Separation problem for a family of Borel and Baire G-powers of shift measures on R
The separation problem for a family of Borel and Baire G-powers of shift measures on R is studied for an arbitrary infinite additive group G using the technique developed in [Kuipers L., Niederreiter H. Uniform distribution of sequences. – New York etc.: John Wiley & Sons, 1974], [Shiryaev A. N....
Gespeichert in:
| Datum: | 2013 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2013
|
| Schriftenreihe: | Український математичний журнал |
| Schlagworte: | |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/165109 |
| 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: | Separation problem for a family of Borel and Baire G-powers of shift measures on R / Z. Zerakidze, G. Pantsulaia, G. Saatashvili // Український математичний журнал. — 2013. — Т. 65, № 4. — С. 470-485. — Бібліогр.: 7 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-165109 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1651092020-02-12T01:27:17Z Separation problem for a family of Borel and Baire G-powers of shift measures on R Zerakidze, Z. Pantsulaia, G. Saatashvili, G. Статті The separation problem for a family of Borel and Baire G-powers of shift measures on R is studied for an arbitrary infinite additive group G using the technique developed in [Kuipers L., Niederreiter H. Uniform distribution of sequences. – New York etc.: John Wiley & Sons, 1974], [Shiryaev A. N. Probability (in Russian). – Moscow: Nauka, 1980] and [Pantsulaia G. R. Invariant and quasiinvariant measures in infinite-dimensional topological vector spaces. – New York: Nova Sci. Publ., Inc., 2007]. Вивчається задача вiдокремлення для сiм’ї борелiвських та берiвських G-степенiв мiр зсуву на R для довiльної нескiнченної адитивної групи G iз використанням пiдходу, розвиненого в роботах [Kuipers L., Niederreiter H. Uniform distribution of sequences. – New York etc.: John Wiley & Sons, 1974], [Ширяев А. Н. Вероятность. – М.: Наука, 1980] та [Pantsulaia G. R. Invariant and quasiinvariant measures in infinite-dimensional topological vector spaces. – New York: Nova Sci. Publ., Inc., 2007]. 2013 Article Separation problem for a family of Borel and Baire G-powers of shift measures on R / Z. Zerakidze, G. Pantsulaia, G. Saatashvili // Український математичний журнал. — 2013. — Т. 65, № 4. — С. 470-485. — Бібліогр.: 7 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165109 517.23 en Український математичний журнал Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Статті Статті |
| spellingShingle |
Статті Статті Zerakidze, Z. Pantsulaia, G. Saatashvili, G. Separation problem for a family of Borel and Baire G-powers of shift measures on R Український математичний журнал |
| description |
The separation problem for a family of Borel and Baire G-powers of shift measures on R is studied for an arbitrary infinite additive group G using the technique developed in [Kuipers L., Niederreiter H. Uniform distribution of sequences. – New York etc.: John Wiley & Sons, 1974], [Shiryaev A. N. Probability (in Russian). – Moscow: Nauka, 1980] and [Pantsulaia G. R. Invariant and quasiinvariant measures in infinite-dimensional topological vector spaces. – New York: Nova Sci. Publ., Inc., 2007]. |
| format |
Article |
| author |
Zerakidze, Z. Pantsulaia, G. Saatashvili, G. |
| author_facet |
Zerakidze, Z. Pantsulaia, G. Saatashvili, G. |
| author_sort |
Zerakidze, Z. |
| title |
Separation problem for a family of Borel and Baire G-powers of shift measures on R |
| title_short |
Separation problem for a family of Borel and Baire G-powers of shift measures on R |
| title_full |
Separation problem for a family of Borel and Baire G-powers of shift measures on R |
| title_fullStr |
Separation problem for a family of Borel and Baire G-powers of shift measures on R |
| title_full_unstemmed |
Separation problem for a family of Borel and Baire G-powers of shift measures on R |
| title_sort |
separation problem for a family of borel and baire g-powers of shift measures on r |
| publisher |
Інститут математики НАН України |
| publishDate |
2013 |
| topic_facet |
