Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes. II
Necessary and suffcient conditions for weak convergence of first-rareevent times for semi-Markov processes, obtained in the first part of this paper [66], are applied to counting processes generating by flows of rare events controlled by semi-Markov processes, random geometric sums, and risk proces...
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irk-123456789-44662009-11-12T12:00:36Z Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes. II Silvestrov, D.S. Drozdenko, M.O. Necessary and suffcient conditions for weak convergence of first-rareevent times for semi-Markov processes, obtained in the first part of this paper [66], are applied to counting processes generating by flows of rare events controlled by semi-Markov processes, random geometric sums, and risk processes. In particular, necessary and sufficient conditions for stable approximation of ruin probabilities including the case of diffusion approximation are given. 2006 Article Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes. II / D.S. Silvestrov, M.O. Drozdenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 187–202. — Бібліогр.: 75 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4466 en Інститут математики НАН України |
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Necessary and suffcient conditions for weak convergence of first-rareevent
times for semi-Markov processes, obtained in the first part of this
paper [66], are applied to counting processes generating by flows of rare
events controlled by semi-Markov processes, random geometric sums,
and risk processes. In particular, necessary and sufficient conditions for
stable approximation of ruin probabilities including the case of diffusion
approximation are given. |
| format |
Article |
| author |
Silvestrov, D.S. Drozdenko, M.O. |
| spellingShingle |
Silvestrov, D.S. Drozdenko, M.O. Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes. II |
| author_facet |
Silvestrov, D.S. Drozdenko, M.O. |
| author_sort |
Silvestrov, D.S. |
| title |
Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes. II |
| title_short |
Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes. II |
| title_full |
Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes. II |
| title_fullStr |
Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes. II |
| title_full_unstemmed |
Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes. II |
| title_sort |
necessary and sufficient conditions for weak convergence of first-rare-event times for semi-markov processes. ii |
| publisher |
Інститут математики НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/4466 |
| citation_txt |
Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes. II / D.S. Silvestrov, M.O. Drozdenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 187–202. — Бібліогр.: 75 назв.— англ. |
| work_keys_str_mv |
AT silvestrovds necessaryandsufficientconditionsforweakconvergenceoffirstrareeventtimesforsemimarkovprocessesii AT drozdenkomo necessaryandsufficientconditionsforweakconvergenceoffirstrareeventtimesforsemimarkovprocessesii |
| first_indexed |
2025-07-02T07:42:17Z |
| last_indexed |
2025-07-02T07:42:17Z |
| _version_ |
1836520191930925056 |
| fulltext |
Theory of Stochastic Processes
Vol. 12 (28), no. 3–4, 2006, pp. 187–202
D. S. SILVESTROV AND M. O. DROZDENKO
NECESSARY AND SUFFICIENT CONDITIONS FOR
WEAK CONVERGENCE OF FIRST-RARE-EVENT TIMES
FOR SEMI-MARKOV PROCESSES. II
Necessary and sufficient conditions for weak convergence of first-rare-
event times for semi-Markov processes, obtained in the first part of this
paper [66], are applied to counting processes generating by flows of rare
events controlled by semi-Markov processes, random geometric sums,
and risk processes. In particular, necessary and sufficient conditions for
stable approximation of ruin probabilities including the case of diffusion
approximation are given.
4. Introduction
In the first part of the present paper [66], necessary and sufficient con-
ditions for weak convergence of first-rare-event times for semi-Markov pro-
cesses. In the second part of the paper, we apply these results to count-
ing processes generating by flows of rare events controlled by semi-Markov
processes, random geometric sums, and risk processes. In particular, we
give necessary and sufficient conditions for stable approximation of ruin
probabilities including the case of diffusion approximation are given.
We use all notations and conditions introduced in the first part of the
present paper as well as continue numbering of sections, theorems and lem-
mas began there.
First-rare-event times for semi-Markov processes considered in the first
part of the present paper reduce to random geometric sums in the case of
degenerated imbedded Markov chain.
