Limit theorems for oscillatory functionals of a Markov process
We study the limit behavior of a family of functionals from a given Markov process which are called oscillatory functionals. The typical oscillatory functional is homogeneneous and non-negative but neither additive nor continuous. We claim that the discontinuity and non-additivity of functionals fro...
Gespeichert in:
| Datum: | 2005 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2005
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/4224 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Limit theorems for oscillatory functionals of a Markov process / T.O. Androshchuk, A.M. Kulik // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 3-13. — Бібліогр.: 6 назв.— англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-4224 |
|---|---|
| record_format |
dspace |
| spelling |
Androshchuk, T.O. Kulik, A.M. 2009-09-02T11:23:47Z 2009-09-02T11:23:47Z 2005 Limit theorems for oscillatory functionals of a Markov process / T.O. Androshchuk, A.M. Kulik // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 3-13. — Бібліогр.: 6 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4224 519.21 We study the limit behavior of a family of functionals from a given Markov process which are called oscillatory functionals. The typical oscillatory functional is homogeneneous and non-negative but neither additive nor continuous. We claim that the discontinuity and non-additivity of functionals from a given family vanish in the limit and, in this framework, prove a generalization of the theorem by E.B. Dynkin on the convergence of a family of W-functionals. This research has been partially supported by the Ministry of Education and Science of Ukraine, project N GP/F8/0086. en Інститут математики НАН України Limit theorems for oscillatory functionals of a Markov process Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Limit theorems for oscillatory functionals of a Markov process |
| spellingShingle |
Limit theorems for oscillatory functionals of a Markov process Androshchuk, T.O. Kulik, A.M. |
| title_short |
Limit theorems for oscillatory functionals of a Markov process |
| title_full |
Limit theorems for oscillatory functionals of a Markov process |
| title_fullStr |
Limit theorems for oscillatory functionals of a Markov process |
| title_full_unstemmed |
Limit theorems for oscillatory functionals of a Markov process |
| title_sort |
limit theorems for oscillatory functionals of a markov process |
| author |
Androshchuk, T.O. Kulik, A.M. |
| author_facet |
Androshchuk, T.O. Kulik, A.M. |
| publishDate |
2005 |
| language |
English |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We study the limit behavior of a family of functionals from a given Markov process which are called oscillatory functionals. The typical oscillatory functional is homogeneneous and non-negative but neither additive nor continuous. We claim that the discontinuity and non-additivity of functionals from a given family vanish in the limit and, in this framework, prove a generalization of the theorem by E.B. Dynkin on the convergence of a family of W-functionals.
|
| issn |
0321-3900 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/4224 |
| citation_txt |
Limit theorems for oscillatory functionals of a Markov process / T.O. Androshchuk, A.M. Kulik // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 3-13. — Бібліогр.: 6 назв.— англ. |
| work_keys_str_mv |
AT androshchukto limittheoremsforoscillatoryfunctionalsofamarkovprocess AT kulikam limittheoremsforoscillatoryfunctionalsofamarkovprocess |
| first_indexed |
2025-11-25T22:31:25Z |
| last_indexed |
2025-11-25T22:31:25Z |
| _version_ |
1850565018517504000 |
| fulltext |
Theory of Stochastic Processes
Vol. 11 (27), no. 3–4, 2005, pp. 3–13
UDC 519.21
TARAS O. ANDROSHCHUK AND ALEXEY M. KULIK
LIMIT THEOREMS FOR OSCILLATORY
FUNCTIONALS OF A MARKOV PROCESS
We study the limit behavior of a family of functionals from a given Markov process
which are called oscillatory functionals. The typical oscillatory functional is homo-
geneneous and non-negative but neither additive nor continuous. We claim that the
discontinuity and non-additivity of functionals from a given family vanish in the limit
and, in this framework, prove a generalization of the theorem by E.B. Dynkin on the
convergence of a family of W -functionals.
Introduction
The well-known theorem by E.B. Dynkin (see [1], Ch. 6.3) provides a characterization
of the L2-convergence of W -functionals from a given Markov process in terms of their
characteristics. In this paper, we consider the families of functionals satisfying some
weaker version of the conditions of the Dynkin theorem. Namely, elements of the family
may fail to be continuous and additive (and therefore to be W -functionals), but their
discontinuity and non-additivity vanish in the limit in an appropriate sense. The limit
results for some families of functionals of such a type have been known for a long time. We
mention three such examples: the normalized numbers of downcrossings of the interval
by a diffusion (see [2], §2.4, 6.5a), the normalized numbers of intersections of a level by
the diffusion (see [3], §6), and the entropy-like metric characteristics of the set of zeros
of the diffusion (see [2], §2.5, 6.5b). Our aim is to give a unified approach to the limit
theorems for functionals of such a type (we informally call them oscillatory functionals),
similar to the one introduced by E.B. Dynkin in [1].
