normed by xT , lie in
a strip of width 2 around zero. We consider below more general sets instead of strips
and derive logarithmic asymtotics for such probabilities. We find conditions under which
these asymptotics are the same as those for sums of independent identically distributed
random variables and homogeneous processes with independent increments. We show
that the asymptotics of small deviations for compound Cox processes are quite different,
if these conditions do not hold.
1. Results
Let X,X1, X2, . . . be a sequence of independent, identically distributed random vari-
ables. Put Sn = X1 +X2 + · · · +Xn for n � 1, S0 = 0.
Let ν(t) be a standard Poisson process, independent with the sequence {Xk}.
Denote ξ(t) = Sν(t). Note that ξ(t) is a compound Poisson process and, therefore, it
is a homogeneous process with independent increments.
Let Λ(t), t > 0, be a random measure, i.e. a.s. (almost surely) Λ(0) = 0, Λ(t) < ∞
for all t > 0 and Λ(t) has non-decreasing trajectories. Assume that the trajectories of
Λ(t) are a.s. continuous and Λ(∞) = ∞ a.s. Suppose that Λ(t) is independent with the
process ν(t) and the sequence {Xk}.
Define the Cox process N(t) and the compound Cox process S(t) by the relations
N(t) = ν(Λ(t)) and S(t) = SN(t).
Let xT be a real function with xT → ∞ as T → ∞.
SMALL DEVIATIONS FOR COMPOUND COX PROCESSES 21
Let g1(t), g2(t), t ∈ [0, 1], be continuous functions such that g1(0) < 0 < g2(0), g1(t)
is non-increasing and g2(t) is non-decreasing. Put
PT = P
(
g1
(
Λ(t)
Λ(T )
)
xT � S(t) − cΛ(t) � g2
(
Λ(t)
Λ(T )
)
xT for all t ∈ [0, T ]
)
.
The probability PT is well defined since trajectories of S(t) are jump functions.
In the sequel, we will describe the asymptotic behaviour of probabilities of small
deviations PT .
In general case, gi(Λ(t)/Λ(T )) in the definition of PT are random. We mentioned above
that the random centering of S(t) is well motivated. So, random normings may also be
used. Nevertheless, there are important cases in which gi(Λ(t)/Λ(T )) are non-random
functions.
The first case is that g1(t) ≡ −1 and g2(t) ≡ 1. In this case the results have been
obtained in Frolov [8].
If N(t) is a Poisson process, then Λ(t) is a continuous increasing function. This is the
second case.
The third case arises when Λ(t) = Λf(t), where Λ is a positive random variable and
f(t) is a continuous function. Then gi(Λ(t)/Λ(T )) = gi(f(t)/f(T )). If, in addition,
f(t) = tβ , β > 0, we have
PT = P
(
g3
(
t
T
)
xT � S(t) − cΛ(t) � g4
(
t
T
)
xT for all t ∈ [0, T ]
)
= P (g3(t)xT � S(T t) − cΛ(T t) � g4(t)xT for all t ∈ [0, 1]) ,
where gi+2(t) = gi(tβ), i = 1, 2. Note that in this last case, one can consider more
general sets in the definition of PT . To this end, consider a family of processes {sT (t) =
S(T t)/xT ; 0 � t � 1, T ∈ (0,∞)}. The trajectories sT (·) belong to the Skorohod space
D[0, 1] and we can introduce P (sT (·) ∈ G), where G ∈ G and G is a class of subsets of
D[0, 1]. One can define this class in the same way as in Mogul’skii [1] with the following
additional assumption: on the first step of the definition on p. 757, functions L1(t) and
L2(t) (L1(t) > L2(t)) have to be non-decreasing and non-increasing correspondingly. Of
course, the class G will be a subclass of the class in Mogul’skii [1], but it will be wide
enough. It seems that the most interesting set is the set G0 = {g ∈ D[0, 1] : g(0) =
0, g3(t) < g(t) < g4(t), for all t ∈ [0, 1]}, where g3(t), g4(t), t ∈ [0, 1], are continuous
functions such that g3(0) < 0 < g4(0), g3(t) is non-increasing and g4(t) is non-decreasing.