Статті |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/165109 |
| citation_txt |
Separation problem for a family of Borel and Baire G-powers of shift measures on R / Z. Zerakidze, G. Pantsulaia, G. Saatashvili // Український математичний журнал. — 2013. — Т. 65, № 4. — С. 470-485. — Бібліогр.: 7 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT zerakidzez separationproblemforafamilyofborelandbairegpowersofshiftmeasuresonr AT pantsulaiag separationproblemforafamilyofborelandbairegpowersofshiftmeasuresonr AT saatashvilig separationproblemforafamilyofborelandbairegpowersofshiftmeasuresonr |
| first_indexed |
2025-07-14T17:54:55Z |
| last_indexed |
2025-07-14T17:54:55Z |
| _version_ |
1837645903894151168 |
| fulltext |
UDC 517.23
Z. Zerakidze (Tbilisi State Univ., Georgia),
G. Pantsulaia (I. Vekua Inst. Appl. Math., Tbilisi State Univ., Georgia),
G. Saatashvili (Georg. Techn. Univ., Tbilisi)
ON THE SEPARATION PROBLEM FOR A FAMILY OF BOREL
AND BAIRE G-POWERS OF SHIFT MEASURES ON R
ПРО ЗАДАЧУ ВIДОКРЕМЛЕННЯ ДЛЯ СIМ’Ї БОРЕЛIВСЬКИХ
ТА БЕРIВСЬКИХ G-СТЕПЕНIВ МIР ЗСУВУ НА R
The separation problem for a family of Borel and Baire G-powers of shift measures on R is studied for an arbitrary
infinite additive group G using the technique developed in [Kuipers L., Niederreiter H. Uniform distribution of sequences.
– New York etc.: John Wiley & Sons, 1974], [Shiryaev A. N. Probability (in Russian). – Moscow: Nauka, 1980] and
[Pantsulaia G. R. Invariant and quasiinvariant measures in infinite-dimensional topological vector spaces. – New York:
Nova Sci. Publ., Inc., 2007]. It is proved that Tn : Rn → R, n ∈ N, defined by
Tn(x1, . . . , xn) = −F−1(n−1#({x1, . . . , xn} ∩ (−∞; 0]))
for (x1, . . . , xn) ∈ Rn is a consistent estimator of a useful signal θ in the one-dimensional linear stochastic model
ξk = θ + ∆k, k ∈ N,
where #(·) is a counting measure, ∆k, k ∈ N, is a sequence of independent identically distributed random variables on R
with a strictly increasing continuous distribution function F , and the expectation of ∆1 does not exist.
Вивчається задача вiдокремлення для сiм’ї борелiвських та берiвських G-степенiв мiр зсуву на R для довiльної
нескiнченної адитивної групи G iз використанням пiдходу, розвиненого в роботах [Kuipers L., Niederreiter H.
Uniform distribution of sequences. – New York etc.: John Wiley & Sons, 1974], [Ширяев А. Н. Вероятность. – М.:
Наука, 1980] та [Pantsulaia G. R. Invariant and quasiinvariant measures in infinite-dimensional topological vector spaces.
– New York: Nova Sci. Publ., Inc., 2007]. Доведено, що Tn : Rn → R, n ∈ N, означений формулою
Tn(x1, . . . , xn) = −F−1(n−1#({x1, . . . , xn} ∩ (−∞; 0]))
при (x1, . . . , xn) ∈ Rn, є консистентною оцiнкою корисного сигналу θ в одновимiрнiй лiнiйнiй стохастичнiй моделi
ξk = θ + ∆k, k ∈ N,
де #(·) — злiченна мiра, ∆k, k ∈ N, — послiдовнiсть незалежних однаково розподiлених випадкових величин на R
iз строго зростаючою неперервною функцiєю розподiлу F , а сподiвання величини ∆1 не iснує.
1. Introduction. In the general theory of statistical decisions there often arises a problem of transition
from a weakly separated family of probability measures to the corresponding strongly separated
family. In 1981, A. Skorokhod [1] proved that if the Continuum Hypothesis is true, then an arbitrary
weakly separated family of probability measures, whose cardinality is not greater than the cardinality
of the continuum, is strongly separable. The validity of the inverse relation was established in [3] (see
also [4]). In particular, it was shown there that if an arbitrary weakly separated family of probability
measures whose cardinality is less than or equal to the cardinality of the continuum is strongly
separated, then the Continuum Hypothesis is true. Applying Martin’s axiom, in 1984 Z. Zerakidze [7]
proved that an arbitrary weakly separated family of Borel probability measures defined in a separable
completely metrizable space (i.e., Polish space) is strongly separated if its cardinality is not greater
than the cardinality of the continuum. In [3], this result is extended to all complete metric spaces
whose topological weights are not measurable in a wider sense.