In this way, our results are connected with asymptotical results for geo-
metric sums. Here, we would like first to mention originating papers by
Rényi (1956), who first formulated conditions of convergence of geometrical
sums to exponential law, and works by Kovalenko (1965), Gnedenko and
Fraier (1969), Szynal (1976), Korolev (1989), Kalashnikov and Vsekhsvy-
atski (1989), Melamed (1989), Kruglov and Korolev (1990), Kartashov
(1991), Gnedenko and Korolev (1996), Kalashnikov (1997), Bening and Ko-
rolev (2002).
It should be noted that the class of possible limiting laws for standard
geometric random sums, with summands independent on geometric random
2000 Mathematics Subject Classification. Primary 60K15, 60F17, 60K20.
Key words and phrases. Weak convergence, semi-Markow processes, first-rare-event
time, limit theorems, necessary and sufficient conditions.
187
188 D. S. SILVESTROV AND M. O. DROZDENKO
indices and possessing regularly varying tail probabilities, was described
by Kovalenko (1965) who also credited for necessary and sufficient condi-
tions for weak convergence of such sums. These results were generalized by
Kruglov and Korolev (1990), in particular, to the case of triangular array
mode. Our results yield necessary and sufficient conditions for weak con-
vergence summands have regularly varying tail probabilities but for more
general geometric sums with summands that can depend on geometric ran-
dom indices via indicators of rare events. Also, our conditions have different
and, as we think, more convenient form than in the works mentioned above.
We give also necessary and sufficient conditions for weak convergence
of counting processes generating by the corresponding flows of rare events
to standard renewal flows. These results are connected with the results re-
lated to studies of convergence for rareficated stochastic flows given in Rényi
(1956), Belyaev (1963), Kovalenko (1965), Mogyoródi (1971, 1972a, 1972b),
Szantai (1971a, 1971b), R̊ade (1972a, 1972b), Zakusilo (1972a, 1972b),
Jagers (1974), Jagers and Lindvall (1974), Tomko (1974), Kallenberg (1975)
and Lindvall (1976), Serfozo (1976, 1984a, 1984b), Gasanenko (1980), and
Böker and Serfozo (1983).
As possible areas of applications, we can point out limit theorems for
different lifetime functionals such as occupation times or waiting times in
queuing systems, lifetime in reliability models, extinction times in popula-
tion dynamic models, ruin times for insurance models, etc. We mention
here works by Gnedenko (1964a, 1964b), Gnedenko and Kovalenko (1964,
1987), Silvestrov (1969), Masol (1973), Kovalenko (1975, 1977, 1980, 1994),
Solov’ev (1971), 1983), Ivchenko, Kashtanov, and Kovalenko (1979), Ko-
valenko and Kuznetsov (1988), Kovalenko, Kuznetsov, Pegg (1997), En-
glund (1999a, 1999b), Anisimov, Zakusilo, and Donchenko (1987), Ass-
mussen (1987, 2003), Kalashnikov and Rachev (1990), Gyllenberg and Sil-
vestrov (1994).
We apply our results to asymptotical analysis of non-ruin probabilities for
risk processes. We study asymptotics for non-ruin probabilities in the crit-
ical case when the initial capital of an insurance company tends to infinity
and simultaneously the safety loading coefficient tends to 1. Two cases are
considered, when the claim distribution belongs to the domain of attraction
of a degenerated or a stable law. The first case corresponds to the classical
model of diffusion approximation, the second one can be referred as the case
of stable approximation for non-ruin probabilities.
As was mention above, we give necessary and sufficient conditions for sta-
ble approximation for non-ruin probabilities, including the case of diffusion
approximation.
Our results are connected with the results related to various sufficient
conditions for diffusion approximation given in Iglehart (1969), Siegmund
(1975), Harrison (1977), Grandell (1977, 1991), Gerber (1979), Asmussen
FIRST-RARE-EVENT TIMES 189
(1987, 1989, 2000, 2003), Glynn (1990), Schmidli (1992, 1997), Gyllenberg
and Silvestrov (1999, 2000b), and Silvestrov (2000b).