The asymptotics of P (sT (·) ∈ G0) may be derived from our results below.
Our first result is the following theorem.
Theorem 1. Assume that there exist a positive, increasing, continuous function f(T ),
f(T ) → ∞ as T → ∞, and a non-negative random variables Λ such that the dis-
tributions of Λ(T )/f(T ) converge weakly to the distribution Λ as T → ∞. Denote
aT = ess inf Λ(T )/f(T ), a = ess inf Λ. Assume that aT → a as T → ∞ and a > 0.
If EX <∞, then put c = EX. Put c = 0 otherwise.
Suppose that the distribution of the random variable ξ1 = ξ(1)− c belongs to a domain
of attraction of a strictly stable law Fα with the index α ∈ (0, 2], i.e. the distributions of
(ξ(n) − cn)/Bn converge weakly to Fα as n → ∞, where {Bn} is a sequence of positive
constants. Assume that Fα is not concentrated on the half of line.
Then for every positive function xT with xT → ∞ and xT = o(B[f(T )]) as T → ∞,
the following relation holds
(1) logPT = −CHαa
f(T )
xα
T
L(xT )
(
1 + o(1)
)
as T → ∞,
22 ANDREI N. FROLOV
where L(x) = xα−2 Eξ21I{|ξ1| < x} is a slowly varying at infinity function, C is an abso-
lute positive constant, depending only on the distribution Fα, Hα =
∫ 1
0 (g2(t)−g1(t))−αdt.
If α = 2, then C = π2/8. Here and in the sequel, I{B} denotes the indicator of the event
B, [x] denotes the integer part of x.
Theorem 1 for g1(t) ≡ −1 and g2(t) ≡ 1 has been obtained in Frolov [8].
Conditions, necessary and sufficient for belonging of the distribution of ξ1 to a domain
of attraction of Fα, are well known. These conditions are usually stated in terms of
asymptotic behaviours for tails or truncated moments (cf., for example, Feller [9], Chapter
XVII, §5). Nevertheless, to check the conditions of Theorem 1, it is more convenient to
apply Theorem 2.6.5, p. 103 from Ibragimov and Linnik [10] where such conditions are
given in terms of an asymptotic behaviour of a characteristic function at zero. Since
Eeitξ1 = exp
{
−itc+ EeitX − 1
}
, one can easily check these conditions.
The simplest example of Λ(t) is Λ(t) = Λf(t), where the random variable Λ and the
function f(t) satisfy the conditions of Theorem 1.(Note that the random variable Λ may
be degenerate and we will deal in this case with the non-homogeneous Poisson process
N(t) and the corresponding process S(t).) We will arrive to another examples, if Λ(t)
will be a stochastic process which satisfies the law of large numbers and has appropriate
trajectories.
Theorem 1 yields that if a > 0, then the behaviour of logPT is the same as that for
probabilities of small deviations for sums of independent identically distributed random
variables and homogeneous processes with independent increments. Now we turn to the
case a = 0. In this case the asymptotic of logPT may be quite different.
Theorem 2. Assume that the conditions of Theorem 1 hold and a = 0.
Then for every positive function xT with xT → ∞ and xT = o(B[f(T )]) as T → ∞,
the following relation holds
(2) logPT = o
(
f(T )
xα
T
L(xT )
)
as T → ∞,
where L(x) is the function from Theorem 1.
In the case g1(t) ≡ −1 and g2(t) ≡ 1, Theorem 2 has been proved in Frolov [8].
Theorem 2 does not give the exact asymptotic of logPT . In the next result we find this
asymptotic which depends on the behaviour of the distribution functions of Λ(T )/f(T )
and Λ at zero.
Theorem 3. Assume that the conditions of Theorem 1 hold and aT = a = 0 for all T .
Put FT (λ) = P (Λ(T ) < λf(T )) and F (λ) = P (Λ < λ).