Below we give some definitions from the theory of stochastic processes.
c© Z. ZERAKIDZE, G. PANTSULAIA, G. SAATASHVILI, 2013
470 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
ON THE SEPARATION PROBLEM FOR A FAMILY OF BOREL AND BAIRE G-POWERS . . . 471
Definition 1.1. Let (Ω,F , p) be a probability space and G be an infinite additive group. A
stochastic process X = (Xg)g∈G : Ω→ RG is called a G-process on (Ω,F , p) if a joint probability
distribution
F
(X)
(g1,...,gn)(x1, . . . , xn) = p({ω : Xg1(ω) ≤ x1, . . . , Xgn(ω) ≤ xn})
with (g1, . . . , gn) ∈ Gn and (x1, . . . , xn) ∈ Rn, does not change when shifted in a group, i.e., the
following equality
F
(X)
(g1,...,gn)(x1, . . . , xn) = F
(X)
(g1+h,...,gn+h)(x1, . . . , xn)
holds for an arbitrary h ∈ G.
Remark 1.1. For G = Rn, n > 1, a G-process coincides with a homogenous field. If G = R,
then a G-process coincides with a stationary process.
Definition 1.2. Let p be a Borel probability measure on R and θ ∈ R. Then a probability
measure pθ defined by
(∀X)(X ∈ B(R)→ pθ(X) = p(X + θ))
is called a θ-shift measure of p.
Definition 1.3. Let p be a Borel probability measure on R and G be an infinite additive group.
Suppose that pg = p for g ∈ G. Then the product measure
∏
g∈G
pg is called the Baire G-power
of p and is denoted by pG. If pG admits a Borel extension, then that extension is called the Borel
G-power of p.
Remark 1.2. Note that the notions of Baire G-power and Borel G-power of p coincide when
the group G is countable.
Example 1.1. Let p be a Borel probability measure on R and G be an infinite additive group.
Then the family of all coordinate projections (Prg)g∈G defined on a probability space
(
RG, Ba(RG),
pGθ
)
is a G-process for every θ ∈ R, where pGθ is the G-power of a shift measure pθ on R.
The main aim of the present paper is to consider the separation problem for a family of G-powers
of shift measures on R, where G is an arbitrary additive group. Note that such measures generate
G-processes on RG.
The attention is focused on two essentially different examples of strongly separated families of
Borel and Baire G-powers of shift measures in R for an arbitrary infinite additive group G.
Our tools of investigation are the techniques developed in [2, 5, 6].
The paper is organized as follows. Some auxiliary notions and facts from the theory of uniformly
distributed sequences and the probability theory are considered in Section 2. Section 3 contains
the formulations and proofs of the obtained results. In Section 4, the existence of some consistent
estimators of a useful signal in the one-dimensional linear stochastic model is proved and some
examples of the corresponding simulations with numerical computations are considered.
2. Some auxiliary notions and facts. We start this section with some standard notions and
definitions from the probability theory.
Let I be an arbitrary nonempty set of parameters. Denote by (RI , τ) the vector space of all real-
valued functions on I equipped with the Tychonoff topology τ. We denote by B(RI) a σ-algebra of
all Borel subsets of the space RI generated by the Tychonoff topology τ.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
472 Z. ZERAKIDZE, G. PANTSULAIA, G. SAATASHVILI
Let (Pri)i∈I be the family of all coordinate projections defined by
(∀i)(∀(xj)j∈I)(i ∈ I & (xj)j∈I ∈ RI → Pri((xj)j∈I) = xi).
A minimal σ-algebra of subsets of RI generated by the class of subsets
{Pr−1
i (X) : i ∈ I & X ∈ B(R)}
is denoted by Ba(RI) and is called a Baire σ-algebra of subsets of RI .