1. Main theorems
In a sequel, we suppose a locally compact metric space (X , ρ) with a homogeneous
Markov process ({Xt, t ≥ 0}, {Mt, t ≥ 0}, {Px, x ∈ X}) to be fixed. Here and below, the
E.B.Dynkin’s notation (see [1], Ch. 3.1) is used. In order to shorten the exposition, we
consider only the case where the lifetime of the process ζ = +∞. The process is claimed
to be a Feller one and to have right continuous trajectories. Therefore (see ([1], Ch. 3.2),
the flow {Mt, t ≥ 0} can be supposed to be right continuous and complete w.r.t. every
probability Px, x ∈ X . We also suppose (this does not restrict generality, but makes
notation more simple) that, for every t, Mt is the smallest σ-algebra, complete w.r.t.
every probability Px, x ∈ X and containing all sets of the type
{ω : Xs1(ω) ∈ Δ1, . . . , Xsn(ω) ∈ Δn}, s1, . . . , sn ≤ t, Δ1, . . . ,Δn ∈ B(X ).
2000 AMS Mathematics Subject Classification. Primary 60J55.
Key words and phrases. Markov process, oscillatory functionals, characteristics.
This research has been partially supported by the Ministry of Education and Science of Ukraine,
project N GP/F8/0086.
3
4 TARAS O. ANDROSHCHUK AND ALEXEY M. KULIK
Let us recall some notions. The family {φs
t , 0 ≤ s ≤ t < +∞} of random variables is
called a functional of the Markov process {Xt} if, for every Γ ∈ B(�) {ω : φs
t (ω) ∈
Γ} ∈ M ≡ ∨
t Mt. A functional {φs
t} is called homogeneous if, for every s ≤ t, h > 0,
θhφs
t = φs+h
t+h almost surely. Here, θh denotes the operator of the time shift ([1], Ch. 3.1).
Definition 1. The family {φε ≡ {φs,ε
t }, ε > 0} is called a QW-family (quasi-W-family)
of functionals if, for every ε {φs,ε
t } is a homogeneous non-negative functional, non-
decreasing and càdlàg w.r.t. to t, and two following conditions hold true:
VD. (”vanishing discontinuity”): almost surely for all 0 ≤ s ≤ t,
φs,ε
t − φs,ε
t−0 ≤ ε;
VN. (”vanishing non-additivity”): almost surely for all 0 ≤ s ≤ t ≤ u,
|φs,ε
u − φs,ε
t − φt,ε
u | ≤ ε;
For a given QW-family {φε}, we denote the corresponding family of characteristics by
{fε}:
fε
t (x) = Exφ0,ε
t = Exφs,ε
t+s, x ∈ X , s, t ≥ 0.
Here and below, Ex denotes the expectation w.r.t. Px. One version of the main result
of the paper is given in the following theorem analogous to Theorem 6.4 [1].
Theorem 1. 1. Let QW-family {φε} be such that
‖fε
t ‖ ≡ sup
x∈X
fε
t (x) < ∞, ε > 0, t ≥ 0, (1)
and there exists a function f = {ft(x), t ≥ 0, x ∈ X} such that
lim
ε→0
sup
0≤u≤t
‖fε
u − fu‖ = 0, t ≥ 0. (2)
Then f is a characteristic of some non-negative homogenous additive functional φ,
and
φs
t = l.i.m.
ε→0
φs,ε
t , 0 ≤ s ≤ t < +∞.
2. Let the following additional condition hold true:
US. (”upper semiadditivity”): almost surely for all 0 ≤ s ≤ t ≤ u,
φs,ε
u − φs,ε
t − φt,ε
u ≥ 0.
Then φ is a V -functional.
The method of proof is analogous to that of Theorem 6.4 [1] and is based on two
auxiliary lemmas corresponding to Lemma 6.4 and Lemma 6.5 [1].
Lemma 1. Let conditions VD, VN and (1) hold true, and let ε > 0 be fixed. Then
Ex[φ0,ε
t ]2 ≤ 2‖fε
t ‖2 + 3ε‖fε
t ‖, t ≥ 0, x ∈ X .
Proof. For a fixed t, let us take a partition S = {0 = s0 < s1 < · · · < sM = t} and
decompose φ0,ε
t into a sum
φ0,ε
t =
M−1∑
j=0
ΦS
j , ΦS
j ≡ φ0,ε
sj+1
− φ0,ε
sj
.
One has
[φ0,ε
t ]2 =
M−1∑
j=0
[ΦS
j ]2 + 2
∑
i<j
ΦS
i ΦS
j = ΣS
1 + 2ΣS
2 .