For every positive function xT with xT → ∞ and xT = o(B[f(T )]) as T → ∞, define
εT by the relation
εT = sup
{
ε > 0 :
ε
− logFT (ε)
� xα
T
CHαf(T )L(xT )
}
,
where the function L(x) and the constants C and Hα are from Theorem 1. Assume that
εT is equivalent to a continuous decreasing function and FT (εT ) → 0 as T → ∞.
Suppose that for every τ > 0, the following relation holds
(3) logFT (τεT ) ∼ logFT (εT ) as T → ∞.
Then
(4) logPT = logFT (εT )
(
1 + o(1)
)
= −εTCHα
f(T )
xα
T
L(xT )
(
1 + o(1)
)
as T → ∞.
SMALL DEVIATIONS FOR COMPOUND COX PROCESSES 23
Here εT → 0 as T → ∞.
Theorem 3 for g1(t) ≡ −1 and g2(t) ≡ 1 and continuous FT (λ) and F (λ) has been
obtained in Frolov [8].
Note that if for all T the distribution functions FT (λ) and F (λ) are continuous and
positive in the non-degenerate interval [0, λ0], then in Theorem 3, εT is a continuous
decreasing function and FT (εT ) → 0 as T → ∞. Indeed, FT (εT ) � Δ + F (εT ), where
Δ = sup0�x�λ0
|FT (x) − F (x)| → 0 as T → ∞ by the weak convergence of FT (λ) to
F (λ) and the continuity of the limit function.
We now show that that the asymptotic of logPT in (1) and (4) are quite different. To
this goal, suppose that FT (x) ≡ F (x) for all T . If, for example, F (x) = xp for x ∈ [0, 1],
where p > 0, then logFT (εT ) ∼ −p log(f(T )
xα
T
L(xT )) as T → ∞. If F (x) = (− log x)−p for
x ∈ (0, e−1], where p > 0, then logFT (εT ) ∼ −p log log(f(T )
xα
T
L(xT )) as T → ∞.
It turns out that the condition (3) can not be omitted in Theorem 3. This follows
from the next result.
Theorem 4. Assume that all the conditions of Theorem 3 hold except the condition (3).
Assume that for all τ > 0, the following relation holds logFT (τεT ) ∼ τp logFT (εT ) as
T → ∞, where p > 0.
Then
(5) logPT = o
(
εT
f(T )
xα
T
L(xT )
)
as T → ∞.
Theorem 4 for g1(t) ≡ −1 and g2(t) ≡ 1 and continuous FT (λ) and F (λ) has been
proved in Frolov [8].
2. Proofs
Put ξ̄(t) = ξ(t) − ct.
In what follows, we will use the following result.
Lemma 1. If the function f(t) satisfies the conditions of Theorem 1, then the probability
P (g1(t)xT � ξ̄(λf(T )t) � g2(t)xT for all t ∈ [0, 1]) is non-increasing in λ.
Proof of Lemma 1. Take λ > 1. Then
P (g1(t)xT � ξ̄(λf(T )t) � g2(t)xT for all t ∈ [0, 1])
= P
(
g1
(
u
λf(T )
)
xT � ξ̄(u) � g2
(
u
λf(T )
)
xT for all u ∈ [0, λf(T )]
)
� P
(
g1
(
u
λf(T )
)
xT � ξ̄(u) � g2
(
u
λf(T )
)
xT for all u ∈ [0, f(T )]
)
� P
(
g1
(
u
f(T )
)
xT � ξ̄(u) � g2
(
u
f(T )
)
xT for all u ∈ [0, f(T )]
)
= P (g1(t)xT � ξ̄(f(T )t) � g2(t)xT for all t ∈ [0, 1]).
In the last inequality we have used that g1(t) is non-increasing and g2(t) is non-decreasing.
We also need the following result on asymptotics of small deviations for compound
Poisson process ξ(t).
Lemma 2. Let g(T ) be an increasing, continuous function with g(T ) → ∞ as T → ∞.