Remark 2.1. Note that Ba(RI) = B(RI) for card(I) ≤ ℵ0, where ℵ0 denotes the cardinality
of the set of all natural numbers. If card(I) > ℵ0, then
Ba(RI) ⊂ B(RI) & B(RI) \Ba(RI) 6= ∅.
As usual, a measure defined on B(RI) is called a Borel measure. Analogously, a measure defined
on Ba(RI) is called a Baire measure.
Definition 2.1. Let µ1 be a Baire measure defined on RI . A Borel measure µ2 defined on RI
is called a Borel extension of µ1 if
(∀X)(X ∈ Ba(RI)→ µ2(X) = µ1(X)).
Example 2.1. Let I be an arbitrary nonempty parametric set and pi be a Borel probability
measure on Ri := R for all i ∈ I. If card(I) > ℵ0, then the probability product-measure
∏
i∈I
pi is
defined on the σ-algebra ∏
i∈I
B(Ri) = Ba(RI).
Accordingly, this measure is an example of a Baire probability measure which is not defined on
B(RI).
Lemma 2.1 ([5, p. 67], Lemma 4.4). Let (E1, τ1) and (E2, τ2) be two topological spaces. De-
note by B(E1) and B(E2) (respectively, by B(E1 ×E2)) the class of all Borel subsets generated by
the topologies τ1 and τ2 (respectively, by τ1 × τ2). If at least one of these topological spaces has a
countable base, then the equality
B(E1)×B(E2) = B(E1 × E2)
holds.
Lemma 2.2 ([5, p. 70], Remark 4.5). Let (pi)i∈I be a family of Borel probability measures on R
with strictly increasing continuous distribution functions. Then there exists only one Borel extension
PI of the Baire product-measure
∏
i∈I
pi.
Corollary 2.1 ([5, p. 75], Corollary 4.2). The product of an arbitrary family (pi)i∈I of nontriv-
ial Gaussian Borel probability measures defined on RI has only one Borel extension.
Corollary 2.2 ([5, p. 75], Corollary 4.3). In the case of the space RI , for Card(I) > ℵ0,
Lemma 2.2 is a generalization of Anderson well-known theorem which gives only the construction of
a Baire product-measure.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
ON THE SEPARATION PROBLEM FOR A FAMILY OF BOREL AND BAIRE G-POWERS . . . 473
Definition 2.2 [2]. A sequence s1, s2, s3, . . . of real numbers from the interval (a, b) is said to
be equidistributed or uniformly distributed on the interval (a, b) if for any subinterval [c, d] of (a, b)
we have
lim
n→∞
#({s1, s2, s3, . . . , sn} ∩ [c, d])
n
=
d− c
b− a
,
where # is a counter measure.
Now let X be a compact Polish space and µ be a probability Borel measure on X. Let R(X) be
a space of all bounded continuous functions on X.
Definition 2.3. A sequence s1, s2, s3, . . . of elements of X is said to be µ-equidistributed or
µ-uniformly distributed on X if for every f ∈ R(X) we have
lim
N→∞
1
N
N∑
n=1
f(sn) =
∫
X
fdµ.
Lemma 2.3 [2, p. 199 – 201]. Let f ∈ R(X), µ∞ := µ∞ and S be a set of all µ-equidistributed
sequences in X∞. Then
(i) µ∞(S) = 1;
(ii) S is a set of the first category;
(iii) S is everywhere dense in the Tychonoff topology.
Corollary 2.3. Let `1 be a Lebesgue measure on (0, 1). Let D be a set of all `1-equidistributed
sequences in (0, 1)∞. Then
(i) `∞1 (D) = 1;
(ii) D is a set of the first category;
(iii) D is everywhere dense in the Tychonoff topology.
Definition 2.4. Let µ be a probability Borel measure on R such that its distribution function
F is continuous. A sequence s1, s2, s3, . . . of elements of R is said to be µ-equidistributed or µ-
uniformly distributed on R if for every interval [a, b] (−∞ ≤ a < b ≤ +∞) we have
lim
n→∞
#([a, b] ∩ {x1, . . . , xn})
n
= F (b)− F (a).