It is not out of picture to note that asymptotical relations related to dif-
fusion approximation for ruin probabilities can be also interpreted in terms
of the queue theory as variants of so-called heavy traffic approximation for
waiting times (see, for instance, Asmussen (1987)).
The results presented in the paper are also connected with results related
to asymptotics of ruin probabilities for risk processes with heavy tailed
claim distributions given by von Bahr (1975), Thorin and Wikstad (1976),
Embrechts, Goldie and Veraverbeke (1979), Embrechts and Veraverbeke
(1982), Assmussen (1996, 2000, 2003), Embrehts (1997), Klüppelberg and
Stadmular (1998), Zinchenko (1999). It should be noted, however, that the
latter group of works relates to so-called sub-critical case when the safety
loading coefficient less than and separated of 1.
The paper is organized in the following way. In Section 2 we give necessary
and sufficient of weak convergence of counting processes generating by the
corresponding flows of rare events. As was mentioned above, the model
of first-rare-event times for semi-Markov processes reduces to the model
of geometrical random sums in the case of degenerated imbedded Markov
chain. In Section 3, we present applications of results obtained in the first
part of the present paper to random geometric sums as well as give necessary
and sufficient conditions for stable approximation for non-ruin probabilities.
5. Flows of rare events
In this section we study conditions of convergence for the flows of rare
events connected with the model considered above.
Let us define recursively random variables
νε(k) = min(n ≥ νε(k − 1) : ζn ∈ Dε), k = 1, 2, . . . ,
where νε(0) = 0. A random variable νε(k) counts the number of transitions
of the imbedded Markov chain ηn up to the k-th appearance of the “rare”
event {ζn ∈ Dε}. Obviously, νε(1) = νε.
Let us also define inter-rare-event times,
κε(k) =
νε(k)∑
n=νε(k−1)
κn, k = 1, 2, . . . .
Let us also introduce random variables showing positions of the imbedded
Markov chain ηn at moments νε(k),
ηε(k) = ηνε(k), k = 0, 1, . . . .
Obviously (ηε(k), κε(k)), k = 0, 1, . . . (here κε(k) = 0) is a Markov re-
newal process, i.e. a homogeneous Markov chain with the phase space
190 D. S. SILVESTROV AND M. O. DROZDENKO
X × [0,∞) and transition probabilities,
(1)
P{ηε(k + 1) = j, κε(k + 1) ≤ t/ηε(k) = i, κε(k) = s}
= P{ηε(k + 1) = j, κε(k + 1) ≤ t/ηε(k) = i}
= Q
(ε)
ij (t) = Pi{ηνε = j, ξε ≤ t}, i, j ∈ X, s, t ≥ 0.
Let us now define random variables,
ξε(k) =
νε(k)∑
n=1
κn =
k∑
n=1
κε(n), k = 0, 1, . . . .
Random variable ξε(k) can be interpreted as the time of k-th appearance
of the rare event time for the semi-Markov process η(t). Obviously, ξε(0) = 0
and ξε(1) = ξε.
Now we can define a counting stochastic process that describes the flow
of rare events,
Nε(t) = max(k ≥ 0 : ξε(k) ≤ tuε), t ≥ 0.
Note that the time scale for this counting process is stretched with the
use of the scale parameter uε according the asymptotic results given in
Theorem 1.
Let us also define the corresponding limiting counting process. Let κ(k),
k = 1, 2, . . . be a sequence of positive i.i.d. random variables with the
distribution Fa,γ(u), and
ξ(k) =
k∑
n=1
κ(n), n = 0, 1, . . . .
Let us also define the standard renewal counting process with i.i.d. inter-
renewal times κ(k), k = 1, 2, . . .,
N(t) = max(k ≥ 0 : ξ(k) ≤ t), t ≥ 0.