LIMIT THEOREMS FOR OSCILLATORY FUNCTIONALS OF A MARKOV PROCESS 5
Since φ0,ε
t is non-decreasing as a function of t, we have ΦS
j ≥ 0. Therefore, if we have
two partitions S ⊂ S̃, then ΣS
1 ≥ ΣS̃
1 , and thus ΣS
2 ≤ ΣS̃
2 . Now let us take a sequence
of partitions Sn such that S1 ⊂ S2 ⊂ . . . and the diameter |Sn| of Sn tends to 0 as
n → +∞. Then, with probability 1,
ΣSn
1 → var2(φ0,ε
· )t =
∑
s≤t
[φ0,ε
s − φ0,ε
s−]2 =: ζt,
and 2ΣSn
2 ↑ [φ0,ε
t ]2 − ζt, n → +∞. Denote ΔSn
i = φ0,ε
t −φ0,ε
sn
i+1
−φ
sn
i+1,ε
t . Due to condition
VN, |ΔSn
i | ≤ ε. Then
ExΣSn
2 = Ex
Mn−1∑
i=0
ΦSn
i [φ
sn
i+1,ε
t + ΔSn
i ] ≤ εEx
Mn−1∑
i=1
ΦSn
i + Ex
Mn−1∑
i=0
ΦSn
i θsn
i+1
φ0,ε
t . (3)
The first summand on the right-hand side of (3) is equal to εfε
t (x). The second summand
is estimated by (see [1], (6.25) and Theorem 3.1)
Ex
Mn−1∑
i=0
(
ΦSn
i EXsn
i+1
φ0,ε
t
)
= Ex
Mn−1∑
i=0
ΦSn
i f0,ε
t (Xsn
i+1
) ≤ ‖fε
t ‖Exφ0,ε
t ≤ ‖fε
t ‖2. (4)
Thus, due to the theorem on monotone convergence,
Ex[φ0,ε
t ]2 − ζt ≤ 2‖fε
t ‖(‖fε
t ‖ + ε). (5)
On the other hand, condition VD implies that ζt ≤ εφ0,ε
t and therefore Exζt ≤ εfε
t (x) ≤
ε‖fε
t ‖, which together with (5) gives the needed statement. The lemma is proved.
Lemma 2. Under the conditions of Lemma 1,
Ex
(
φs,ε
t − φs,ε̃
t
)2
≤
[
2 sup
u≤ t−s
‖fε
u − f ε̃
u‖ + 5 max(ε, ε̃)
]
×
[
fs,ε
t (x) + fs,ε̃
t (x)
]
,
0 ≤ s ≤ t, ε, ε̃ > 0, where fs,ε
t (x) ≡ Exφs,ε
t .
Proof. We consider only the case where s = 0, the general case is analogous. In order
to simplify the notation, we write φs
t = φs,ε
t , φ̃s
t = φs,ε̃
t , δ = max(ε, ε̃). For a fixed t and a
fixed partition S of [0, t], we denote Φj = φ0
sj+1
− φ0
sj
, Φ̃j = φ̃0
sj+1
− φ̃0
sj
. We have
(φ0
t − φ̃0
t )
2 = ΣS
3 + 2ΣS
4 , ΣS
3 =
M−1∑
j=0
(Φj − Φ̃j)2,
ΣS
4 =
M−1∑
i=0
(Φi − Φ̃i)
[
(φ0
t − φ0
si+1
) − (φ̃0
t − φ̃0
si+1
)
]
.
Denoting also Δi = φ0
t − φ0
si+1
− φ
si+1
t , Δ̃i = φ̃0
t − φ̃0
si+1
− φ̃
si+1
t , we have that Δi, Δ̃i ∈
[−δ, δ]. The expectation of ΣS
4 can be expressed and estimated analogously to (3),(4):
ExΣS
4 = Ex
M−1∑
i=0
(Φi − Φ̃i)(Δi − Δ̃i) + Ex
M−1∑
i=1
(Φi − Φ̃i)[ft−si+1(Xsi+1)− f̃t−si+1(Xsi+1)],
(6)
where f̃ denotes the characteristic of φ̃. The first summand in (6) is estimated by
Ex
M−1∑
i=0
(Φi + Φ̃i)(|Δi| + |̃Δi|) ≤ 2δEx(φ0
t + φ̃0
t ) ≤ 2δ[ft(x) + f̃t(x)].
6 TARAS O. ANDROSHCHUK AND ALEXEY M. KULIK
The second summand is estimated by
Ex
M−1∑
i=0
(Φi + Φ̃i)
∣∣∣ft−si+1(Xsi+1) − f̃t−si+1(Xsi+1)
∣∣∣ ≤ sup
u≤t
‖fu − f̃u‖ × [ft(x) + f̃t(x)].
Therefore,
Ex(φ0
t − φ̃0
t )
2 ≤
(
4δ + 2 sup
u≤t
‖fu − f̃u‖
)
× [ft(x) + f̃t(x)] + lim sup
|S|→0
ExΣS
3 . (7)
We have ΣS
3 ≤ 2(ζS
t + ζS
t ), where
ζS
t =
M−1∑
j=0
Φ2
j , ζ̃S
t =
M−1∑
j=1
Φ̃2
j .
For every S, the variables ζS
t , ζ̃S
t are majorized by the variables [φ0
t ]
2, [φ0
t ]
2 which are
integrable due to Lemma 1. On the other hand,
ζS
t → ζt =
∑
s≤t
[φ0
s − φ0
s−]2, ζ̃S
t → ζ̃t =
∑
s≤t
[φ̃0
s − φ̃0
s−]2, |S| → 0,
Exζt ≤ εft(x), Exζ̃t ≤ ε̃f̃t(x), thus
lim sup
|S|→0
ExΣS
3 ≤ δ[ft(x) + f̃t(x)].
This together with (7) gives the needed estimate. The lemma is proved.
Proof of the theorem. It follows immediately from the statement of Lemma 2 that, for
every s, t,
lim
ε, ε̃→0
sup
x∈X
Ex
(
φs,ε
t − φs,ε̃
t
)2
= 0
Therefore (see Lemma 6.3 [1]), there exists the limit φ in mean square of functionals φε
as ε → 0.