Lemma 2.4. Let (xk)k∈N be an `1-equidistributed sequence in (0, 1) and F be a strictly
increasing continuous distribution function on R. Let p be a Borel probability measure on R defined
by F. Then (F−1(xk))k∈N is a p-equidistributed sequence on R.
Proof. We have
lim
n→∞
#([a, b] ∩ {F−1(x1), . . . , F−1(xn)})
n
=
= lim
n→∞
#([F (a), F (b)] ∩ {x1, . . . , xn})
n
= F (b)− F (a).
Corollary 2.4. Let F be a strictly increasing continuous distribution function on R and p be
a Borel probability measure on R defined by F. Then for a set of all p-equidistributed sequences
DF ⊂ R∞ we have
(i) DF = {(F−1(xk))k∈N : (xk)k∈N ∈ D}, where D is from Corollary 2.3;
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
474 Z. ZERAKIDZE, G. PANTSULAIA, G. SAATASHVILI
(ii) p∞(DF ) = 1;
(iii) DF is a set of the first category;
(iv) DF is everywhere dense in the Tychonoff topology.
Let (µθ)θ∈Θ be a sequence of probability measures defined on a measurable space (E,S). For
θ ∈ Θ, we denote by µθ the completion of the measure µθ and by dom(µθ) the σ-algebra of all
µθ-measurable subsets of E.
Definition 2.5. We say that the family (µθ)θ∈Θ is strongly separated if there exists a family
(Zθ)θ∈Θ of elements of the σ-algebra ∩θ∈Θdom(µθ) such that
(i) µθ(Zθ) = 1 for θ ∈ Θ;
(ii) Zθ1 ∩ Zθ2 = ∅ for all different parameters θ1 and θ2 from Θ;
(iii) ∪θ∈ΘZθ = E.
Definition 2.6. Let (µθ)θ∈Θ be a family of pairwise singular probability measures on a mea-
surable space (E,S), where Θ is equipped with a σ-algebra L(Θ) that contains all singletons of Θ
and S1 := ∩θ∈Θdom(µθ). We say that a measurable mapping θ̃ : E → Θ is a consistent estimator
of the parameter θ if
(∀θ)
(
θ ∈ Θ→ µθ({x : θ̃(x) = θ}) = 1
)
.
Lemma 2.5. Let (µθ)θ∈Θ be a family of pairwise singular probability measures on a measur-
able space (E,S), where Θ is equipped with σ-algebra L(Θ) that contains all singletons of Θ and
S1 := ∩θ∈Θdom(µθ). Then the following sentences are equivalent:
(a) there is a consistent estimator θ̃ : E → Θ of the parameter θ;
(b) the family of measures (µθ)θ∈Θ is strongly separated.
Proof. Let us show the validity of the implication (a) → (b). The existence of a consistent
estimator θ̃ : E → Θ of the parameter θ implies that
(∀θ)
(
θ ∈ Θ→ µθ({x : θ̃(x) = θ}) = 1
)
.
Setting Zθ = {x : θ̃(x) = θ} for θ ∈ Θ, we get:
(i) µθ(Zθ) = µθ({x : θ̃(x) = θ}) = 1 for θ ∈ Θ;
(ii) Zθ1 ∩ Zθ2 = ∅ for all different parameters θ1 and θ2 from Θ because
{x : θ̃(x) = θ1} ∩ {x : θ̃(x) = θ2} = ∅;
(iii) ∪θ∈ΘZθ = {x : θ̃(x) ∈ Θ} = E.
Let us show the validity of the implication (b)→ (a).
Since the family (µθ)θ∈Θ is strongly separated there exists a family (Zθ)θ∈Θ of elements of
σ-algebra S1 := ∩θ∈Θdom(µθ) such that:
(i) µθ(Zθ) = 1 for θ ∈ Θ;
(ii) Zθ1 ∩ Zθ2 = ∅ for all different parameters θ1 and θ2 from Θ;
(iii) ∪θ∈ΘZθ = E.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
ON THE SEPARATION PROBLEM FOR A FAMILY OF BOREL AND BAIRE G-POWERS . . . 475
For x ∈ E, we put θ̃(x) = θ, where θ is a unique parameter from the set Θ for which x ∈ Zθ.