Theorem 2. Let conditions A, B, C, and D hold. Then, the class
of all possible non-zero limiting counting processes (in the sense of weak
convergence of finite-dimensional distributions) for the counting processes
Nε(t), t ≥ 0 coincides with the class of standard renewal counting process
N(t), t ≥ 0 with the distribution function of inter-renewal time Fa,γ(u) where
0 < γ ≤ 1, a > 0. Conditions Eγ and Fa,γ are necessary and sufficient
for such convergence in the case when the corresponding limiting counting
process has the distribution function of inter-renewal time Fa,γ(u).
Proof. Obviously,
F
(ε)
i (u) = Pi{ξε ≤ u} =
∑
j∈X
Q
(ε)
ij (t), u ≥ 0.
FIRST-RARE-EVENT TIMES 191
Using Markov property of the Markov renewal process (ηε(k), κε(k)) we
get the following formula for joint distributions of the properly normalized
inter-renewal times for the counting process Nε(t),
(2)
Pi{κε(k)/uε ≤ tk, k = 1, . . . , n}
=
∑
j∈X
Pi{κε(k)/uε ≤ tk, k = 1, . . . , n − 1, ηε(n − 1) = j}
× F
(ε)
j (tnuε), i ∈ X, t1, . . . , tn ≥ 0, n = 1, 2, . . . .
According Theorem 1, under A, B, C, and D, conditions Eγ and Fa,γ
imply that (a) F
(ε)
j (tnuε) → Fa,γ(tn) as ε → 0, tn ≥ 0, j ∈ X.
Using (a) and relation (2) we get that, under A, B, C, and D, conditions
Eγ and Fa,γ imply that, for every i ∈ X, n = 1, 2, . . . , t1, . . . , tn ≥ 0,
(3)
Pi{κε(k)/uε ≤ tk, k = 1, . . . , n}
→
n∏
k=1
Fa,γ(tk) as ε → 0.
Relation (3) means that the inter-renewal times κε(k)/uε, k = 1, 2, . . .
are asymptotically independent. Note that the multivariate distribution
function on the right hand side in (3) is continuous. Due to this fact,
relation (3) implies in an obvious way that, for every i ∈ X, real 0 = t0 ≤
t1 ≤ · · · ≤ tn, integer 0 ≤ r1 ≤ · · · ≤ rn, n = 1, 2, . . .,
(4)
Pi{Nε(tk) ≥ rk, k = 1, . . . , n}
= Pi{ξε(rk)/uε ≤ tk, k = 1, . . . , n}
→ Pi{ξ(rk) ≤ tk, k = 1, . . . , n}
= Pi{N(tk) ≥ rk, k = 1, . . . , n} as ε → 0,
The statement of necessity follows is trivial and follows from the following
formula
Pi{Nε(t) ≥ 1} = Pi{ξε/uε ≤ t}, t ≥ 0.
The proof is complete. �
What is interesting, that under A, B, C, and D, conditions Eγ and Fa,γ
are not sufficient for weak convergence of transition probabilities of the
Markov renewal process (ηε(k), κε(k)) that forms the counting process Nε(t).
It follows from the following lemma which describes the asymptotic be-
havior of so-called “absorbing” probabilities,
Q
(ε)
ij (∞) = Pi{ηνε = j}, i, j ∈ X.
Let us denote
piε(r) = Pi{ζ1 ∈ Dε, η1 = r}, i, r ∈ X,
192 D. S. SILVESTROV AND M. O. DROZDENKO
and
pε(r) =
m∑
i=1
πipiε(r), j ∈ X.
By the definition,
(5) pε =
∑
i∈X
πiPi{ζ1 ∈ Dε} =
∑
r∈X
pε(r).
Lemma 7. Let conditions B, C hold. Then, for every i ∈ X,
(6) Q
(ε)
ir (∞) − pε(r)
pε
→ 0 as ε → 0, r ∈ X.