24 ANDREI N. FROLOV
If the conditions of Theorem 1 hold, then for every positive function xT with xT → ∞
and xT = o(B[g(T )]) as T → ∞, the following relation holds
(6)
logP
(
g1(t)xT � ξ̄(g(T )t) � g2(t)xT for all t ∈ [0, 1]
)
= −CHα
g(T )
xα
T
L(xT )
(
1 + o(1)
)
as T → ∞, where L(x) and C,Hα are the function and the constants from Theorem 1.
Proof of Lemma 2. Take a sequence {Tk} such that Tk ↗ ∞ as k → ∞.
Since ξ(t) is a homogeneous process with independent increments, we have by Theorem
4 from Mogul’skii [1] and Lemma 1
P
(
g1(t)xTk
� ξ̄(g(Tk)t) � g2(t)xTk
for all t ∈ [0, 1]
)
�P
(
g1(t)xTk
� ξ̄([g(Tk)]t) � g2(t)xTk
for all t ∈ [0, 1]
)
= exp
{
−CHα
[g(Tk)]
xα
Tk
L(xTk
)(1 + o(1))
}
as k → ∞.
The lower bound for the probability in (6) may be derived in the same way. Taking into
account that the sequence {Tk} may be chosen arbitrarily, we get (6).
Proof of Theorem 1. Put FT (λ) = P (Λ(T ) < λf(T )).
Taking into account Lemma 1 and the independence of ξ̄(t) and Λ(t), we have
PT = P
(
g1
(
Λ(t)
Λ(T )
)
xT � ξ̄(Λ(t)) � g2
(
Λ(t)
Λ(T )
)
xT for all t ∈ [0, T ]
)
= P (g1(t)xT � ξ̄(Λ(T )t) � g2(t)xT for all t ∈ [0, 1])
=
∫ ∞
0
P (g1(t)xT � ξ̄(λf(T )t) � g2(t)xT for all t ∈ [0, 1])dFT (λ)(7)
=
∫ ∞
aT
P (g1(t)xT � ξ̄(λf(T )t) � g2(t)xT for all t ∈ [0, 1])dFT (λ)
� P (g1(t)xT � ξ̄(aT f(T )t) � g2(t)xT for all t ∈ [0, 1]).
Take ε ∈ (0, a). Then aT � a − ε for all sufficiently large T . By Lemma 1 it follows
that
(8) PT � P (g1(t)xT � ξ̄((a− ε)f(T )t) � g2(t)xT for all t ∈ [0, 1]).
for all sufficiently large T .
The norming constants Bn may be chosen such that Bn = n1/αL1(n), where L1(x) is
a slowly varying at infinity function (cf., for example, [10], p. 48). It follows that the
condition xT = o(B[f(T )]) as T → ∞ is equivalent to the condition xT = o(B[bf(T )]) as
T → ∞, where b is an arbitrary fixed positive constant. By Lemma 2
logP
(
g1(t)xT � ξ̄((a− ε)f(T )t) � g2(t)xT for all t ∈ [0, 1]
)
= −CHα(a− ε)
f(T )
xα
T
L(xT )
(
1 + o(1)
)
as T → ∞. The latter and (8) yield that
lim sup
T→∞
xα
T
f(T )L(xT )
logPT � −CHα(a− ε).
Taking in the last inequality the limit as ε→ 0, we get
(9) lim sup
T→∞
xα
T
f(T )L(xT )
logPT � −CHαa.