By virtue of general results about the convergence of functionals ([1], Ch. 6.2)), the
functional φ is a non-negative and homogeneous functional of the process X· (note that
we have changed slightly the terminology from [1] Ch.6; our functionals are almost non-
negative and almost homogenous in the terminology of E.B. Dynkin). Taking limits
on the both parts of condition VN, we obtain the additivity of the limit functional φ.
The function f from condition (2) of the Theorem is obviously the characteristic of the
functional φ. Statement 1 is proved.
To prove Statement 2, it is sufficient to prove (see Theorem 6.2 [1]) that, for an
arbitrary starting distribution μ of the process X , the following convergence takes place:
lim
|S|→0
Eμ
M−1∑
i=0
(φsi
si+1
)2 = 0. (8)
Obviously,
Ex(φsi
si+1
)2 ≤ 2Ex[φsi
si+1
− φsi,ε
si+1
]2 + 2Ex[φsi,ε
si+1
]2.
From the definition of the functional φ and the statement of Lemma 2, we have
Ex[φsi
si+1
− φsi,ε
si+1
]2 = lim
ε̃→0
Ex[φsi,ε̃
si+1
− φsi,ε
si+1
]2 ≤
≤ lim
ε̃→0
[2 sup
u≤si+1−si
||f ε̃
u − fε
u|| + 5 max(ε, ε̃)] × [fsi,ε̃
si+1
(x) + fsi,ε
si+1
(x)] =
= [2 sup
u≤si+1−si
||fu − fε
u|| + 5ε]× [fsi
si+1
(x) + fsi,ε
si+1
(x)].
LIMIT THEOREMS FOR OSCILLATORY FUNCTIONALS OF A MARKOV PROCESS 7
Thus
Ex(φsi
si+1
)2 ≤ 2
[
2 sup
u≤si+1−si
||fu − fε
u|| + 5ε
]
×
[
fsi
si+1
(x) + fsi,ε
si+1
(x)
]
+ 2Ex
[
φsi,ε
si+1
]2
.
Taking the sum over i = 0, M − 1 and noting that the inequality fs,ε
t + f t,ε
u ≤ fs,ε
u holds
under condition US for s < t < u, we have
Ex
M−1∑
i=0
(φsi
si+1
)2 ≤
[
4 sup
u≤t−s
||fu − fε
u|| + 10ε
]
× [fs
t (x) + fs,ε
t (x)] + 2Ex
M−1∑
i=0
[
φsi,ε
si+1
]2
. This implies that, for an arbitrary probability distribution μ,
Eμ
M−1∑
i=0
(φsi
si+1
)2 ≤
[
4 sup
u≤t−s
||fu − fε
u|| + 10ε
]
× ‖fs
t + fs,ε
t ‖ + 2Eμ
M−1∑
i=0
[
φsi,ε
si+1
]2
.
We have that
∑M−1
i=0
[
φsi,ε
si+1
]2
≤ ΣS
3 (see notations in the proof of Lemma 1). Due to the
estimate of the second moment of ΣS
1 given in this proof,
lim sup
|S|→0
Eμ
M−1∑
i=0
(φsi
si+1
)2 ≤
[
4 sup
u≤t−s
||fu − fε
u|| + 10ε
]
× ‖fs
t + fs,ε
t ‖ + 2ε‖fε
t ‖.
Now taking ε → 0, we obtain (8), which completes the proof of the theorem.
Let us formulate another version of Theorem 1 under slightly different suppositions
on the family of functionals. Let us suppose functionals φε to be homogeneous not at an
arbitrary moment of the time but at the points of some partitions Sε = {0 = sε
0 < sε
1 <
. . . } of �+:
φ
sε
i ,ε
sε
i+1
= θsε
i
φ0,ε
sε
i+1−sε
i
. (9)
Let also |Sε| → 0 as ε → 0. Such a supposition enlarges the class of families which can
be treated in our approach, but now we cannot use the construction of Lemma 1 in order
to estimate EΣS
1 (and EΣS
3 in a sequel). Therefore, we impose the following additional
condition: there exists a non-random r(ε) such that lim
ε→0
r(ε) = 0 and
φ
sε
i ,ε
sε
i+1
≤ r(ε), i ≥ 0. (10)
Theorem 2. Let the family {φε} satisfy conditions (1),(2),(9),(10), VN,VD, and let
the function f be continuous w.r.t. (x, t) ∈ X × �+.
Then f is a characteristic of some non-negative homogenous additive functional φ,
and
φs
t = l.i.m.
ε→0
φs,ε
t , 0 ≤ s ≤ t < +∞.
Under additional condition US, φ is a V -functional.
Proof. We will prove the theorem under the additional supposition that X is compact
(this, due to standard localization arguments, does not restrict generality). Then, for
every δ > 0,
(t, δ) ≡ sup
u≤t
sup
x
Px(ρ(Xu, x) > δ) → 0, t → 0 + .