Proof. Let us define the probability that the first rare event will occur when
the state of the imbedded Markov chain will be r and before the first hitting
of the imbedded Markov chain in the state i, under condition that the initial
state of this Markov chain η0 = j,
qjiε(r) = Pj{νε ≤ τi, ηνε = r}, i, j, r ∈ X.
Taking into account that the Markov renewal process (ηn, κn, ζn) regen-
erates at moments of return to every state i and νε is a Markov moment
for this process, we can get following cyclic representation for absorbing
probabilities Q
(ε)
ir (∞),
Q
(ε)
ir (∞) =
∞∑
n=0
Pi{τi(n) < νε ≤ τi(n + 1), ηνε = j}
=
∞∑
n=0
(1 − qiε)
nqiiε(r) =
qiiε(r)
qiε
, i, r ∈ X.(7)
The probabilities qjiε(r), j ∈ X satisfy, for every i, r ∈ X, the following
system of linear equations similar with system (20) given in the proof of
Lemma 1,
(8)
{
qjiε(r) = pjε(r) +
∑
k �=i
p
(ε)
jk qkiε(r)
j ∈ X
This system has the matrix of coefficients iP
(ε) as the system of linear
equations (20) mentioned above and differs of this system only by the free
terms. Thus by repeating reasoning given in the proof of Lemma 1 we can
get the following formula similar with formula (24) also given in the proof
of Lemma 1,
(9) qiiε(r) =
m∑
k=1
Eiδikε pkε(r).
Recall that it was shown in the proof of Lemma 1 that (b) Eiδikε → πk/πi
as ε → 0, for i, k ∈ X. Using (b), relation (c) πiqiε/pε → 1 given in
FIRST-RARE-EVENT TIMES 193
Lemma 1, and inequality (d) pε(r) ≤ pε, following from formula (5), we get,
for every i, r ∈ X,
∣∣∣qiiε(r) − pε(r)
πi
∣∣∣
qiε
≤
m∑
k=1
∣∣∣∣Eiδikε − πk
πi
∣∣∣∣ · πipkε(r)∑m
j=1 πjpjε
· pε
πiqiε
≤
m∑
k=1
∣∣∣∣Eiδikε − πk
πi
∣∣∣∣ · πi
πk
· pε
πiqiε
→ 0 as ε → 0.(10)
Using (c) and (d) ones more time we get,
∣∣∣∣pε(r)
πiqiε
− pε(r)
pε
∣∣∣∣ ≤ pε(r)
pε
·
∣∣∣qiε − pε
πi
∣∣∣
πiqiε
≤
∣∣∣qiε − pε
πi
∣∣∣
pε
· pε
πiqiε
→ 0 as ε → 0.(11)
Formula (7) together with relations (10) and (11) imply in an obvious
way relation (6). This completes the proof. �
Let us introduce the following balancing condition:
L:: pε(j)
pε
→ Qj as ε → 0, j ∈ X.
Constants Qj , automatically satisfy the following conditions (e1) Qj ≥
0, j ∈ X, and (e2)
∑
j∈X Qj = 1.
Lemma 7 implies the following statement.
Lemma 8. Let conditions B, C hold. Then, condition L is necessary
and sufficient for the following relation to hold (for some or every i ∈ X,
respectively, in the statements of necessity and sufficiency),
(12) Q
(ε)
ir (∞) → Qr as ε → 0, r ∈ X.
The following theorem shows that the-first-rare-event times ξε and ran-
dom functional ηνε are asymptotically independent, and completes the de-
scription of the asymptotic behavior of the transition probabilities Q
(ε)
ij (t)
for the Markov renewal process (ηε(k), κε(k)).
Theorem 3. Let conditions A, B, C, and D hold. Then, conditions Eγ,
Fa,γ and L are necessary and sufficient for the asymptotic relation (2) given
in Theorem 1 and the asymptotic relation (12) given in Lemma 8 to hold.