SMALL DEVIATIONS FOR COMPOUND COX PROCESSES 25
Take ε > 0. Since aT → a as T → ∞, we get from (7) and Lemma 1 that
PT �
∫ (1+ε)2aT
aT
P (g1(t)xT � ξ̄(λf(T )t) � g2(t)xT for all t ∈ [0, 1])dFT (λ)
� P (g1(t)xT � ξ̄((1 + ε)2aT f(T )t) � g2(t)xT for all t ∈ [0, 1])FT ((1 + ε)2aT )
� P (g1(t)xT � ξ̄((1 + ε)3af(T )t) � g2(t)xT for all t ∈ [0, 1])FT ((1 + ε)a)
for all sufficiently large T . Choose ε such that (1 + ε)a is a point of continuity for
F (λ) = P (Λ < λ). By the weak convergence of FT (λ) to F (λ) we have
(10) PT � 1
2
P (g1(t)xT � ξ̄((1 + ε)3af(T )t) � g2(t)xT for all t ∈ [0, 1])F ((1 + ε)a)
for all sufficiently large T . Since xT = o(B[(1+ε)3af(T )]) as T → ∞, by Lemma 2
logP
(
g1(t)xT � ξ̄((1 + ε)3af(T )t) � g2(t)xT for all t ∈ [0, 1]
)
= −C(1 + ε)3aHα
f(T )
xα
T
L(xT )
(
1 + o(1)
)
as T → ∞. It follows from (10) that
(11) lim inf
T→∞
xα
T
f(T )L(xT )
logPT � −CaHα(1 + ε)3.
Taking in the last inequality the limit as ε→ 0, we get
lim inf
T→∞
xα
T
f(T )L(xT )
logPT � −CaHα,
which together with (9) yields (1).
Proof of Theorem 2. It is clear that logPT � 0 and we need only prove the lower bound.
Take ε > 0. Using (7) and Lemma 1, we have
PT �
∫ (1+ε)2ε
ε
P (g1(t)xT � ξ̄(λf(T )t) � g2(t)xT for all t ∈ [0, 1])dFT (λ)
� P (g1(t)xT � ξ̄((1 + ε)2εf(T )t) � g2(t)xT for all t ∈ [0, 1])(FT ((1 + ε)2ε) − FT (ε))
for all sufficiently large T . In the same way as in the proof of Theorem 1, the last implies
(11) with ε instead of a. Taking the limit as ε→ 0, we get (2).
Proof of Theorem 3. Put bT = CHα
f(T )
xα
T
L(xT ). The condition xT = o(B[f(T )]) as T → ∞
and formulae for the norming constants Bn, which may be chosen to satisfy
(12)
nL(Bn)
Bα
n
→ d as n→ ∞,
(cf., for example, [9], Chapter XVII, §5), imply that bT → ∞ as T → ∞.
We will first prove that εT → 0 as T → ∞.
Suppose that there exists a sequence {Tk} such that Tk ↗ ∞ as k → ∞ and
εTk
> ε > 0 for all sufficiently large k, where ε is a point of continuity of F (λ).
Then − logFTk
(εTk
) � − logFTk
(ε) � − log(F (ε)/2) < ∞ for all sufficiently large k
in view of the weak convergence of FT (λ) to F (λ). Hence 1/bTk
� −εTk
/ logFTk
(εTk
) >
−ε/ log(F (ε)/2) > 0 which contradicts to the relation bT → ∞ as T → ∞.
By the definition of εT , we get −3εT/ log(FT (3εT )) > 1/bT . This and (3) imply
that −2εT/ log(FT (εT )) � 1/bT for all sufficiently large T . The latter and the relation
26 ANDREI N. FROLOV
FT (εT ) → 0 as T → ∞ give 1/bT = o(εT ) as T → ∞. The last relation, the definition of
bT and (12) with n = [f(T )εT τ ], τ > 0, yield that
xα
TL(B[f(T )εT τ ])
Bα
[f(T )εT τ ]L(xT )
→ 0 as T → ∞.
Using the well known representation of a slowly varying function (cf., for example, [9]
Chapter VIII, §9), we have that
L(B[f(T )εT τ ])
L(xT )
=
l(B[f(T )εT τ ])
l(xT )
exp
{∫ B[f(T )εT τ]
xT
�(u)
u
du
}
,
where l(u) → l <∞ and �(u) → 0 as u→ ∞. This yields that
L(B[f(T )εT τ ])
L(xT )
� 1
2
(
xT
B[f(T )εT τ ]
)−α/2
for all sufficiently large T . It follows that xT = o(B[f(T )εT τ ]) as T → ∞, where τ > 0.