The existence of such a unique parameter θ can be proved by using conditions (ii), (iii).
Now let Y ∈ L(Θ). Then {x : θ̃(x) ∈ Y } = ∪θ∈Y Zθ. We have to show that {x : θ̃(x) ∈
∈ Y } ∈ dom(µθ0) for each θ0 ∈ Θ.
If θ0 ∈ Y, then
{x : θ̃(x) ∈ Y } = ∪
θ∈Y
Zθ = Zθ0 ∪ ∪
θ∈Y \θ0
Zθ.
On the one hand, from the validity of the condition (b) it follows that
Zθ0 ∈ S1 = ∩
θ∈Θ
dom(µθ) ⊆ dom(µθ0).
On the other hand, the validity of the condition
∪
θ∈Y \θ0
Zθ ⊆ (E \ Zθ0)
implies that
µθ0(∪θ∈Y \θ0Zθ) = 0.
The latter equality yields that
∪
θ∈Y \θ0
Zθ ∈ dom(µθ0).
Since dom(µθ0) is a σ-algebra, we deduce that
{x : θ̃(x) ∈ Y } = Zθ0 ∪ ∪
θ∈Y \θ0
Zθ ∈ dom(µθ0).
If θ0 /∈ Y, then
{x : θ̃(x) ∈ Y } = ∪
θ∈Y
Zθ ⊆ (E \ Zθ0)
and we claim that µθ0({x : θ̃(x) ∈ Y }) = 0. The latter relation implies that
{x : θ̃(x) ∈ Y } ∈ dom(µθ0).
Thus we have shown the validity of the condition
{x : θ̃(x) ∈ Y } ∈ dom(µθ0)
for an arbitrary θ0 ∈ Θ. Hence
{x : θ̃(x) ∈ Y } ∈ ∪
θ0∈Θ
dom(µθ0) = S1.
Since L(Θ) contains all singletons of Θ, we claim that
(∀θ)
(
θ ∈ Θ→ µθ({x : θ̃(x) = θ}) = µθ(Zθ) = 1
)
.
Lemma 2.5 is proved.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
476 Z. ZERAKIDZE, G. PANTSULAIA, G. SAATASHVILI
Remark 2.2. Let E be a Polish space and (µθ)θ∈Θ be a family of Borel probability measures on
E. Then the σ-algebra S1 = ∩θ∈Θdom(µθ) contains a class of all universally measurable1 subsets.
Note that each universally measurable consistent estimator θ̃ : E → Θ (if such an estimator exists)
will be measurable also in the sense of Definition 2.6.
Definition 2.7. Let µ be a Borel measure on R and θ ∈ R. Let I be a non-empty parameter set.
A measure µIθ defined by µIθ =
∏
i∈I
λi with λi = µθ for i ∈ I, where µθ denotes a θ-shift measure
of µ (i.e., µθ(X) = µ(X + θ) for X ∈ B(R)), is called a Baire I-power of the θ-shift measure µθ
on R.
Definition 2.8. Let µ be a Borel measure on R, θ ∈ R and I be a non-empty parameter set.
Assume that PI is a Borel extension of the Baire I-power of a θ-shift measure in RI . Then PI is
called a Borel I-power of the θ-shift measure µθ on R.
Lemma 2.6. Let F be a strictly increasing continuous distribution function on R and let p be
a Borel probability measure on R defined by F. Then the family of Baire (equivalently, Borel) N-
powers (pNθ )θ∈R of shift measures (pθ)θ∈R, where N denotes a set of all natural numbers, is strongly
separated.
Proof. For θ ∈ R, we denote by Dθ the set of all pθ-equidistributed sequences in RN. Let us
show that Dθ1 ∩Dθ2 = ∅ for −∞ < θ1 < θ2 < +∞. For (xk)k∈N ∈ Dθ1 , we have
lim
n→∞
#((−∞, 0] ∩ {x1, . . . , xn})
n
= Fθ1(0) = F (θ1).
Analogously, for all (xk)k∈N ∈ Dθ2 , we have
lim
n→∞
#((−∞, 0] ∩ {x1, . . . , xn})
n
= Fθ2(0) = F (θ2).