In this case, for every u ∈ [0,∞], i, r ∈ X,
(13) Q
(ε)
ir (uuε) → Fa,γ(u)Qr as ε → 0.
194 D. S. SILVESTROV AND M. O. DROZDENKO
Proof. The first statement of the theorem follows from Theorem 1 and
Lemma 8. Let us prove that conditions Eγ, Fa,γ and L imply the asymptotic
relation (13).
Let us introduce, for i, r ∈ X, Laplace transforms,
Φirε(s) = Ei exp{−sξε}χ(ηνε = r), s ≥ 0,
and
ψ̃irε(s) = Ei{exp{−sβ̃iε}χ(ηνε = r)/νε ≤ τi}, s ≥ 0.
Analogously to formula (4) given in the proof of Theorem 1 the following
representation can be written down for Laplace transforms Φirε(s),
Φirε(s) =
∞∑
n=0
(1 − qiε)
nqiεψiε(s)
nψ̃irε(s)
=
qiεψ̃irε(s)
1 − (1 − qiε)ψiε(s)
=
1
1 + (1 − qiε)
(1−ψiε(s))
qiε
· ψ̃irε(s), s ≥ 0.(14)
Let us now show that under, conditions A, B, and C, for every s ≥ 0
and i, r ∈ X,
(15) ψ̃irε(s/uε) − qiiε(r)/qiε → 0 as ε → 0.
Indeed, using Lemma 2 we get, for any δ > 0,
(16)
qiiε(r)
qiε
− ψ̃irε(
s
uε
)
= Ei{(1 − exp{−s
β̃iε
uε
})χ(ηνε = r)/νε ≤ τi}
≤ (1 − esδ) + Pi{ β̃iε
uε
≥ δ/νε ≤ τi} → (1 − esδ) as ε → 0.
Relation (16) implies, due to possibility of an arbitrary choice of δ > 0,
relation (15).
Using formula (14) and relation (29), given in Lemma 2, and (15), The-
orem 1 and Lemma 8 we get, for every s ≥ 0 and i, r ∈ X,
(17)
lim
ε→0
Φirε(s/uε)
= lim
ε→0
1
1 + (1 − qiε)
(1−ψiε(s))
qiε
· ψ̃irε(s/uε)
= lim
ε→0
Φiε(s/uε) · Q(ε)
ir (∞)
=
1
1 + asγ
· Qr.
Relation (17) equivalent to relation (13). �
FIRST-RARE-EVENT TIMES 195
6. Geometric sums and stable approximation
for non-ruin probabilities
In this section we apply our results to so-called geometric random sums.
This is reduction of our model for the case when the imbedded Markov
chain ηn has the degenerated set of states X = {1}.
In this case, the first-rare-event time ξε =
∑νε
n=1 κn is a geometric sum.
Indeed, (κn, ζn), n = 1, 2, . . . is a sequence of i.i.d. random vectors. There-
fore the random variable
νε = min(n ≥ 1 : ζn ∈ Dε)
has a geometric distribution with the success probability
pε = P{ζn ∈ Dε}.
However, the geometric random index νε and random summands κn, n =
1, 2, . . . are, in this case, dependent random variables. They depend via the
indicators of rare events χnε = χ(ζn ∈ Dε), n = 1, 2, . . .. More precisely,
(κn, χnε, n = 1, 2, . . .) is a sequence of i.i.d. random vectors.
Conditions A and B take, in this case, the following form:
A′:: lim
t→∞
lim
ε→0
P{κ1 > t/ζ1 ∈ Dε} = 0;
and
B′:: 0 < pε = P{ζ1 ∈ Dε} → 0 as ε → 0.
Condition C holds automatically.
Condition D remains and should be imposed on the function p−1
ε , defined
in condition B′, and the normalization function uε.
Conditions Eγ and Fa,γ remain and should be imposed on the distribution
function G(t) = P{κ1 ≤ t} (no averaging is involved).