By W γ(·) W (·), we denote, respectively, the moduli of continuity of the family {fε, ε ≤ γ}
and of f :
W γ(δ) = sup
ε≤γ,ρ(x,y)≤δ
|fε(x) − fε(y)|, W (δ) = sup
ρ(x,y)≤δ
|f(x) − f(y)|.
One can see that W γ(δ) → W (δ), γ → 0 for every δ > 0.
8 TARAS O. ANDROSHCHUK AND ALEXEY M. KULIK
The main point of the proof is an estimate analogous to one given in Lemma 2. In
order to provide such an estimate, let us introduce some notations. Suppose ε, ε̃ > 0 to
be fixed. For u > 0, we denote
r(u) = max{i : sε
i ≤ u}, r̃(u) = max{i : sε̃
i ≤ u}, ν(u) = sε
r(u), ν̃(u) = sε̃
r̃(u).
Denote also si = sε
i , s̃i = sε̃
i , q(i) = r̃(si), q̃(i) = r(s̃i). Note that if either j > q(i) or
i > q̃(j), then the segments (si, si+1) and (s̃j , s̃j+1) do not intersect. Let us estimate
Ex(φ0
t − φ̃0
t )
2 (see notations in the proof of Lemma 2). We suppose that, for a given t,
ν(t) = ν̃(t) = t (this does not restrict generality since φ0
t −φ0
ν(t), φ̃
0
t − φ̃0
ν̃(t) ∈ [0, max(ε +
r(ε), ε̃ + r(ε̃))]). We put Φj = φ0
sε
j+1
− φ0
sε
j
, Φ̃j = φ̃0
s̃j+1
− φ̃0
s̃j
, M = r(t), M̃ = r̃(t).
One has
(φ0
t − φ̃0
t )
2 =
(
M−1∑
i=0
Φi
)2
+
⎛
⎝M̃−1∑
j=0
Φ̃j
⎞
⎠
2
− 2
M−1∑
i=0
M̃−1∑
j=0
ΦiΦ̃j = Σ5 + 2Σ6,
where
Σ5 =
M−1∑
i=0
Φ2
i +
M̃−1∑
j=0
Φ̃2
j − 2
∑
i,j:j≤q(i),i≤q̃(j)
ΦiΦ̃j ≤
M−1∑
i=0
Φ2
i +
M̃−1∑
j=0
Φ̃2
j ,
Σ6 =
⎡
⎣∑
i<l
ΦiΦl −
∑
q(i)<j
ΦiΦ̃j
⎤
⎦ +
⎡
⎣∑
j<k
Φ̃jΦ̃k −
∑
q̃(j)<i
ΦiΦ̃j
⎤
⎦ =
=
M−1∑
i=0
Φi
[
(φ0
t − φ0
si+1
) − (φ̃0
t − φ̃0
s̃q(i)+1
)
]
+
M̃−1∑
j=0
Φ̃i
[
(φ̃0
t − φ̃0
s̃j+1
) − (φ0
t − φ0
sq̃(j)+1
)
]
.
(11)
Due to conditions VN and (10),
ExΣ5 ≤ (r(ε)+ε)Ex
M−1∑
i=0
Φi+(r(ε̃)+ε̃)Ex
M̃−1∑
j=0
Φ̃j ≤ max(r(ε)+ε, r(ε̃)+ε̃)[f0
t (x)+f̃0
t (x)].
Next, analogously to (3),(4) we have
Ex
M−1∑
i=0
Φi
[
(φ0
t − φ0
si+1
) − (φ̃0
t − φ̃0
s̃q(i)+1
)
]
≤ 2 max(ε, ε̃)[ft(x) + f̃t(x)]+
+Ex
M−1∑
i=0
Φi|ft−si+1(Xsi+1) − f̃t−s̃q(i)+1 (Xs̃q(i)+1 )|. (12)
Note that |si+1 − s̃q(i)+1|, |s̃j+1 − sq̃(j)+1| ≤ θ ≡ |Sε| + |S ε̃|. Thus, the second summand
in (12) can be estimated by
ft(x)
[
sup
u≤t
‖fu − f̃u‖ + W γ(θ + δ) + F ·
(θ, δ)
]
,
where F = supε supu≤t ‖fε
t ‖, γ = max(ε, ε̃), and δ > 0 is arbitrary. Writing the same in-
equalities for the second summand on the right-hand side of (11), we obtain the following
estimate analogous to one given in Lemma 2:
Ex(φ0
t − φ̃0
t )
2 ≤ [ft(x) + f̃t(x)]×
×
[
sup
u≤t
‖fu − f̃u‖ + 5 max(ε, ε̃) + max(r(ε), r(ε)) + 2W γ(θ + δ) + 2F ·
(θ, δ)
]
.
LIMIT THEOREMS FOR OSCILLATORY FUNCTIONALS OF A MARKOV PROCESS 9
The same estimate, of course, can be written also for the L2-distance between φs
t and
φ̃s
t . Therefore, for every s, t,
lim sup
ε, ε̃→0
sup
x∈X
Ex
(
φs,ε
t − φs,ε̃
t
)2
≤ 4F · W (2δ).
After taking δ → 0, we obtain that there exists a mean square limit φ of the family {φε}.