Without loss of generality we will assume in the rest of the proof that εT is continuous.
Otherwise, one can replace εT by an equivalent function in the sequel.
We have from (7) and Lemma 1
PT =
∫ εT
0
P (g1(t)xT � ξ̄(λf(T )t) � g2(t)xT for all t ∈ [0, 1])dFT (λ)
+
∫ ∞
εT
P (g1(t)xT � ξ̄(λf(T )t) � g2(t)xT for all t ∈ [0, 1])dFT (λ)
� FT (εT ) + P (g1(t)xT � ξ̄(εT f(T )t) � g2(t)xT for all t ∈ [0, 1])
� e−εT bT + P (g1(t)xT � ξ̄(εT f(T )t) � g2(t)xT for all t ∈ [0, 1]).
Applying Lemma 2, we get the upper bound in (4).
We now turn to the lower bound. Take τ > 0. By Lemmas 1 and 2
PT �
∫ τεT
0
P (g1(t)xT � ξ̄(λf(T )t) � g2(t)xT for all t ∈ [0, 1])dFT (λ)
� P (g1(t)xT � ξ̄(τεT f(T )t) � g2(t)xT for all t ∈ [0, 1])FT (τεT )
= e−τεT bT (1+o(1))FT (τεT )
as T → ∞. By the definition of εT we get logFT ((1 + τ)εT ) > −(1 + τ)εT bT . It follows
from the last inequality and (3) that
PT � e−τεT bT (1+o(1))+log FT ((1+τ)εT )(1+o(1)) � e−(1+2τ)εT bT (1+o(1))(13)
as T → ∞. Since τ may be chosen arbitrarily small, we get the lower bound in (4).
Proof of Theorem 4. As in the proof of Theorem 2, we need only prove the lower bound.
In the same way as in proof of (13), we get
PT � e−(τp(1+τ)1−p+τ)εT bT (1+o(1))
as T → ∞ which yields (5).
SMALL DEVIATIONS FOR COMPOUND COX PROCESSES 27
Bibliography
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Lecture Notes in Mathematics, Vol. 1648, Springer, Berlin (1996), 165-294.
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|
| id | nasplib_isofts_kiev_ua-123456789-4548 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-07T17:03:40Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Frolov, A.N. 2009-12-03T16:34:21Z 2009-12-03T16:34:21Z 2008 On asymptotic behaviour of probabilities of small deviations for compound Cox processes / A.N. Frolov // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 19–27. — Бібліогр.: 10 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4548 519.21 We derive logarithmic asymtotics for probabilities of small deviations for compound Cox processes in the space of trajectories. We find conditions under which these asymptotics are the same as those for sums of independent identically distributed random variables and homogeneous processes with independent increments. We show that if these conditions do not hold, the asymptotics of small deviations for compound Cox processes are quite different. This article was partially supported by RFBR, grant 05–01–00486. en Інститут математики НАН України On asymptotic behaviour of probabilities of small deviations for compound Cox processes Article published earlier |
| spellingShingle | On asymptotic behaviour of probabilities of small deviations for compound Cox processes Frolov, A.N. |
| title | On asymptotic behaviour of probabilities of small deviations for compound Cox processes |
| title_full | On asymptotic behaviour of probabilities of small deviations for compound Cox processes |
| title_fullStr | On asymptotic behaviour of probabilities of small deviations for compound Cox processes |
| title_full_unstemmed | On asymptotic behaviour of probabilities of small deviations for compound Cox processes |
| title_short | On asymptotic behaviour of probabilities of small deviations for compound Cox processes |
| title_sort | on asymptotic behaviour of probabilities of small deviations for compound cox processes |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4548 |
| work_keys_str_mv | AT frolovan onasymptoticbehaviourofprobabilitiesofsmalldeviationsforcompoundcoxprocesses |