A standard geometric sum is particular case of the model described above,
which corresponds to the case, when two sequences of random variables
κn, n = 1, 2, . . . and ζn, n = 1, 2, . . . are independent. In this case, the
random index νε and summands κn, n = 1, 2, . . . are also independent.
Note that a standard geometric sum with any distribution of summands
G(t) and parameter of geometric random index pε ∈ (0, 1] can be modelled
in this way. Indeed, it is enough to consider the geometric sum
ξε =
νε∑
n=1
κn
defined above, where (a1) κn, n = 1, 2, . . . is a sequence of i.i.d. random
variables with the distribution function G(t);
(a2) νε = max(n ≥ 1: ζn ∈ Dε),
where ζn, n = 1, 2, . . . is a sequence of i.i.d. random variables uniformly
distributed in the interval [0, 1] and domains Dε = [0, pε); (a3) two sequences
of random variables κn, n = 1, 2, . . . and ζn, n = 1, 2, . . . are independent.
196 D. S. SILVESTROV AND M. O. DROZDENKO
In the case of standard geometric sums, condition A holds automatically.
Theorem 1 reduces in this case to the result equivalent to those obtained by
Kovalenko (1965). The difference is in the form of necessary and sufficient
conditions. Conditions Gγ and Ha,γ, based on Laplace transforms of distri-
butions G(t), were used in this work. Our results are based on conditions
Eγ and Fa,γ , based on distributions G(t), and have, as we think, a more
transparent form.
Let us illustrate applications of Theorem 1 by giving necessary and suf-
ficient conditions for stable approximation of non-ruin probabilities. Let us
consider a process used in classical risk theory to model the business of an
insurance company,
Xε(t) = cεt −
Nλ(t)∑
n=1
Zn, t ≥ 0.
Here, a positive constant cε (depending on parameter ε > 0) is the gross
premium rate, Nλ(t), t ≥ 0 is a Poissonian process with parameter λ count-
ing the number of claims on an insurance company in the time-interval [0, t],
and Zn, n = 1, 2, . . . is a sequence of nonnegative i.i.d. random variables,
which are independent on the process Nλ(t), t ≥ 0. The random variable Zk
is the amount of the kth claim.
An important object for studies in this model is the non-ruin probabilities
on infinite time interval for a company with an initial capital u ≥ 0,
Fε(u) = P
{
u + inf
t≥0
X(t) ≥ 0
}
, u ≥ 0.
Let H(x) = P{Z1 ≤ x} be a claim distribution function. We assume the
standard condition:
M:: μ =
∫ ∞
0
sH(ds) < ∞.
The crucial role in is plaid by the so-called safety loading coefficient
αε = λμ/cε.
If αε ≥ 1 then Fε(u) = 0, u ≥ 0. The only non-trivial case is when αε < 1.
We assume the following condition:
N:: αε < 1 for ε > 0 and αε → 1 as ε → 0.
According to Pollaczek–Khinchine formula (see, for example, Asmussen
(2000)), the non-ruin distribution function Fε(u) coincides with distribution
function of a geometric random sum which is slightly differ on the standard
geometric sums considered above. Namely,
(18) Fε(u) = P{ξ′ε =
νε−1∑
n=1
κn ≤ u}, u ≥ 0,
FIRST-RARE-EVENT TIMES 197
where (a) κn, n = 1, 2, . . . is a sequence of non-negative i.i.d. random vari-
ables with distribution function
G(u) =
1
μ
∫ u
0
(1 − H(s))ds, u ≥ 0
(so-called steady claim distribution);
(b) νε = min(n ≥ 1, χnε = 1);
and (c) χnε, n = 1, 2, . . ., is a sequence of i.i.d. random variables taking
values 1 and 0 with probabilities pε = 1 − αε and 1 − pε; (d) random
sequences κn, n = 1, 2, . . . and χnε, n = 1, 2, . . . are independent.