The proof of the fact that φ is a V -functional is analogous to the same proof in Theorem
1. The theorem is proved.
Remark. If the function f in Theorems 1,2 is already known to be a W -function, i.e.
a characteristic of some W -functional φ̃, then one can show the mean square convergence
of the family {φε} to the functional φ̃ without use of the additional condition US. This
is an important improvement since condition US is a quite significant restriction, while,
in typical examples, f is known and is a W -function. However, we omit the proof of this
statement since it can be given in a completely analogous way to the proofs of the first
parts of Theorems 1,2.
2. Examples
In this section, we illustrate the general statements obtained before by two examples
of oscillatory functionals.
2.1. Number of passings through a template. Let X = �, and let a1, . . . , aM ∈ �
be fixed (ai �= ai+1, i = 1, . . . , M − 1, a1 �= aM ). The set {τi} of random times τ1, . . . , τM
such that τ1 < · · · < τM and Xτi = aM is called a passing of the process X· through
the template {ai}. Two such sets {τi} and {τ̃i} are called adjusted if either τM ≤ τ̃1 or
τ̃M ≤ τ1. For every s < t, we denote, by N
{ai}
s,t , the maximal number of adjusted passings
of the process X· through the template {ai} happened on the time interval [s, t].
Now we take a family of templates {εai}, ε > 0 and define the functionals φε as the
properly normalized numbers of passings of the given process X· through these templates:
φε,s
t = r(ε)N{εai}
s,t , s < t. (13)
The functionals {φε} describe the local (”oscillatory”) behavior of the process X· near
the point 0. Let us consider a specific example of the process X· and use Theorem 1 in
order to give the detailed description of such a behavior.
Let X· be a skew Brownian motion with skewing parameter q ∈ (−1, 1). It is a
homogeneous Markov process with its transition probability density being equal to
p(t, x, y) =
1√
2πt
[
e−
(x−y)2
2t + q sign y · e− (|x|+|y|)2
2t
]
.
This process was introduced in [2], Ch. 4.2, Problem 1, and can be described in different
terms: in terms of its scale and speed functions ([2]), as the solution to an SDE with
the delta function in the drift term ([4]), and as the simplest example of a generalized
diffusion process ([5]). One of the possible constructions of the process is the following
one (see [2]): take a Wiener process X0
· , consider the set of its excursions at the point
0, and then put on every excursion independently (both from X0
· and other excursions)
X· = X0
· with probability p+ = 1+q
2 and X· = −X0
· with probability p− = 1−q
2 . This
construction shows that 0 is the ”point of asymmetry” for the skew Brownian motion
and motivates the study of the local behavior of the process near this point.
We restrict our considerations, supposing that
M = 2n, a1, . . . , a2n−1 > 0, a2, . . . , a2n < 0.
10 TARAS O. ANDROSHCHUK AND ALEXEY M. KULIK
Theorem 3. Let r(ε) = ε in the given before definition (13). Then
φε,s
t
L2−→
[A+
p+
+
A−
p−
]−1
Ls
t , ε → 0,
where A+ = a1 + a3 + · · · + a2n−1, A− = −a2 − a4 − · · · − a2n, and Ls
t is the symmetric
local time of the process X· at the point 0 defined as the mean square limit
Ls
t = lim
Δ→0
1
2Δ
∫ t
s
1IXr∈[−Δ,Δ] dr.
Proof. One can easily verify that the family {φε} is a QW-family (see Definition 1). In
order to use Theorem 1, we need to verify conditions (1),(2). We consider the functions
V ε(λ, x) ≡
∫ ∞
0
e−λtfε
t (x) dt, λ > 0, x ∈ �,
and show that, for every Λ1 ≤ Λ2, Λ1,2 ∈ (0, +∞), V ε →
[
A+
p+
+ A−
p−
]−1
V, ε → 0 uniformly
on [Λ1, Λ2]×�, where V (λ, x) =
∫ ∞
0 e−λtft(x) dt, ft(x) is the characteristic of L0
t . Since
fε and f are functions non-decreasing in t, this will imply (1),(2). The explicit expressions
for the functions V ε and V are given in the following lemma.
Lemma 3. Denote, by τy, the moment of the first visit of the point y ∈ � by the process
X· and put v(λ, x, y) = Ex exp[−λτy ]. Then
V (λ, x) = v(λ, x, 0) · 1√
2λ3
, (14)
(15)
V ε(λ, x) =
εv(λ, x, εa1)
λ
× v(λ, εa1, εa2)v(λ, εa2, εa3) . . . v(λ, εaM−1, εaM )
1 − v(λ, εa1, εa2)v(λ, εa2, εa3) . . . v(λ, εaM−1, εaM )v(λ, εaM , εa1)
.
Proof. Equality (14) is a consequence of the strong Feller property of X (note that the
distribution of Lt, while X starts from 0, coincides with the distribution of the local time
of the Wiener process at the point 0). Again, it follows from the strong Feller property
that
V ε(λ, x) = v(λ, x, εa1)V ε(λ, εa1). (16)
The calculation of V ε(λ, εa1) uses the standard renewal theory technique. Denoting, by
θ, the first moment when the passing through the given template happens, we note that
Xθ = εaM with probability 1. The distribution of θ is a convolution of the distributions
of subsequent times of the first visits of the points εa2, εa3, . . . , εaM by the process X .
Therefore its Laplace transform is
Θ(λ) ≡ Eεa1e
−λθ = v(λ, εa1, εa2)v(λ, εa2, εa3) . . . v(λ, εaM−1, εaM ).
Now, writing down the renewal equation at the moment θ, we obtain that
V ε(λ, εa1) = εΘ(λ) + Θ(λ) · V ε(λ, εaM ).
Substituting, instead of V ε(λ, εaM ), its expression (16) through V ε(λ, εa1) and then
solving the linear equation for V ε(λ, εa1), we obtain (14). The lemma is proved.
Let us return to the proof of Theorem 3. For every fixed α, β with α · β < 0, one has
that v(λ, εα, εβ) = v(λ, εα, 0)v(λ, 0, εβ). The distribution of the first visit of the process
X to 0 is the same with the distribution of the first visit to 0 of the Brownian motion,
which follows from the construction of X . Therefore, v(λ, εα, 0) = e−ε
√
2λ|α| (see [2], Ch.
1.7).
LIMIT THEOREMS FOR OSCILLATORY FUNCTIONALS OF A MARKOV PROCESS 11
In order to calculate v(λ, 0, εβ), consider the first moment θ when |X | = ε|β|. Due to
the above-described excursion-based construction of the process X , one can say that the
distribution of the moment θ is the same with the distribution of the first moment when
|W·| = ε|β| (W· is a Wiener process.) The Laplace transformation of this distribution
is Q(λ) = 2 ·
[
eε
√
2λ|β| + e−ε
√
2λ|β|
]−1
(see [2], Ch. 1.7). Moreover, independently of θ,
the variable Xθ takes the values ε|β| or −ε|β| with probabilities p+ and p−, respectively.
Writing down the renewal equation at the moment θ, we obtain the equation
v(λ, 0, εβ) = pβQ(λ) + p−βQ(λ)v(λ,−εβ, 0)v(λ, 0, εβ),
where py = 1+q·sign y
2 , i.e.
v(λ, 0, εβ) =
pβQ(λ)
1 − p−βe−ε
√
2λ|β|Q(λ)
.
Therefore, we have that
v(λ, εα, εβ) = 1 −
√
2λ
[
|α| + p−β
pβ
|β|
]
ε + o(ε), ε → 0+,
uniformly for λ ∈ [Λ−, Λ+]. It is easy to verify that v(λ, x, εa1) → v(λ, x, 0) and
v(λ, εai, εaj) → 1, ε → 0+, uniformly for λ ∈ [Λ−, Λ+], x ∈ �. Thus,
V ε(λ, x) → v(λ, x, 0) · 1
λ · √2λ · C , ε → 0+,
where
C = (a1 +
p+
p−
|a2|) + (|a2| + p−
p+
a3) + · · · + (|a2n| + p−
p+
a1)
= A+
(
1 +
p−
p+
)
+ A−
(
1 +
p+
p−
)
=
A+
p+
+
A−
p−
.
The theorem is proved.
2.2. Number of intersections of a level by the diffusion. Let X = �. For a given
process X·, consider the sequence of functionals {ηn},
ηn,s
t =
∑
k:s< k
n≤t
1IX k−1
n
·X k
n
<0.
The limit behavior of the sequence {ηn} as n → ∞ essentially depends on the proper-
ties of the trajectories of the process X·. If these trajectories are smooth, then (under
some additional non-degeneracy condition on the derivative) ηn tends to the number of
intersections of the level 0 by the trajectory of X·. The same feature holds true for some
L2-differentiable stationary processes, for instance see [6], Ch.7 for the classical Rice for-
mula for normal stationary processes. For diffusions, the situation is quite different, and
typically ηn → +∞. Let us study thoroughly the specific example, when X· is the skew
Brownian motion. The following statement is a corollary of Theorem 1, [3], §6.
Proposition 1. For every t > 0, the sequence {n− 1
2 ηn,s
t } converges in distribution to√
2
π (1 − q2)Lt.
The technique developed in the previous section allows us to improve this result.
12 TARAS O. ANDROSHCHUK AND ALEXEY M. KULIK
Theorem 4.
n− 1
2 ηn,s
t
L2−→
√
2
π
(1 − q2)Ls
t , n → ∞.
Proof. We apply Theorem 2 (note that Theorem 1 cannot be used here since, for
a fixed n, the functional ηn is not homogeneous). The family {φn ≡ n− 1
2 ηn} satisfies
conditions (9),(10), VN,VD with ε = n− 1
2 , Sε = {0, 1
n , 2
n , . . . }. One can easily verify
also that the characteristic f of φ· ≡
√
2
π (1−q2)L· is continuous. Let us verify conditions
(1),(2). Due to the strong Feller property of X·, in order to do this, it is sufficient to
prove that, for every T > 0,
sup
t≤T,θ∈[0, 1
n )
∣∣∣n− 1
2 E
[
η
n, 1
n
t
∣∣∣Xθ = 0
]
− 2(1 − q2)
√
t
π
∣∣∣ → 0, n → +∞ (17)
(we recall that E0Lt =
∫ t
0
1√
2πs
ds =
√
2t
π ). We have
E[ηn, 1
n
t |Xθ = 0] =
[tn]−1∑
k=1
[∫ 0
−∞
p(
k
n
− θ, 0, y)P+
n (y) dy +
∫ ∞
0
p(
k
n
− θ, 0, y)P−
n (y) dy
]
,
(18)
where p(·, ·, ·) is the transition probability density of the process X· and Φ±
n (y) =
Py( sign X 1
n
= ±1). The easy calculation gives
{
P+
n (y) = (1 + q)Φ(−y
√
n), y < 0,
P−
n (y) = (1 − q)Φ(y
√
n), y > 0,
Φ(z) ≡
∫ ∞
z
e−
u2
2√
2π
du.