As was mentioned above condition A holds automatically in this case,
and, therefore, due to Remark 12, Theorem 1, which specification to the
geometric sums was described above, can be applied to the geometric ran-
dom sums ξ′ε.
Conditions A and C can be omitted. Condition B is equivalent to con-
dition N. Condition D takes the following form:
D′:: uε, p
−1
ε = (1 − αε)
−1 ∈ W.
Conditions Eγ and Fa,γ (a > 0 and 0 < γ ≤ 1) take, in this case, the
following form:
E′
γ::
t
∞
t
(1−H(s))ds
t
0
s(1−H(s))ds
→ 1−γ
γ
as t → ∞.
F′
a,γ::
uε
0
s(1−H(s))ds
(1−αε)μuε
→ a γ
Γ(2−γ)
as ε → 0.
Let us summarize the discussion above in the form of the following theo-
rem which gives necessary and sufficient conditions for stable approximation
of non-ruin probabilities.
Theorem 4. Let conditions M, N, and D′ hold. Then the class of all
possible non-concentrated in zero limiting (in the sense of weak convergence)
distribution functions F (u), such that the non-ruin distribution functions
Fε(uuε) ⇒ F (u) as ε → 0, coincides with the class of distributions Fa,γ(u)
with Laplace transforms 1
1+asγ , 0 < γ ≤ 1, a > 0. Conditions E′
γ and
F′
a,γ are necessary and sufficient for weak convergence Fε(uuε) ⇒ Fa,γ(u)
as ε → 0.
In conclusion let us give and comment some sufficient conditions providing
diffusion and stable approximations for ruin probabilities.
198 D. S. SILVESTROV AND M. O. DROZDENKO
The case γ = 1 corresponds to so called diffusion approximations of risk
processes. The traditional way for obtaining a diffusion type asymptotic is
based on approximation of risk process in a proper way by a Wiener process
with a shift. Typical conditions assume finiteness of the second moment of
the claim distribution H(t) (or equivalently, finiteness of expectation for the
steady claim distribution G(t) = 1
μ
∫ t
0
(1 − H(s))ds):
O:: μ2 =
∫ ∞
0
z2H(ds) < ∞.
Obviously, condition O is sufficient for condition E′
1 to hold. Note how-
ever that the necessary and sufficient condition E′
1 does not require the
finiteness of the second moment of the claim distribution.
In this case, condition F′
a,1 takes the following simple equivalent form:
Pa:: (1 − αε)uε → b = μ2/2μa as ε → 0.
As was mentioned in Remark 1, the corresponding limiting distribution
is in this case the exponential one, with the parameter a. This is consistent
with the classical form of diffusion approximation for ruin probabilities.
Case γ ∈ (0, 1) corresponds to so-called stable approximation for ruin
probabilities. Remark 1 explains the sense of using the term “stable”.
Remind that, according Lemma 6, condition E′
γ is equivalent to the fol-
lowing condition which require regular variation for the tail probabilities of
the steady claim distribution G(t) = 1
μ
∫ t
0
(1 − H(s))ds:
Rγ::
1
μ
∫ ∞
t
(1 − H(s))ds ∼ t−γL(t)
Γ(1−γ)
as t → ∞.
As follows from theorems about regularly varying functions (see, for ex-
ample, Feller (1966)), the following condition imposing requirement of reg-
ular variation for the tail probabilities of the claim distribution H(t) is
sufficient for condition Rγ to hold:
R′
γ:: 1 − H(t) ∼ t−(γ+1)L(t)γμ
Γ(1−γ)
as t → ∞.
As far as condition F′
a,γ is concerned, it, in this case, can be formulated
in the following form equivalent to condition Ha,γ:
Sa,γ::
L(uε)
(1−αε)uγ
ε
→ a as ε → 0.
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Department of Mathematics and Physics, Mälardalen University, Box
883, SE-721 23 Väster̊as, Sweden
E-mail address: dmitrii.silvestrov@mdh.se
E-mail address: myroslav.drozdenko@mdh.se
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