Therefore, using the explicit expression for p(·, ·, ·), we obtain
n− 1
2 E[ηn, 1
n
t |Xθ = 0] = n− 1
2
[tn]−1∑
k=1
∫ ∞
−∞
(1 − q)(1 + q)√
2π( k
n − θ)
e
− y2
2( k
n
−θ) Φ(|y|√n) dy =
=
1 − q2
n
[tn]−1∑
k=1
1√
2π( k
n − θ)
∫ +∞
−∞
e−
w2
2(k−θn) Φ(|w|) dw.
We have that∫ ∞
−∞
Φ(|w|) dw =
∫ ∞
0
∫ ∞
w
√
2
π
e−
u2
2 du dw =
√
2
π
∫ ∞
0
ue−
u2
2 du =
√
2
π
, (19)
1√
2π( k
n − θ)
∫ +∞
−∞
e−
w2
2(k−θn) Φ(|w|) dw ≤ √
n for every k ≥ 1, θ ∈ [0,
1
n
), (20)
sup
t≤T,θ∈[0, 1
n )
∣∣∣∣∣∣
[tn]−1∑
k=k0
1√
2π( k
n − θ)
−
∫ t
0
1√
2πs
ds
∣∣∣∣∣∣ → 0, n → +∞ for every k0 > 1. (21)
For a given δ > 0, let us take k0 > 1 such that∫ +∞
−∞
(1 − e
− w2
2(k0−1) )Φ(|w|) dw < δ.
Then (19)-(21) yield that
lim sup
n→+∞
sup
t≤T,θ∈[0, 1
n )
∣∣∣n− 1
2 E
[
η
n, 1
n
t
∣∣∣Xθ = 0
]
− 2(1 − q2)
√
t
π
∣∣∣ < δ.
LIMIT THEOREMS FOR OSCILLATORY FUNCTIONALS OF A MARKOV PROCESS 13
Since δ is arbitrary, this proves (17). The theorem is proved.
Remark. The functionals studied in subsections 2.1 and 2.2 can be considered as two
possible answers to the question about how to construct the approximating aggregates for
the number of intersections of the level by the diffusion or generalized diffusion process.
In the first case, the level is made more ”thick”, and the time is discretized in the second
case. For the skew Brownian motion, these two constructions give, after the appropriate
normalization, the same (up to a constant) object, namely, the local time of the process.
It should be mentioned that the situation can be essentially different for other processes.
Let us give an example.
Let Y be a skew Brownian motion, and let � be its symmetric local time at the point 0.
We take a > 0 and define θt = t + a�t, σt = [θ−1]t ≡ inf{u|θu ≥ u} and Xt ≡ Yσt , t ≥ 0.
The process X· is a Markov one (for more details see [3], §5) which spends a positive
time at the point 0. Due to the latter fact, the point 0 is called sticky. The following
proposition shows that two constructions described before give essentially different results
for the process X·.
Proposition 2. 1) In the notations of subsection 2.1,
φε,s
t
L2−→
[A+
p+
+
A−
p−
]−1 ◦
L
s
t , ε → 0,
where
◦
L
s
t = lim
Δ→0
1
2Δ
∫ t
s
1IXr∈[−Δ,0)∪(0,Δ] dr.
2) In the notations of subsection 2.2, the sequence {ηn,s
t } converges in distribution to
some integer-valued random variable for every s, t.
Statement 2) was proved in [3], §6. One can prove statement 1) by either repeating
the proof of Theorem 3 for the process with a sticky point or using the result of Theorem
3 and making a random time change.
Bibliography
1. E.B.Dynkin, Markov processes, Moscow: Fizmatgiz, 1963. (Russian.)
2. K.Itô, H.P.McKean, Diffusion Processes and Their Sample Paths, Berlin: Springer, 1965.
3. M.I.Portenko,, Diffusion in media with semipermeable membranes, Kyiv: Institute of Mathe-
matics, National Academy of Science of Ukraine, 1994. (Ukrainian)
4. J.M.Harrison, L.A.Shepp, On skew Brownian motion, Annals of Probability 9 (1981), no. 2,
309-313.
5. M.I.Portenko, Generalized diffusion processes, Providence, Rhode Island: AMS, 1990.
6. M.R.Leadbetter, G.Lindgren, H.Rootzen, Extremes and Related Properties of Random Se-
quences and Processes, Berlin: Springer, 1986.
E-mail : kulik@imath.kiev.ua
|