Since F is a strictly increasing continuous distribution function on R, we deduce that F (θ1) < F (θ2).
The latter relation implies that Dθ1 ∩Dθ2 = ∅.
For θ ∈ R, let Yθ be a Fσ-subset of Xθ such that pNθ (Yθ) = 1.
For θ ∈ R \ {0}, we set Zθ = Yθ and
Z0 = Y0 ∪
(
RN \ ∪
θ∈R
Zθ
)
.
Let us show that Zθ ∈ S for θ ∈ R. It is clear that Zθ ∈ B(RN) ⊆ S for θ ∈ R \ {0}.
We have
Z0 = Y0 ∪
(
RN \ ∪
θ∈R\{0}
Zθ
)
.
On the one hand, Y0 ∈ B(RN) ⊂ S since Y0 is Fσ-set. On the other hand, we have
RN \ ∪
θ∈R\{0}
Zθ ⊂ RN \ Zθ
1Let (µi)i∈I be a class of all Borel probability measures defined on a Polish space E. A σ-algebra U(E) defined by
U(E) = ∩i∈Idom(µi), where µi denotes a usual completion of µi for i ∈ I, is called a class of all universally measurable
subsets of E.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
ON THE SEPARATION PROBLEM FOR A FAMILY OF BOREL AND BAIRE G-POWERS . . . 477
for each θ ∈ R. Hence
RN \ ∪
θ∈R\{0}
Zθ ∈ dom(µθ)
for θ ∈ R. Finally, we get
RN \ ∪
θ∈R\{0}
Zθ ∈ ∩
θ∈R
dom(µθ) = S.
Since S is a σ-algebra we claim that
Z0 = Y0 ∪
(
RN \ ∪
θ∈R
Zθ
)
∈ S,
because Y0 ∈ S and (RN \ ∪θ∈R Zθ) ∈ S.
Now it is not difficult to verify that
(i) pNθ (Zθ) = 1;
(ii) Zθ1 ∩ Zθ2 = ∅;
(iii) ∪θ∈RXθ = RN.
Lemma 2.6 is proved.
Lemma 2.7 (The strong law of large numbers). Let (Ω,F , p) be a probability space and X1,
X2, . . . be an infinite sequence of independent and identically distributed random variables on
(Ω,F , P ) with finite expectation value m ∈ R, where m = E(X1) = E(X2) = . . . . Then
p
ω : ω ∈ Ω & lim
n→∞
∑n
k=1
Xk(ω)
n
= m
= 1.
Lemma 2.8. Let F be a distribution function defined on the real axis R, such that the integral∫
R
xdF (x) is finite. Suppose that p is a Borel probability measure on R defined by F. Then the family
of Baire (equivalently, Borel) N-powers (pNθ )θ∈R of shift measures (pθ)θ∈R is strongly separable.
Proof. For θ ∈ R, we define Dθ with
Dθ =
(xi)i∈N : (xi)i∈N ∈ RN & lim
n→∞
∑n
k=1
xk
n
= θ +m
= 1,
where the k-th projection Prk is defined by Prk((xi)i∈N) = xk for (xi)i∈N ∈ RN.
By Lemma 2.7 we conclude that p∞θ (Dθ) = 1 for θ ∈ R.
It is obvious that Dθ1 ∩Dθ2 = ∅ for different θ1, θ2 ∈ R.
The application of the argument used in the proof of Lemma 2.6 ends the proof of Lemma 2.8.
Lemma 2.9. Let G be an infinite additive group. Let (pk)k∈N be a sequence of probability
Borel measures defined on R and let (αk)k∈N be a sequence of positive real numbers such that∑
k∈N
αk = 1. Let pGk be a Borel (or a Baire) G-power of the measure pk for k ∈ N and µ =
=
∑
k∈N
αkp
G
k . Then the family of all coordinate projections X = (Prg)g∈G defined on a probabil-
ity space (RG, B(RG), µ)
(
or (RG, Ba(RG), µ)
)
is a G-process.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
478 Z. ZERAKIDZE, G. PANTSULAIA, G. SAATASHVILI
Proof. For n ∈ N, (g1, . . . , gn) ∈ Gn, (x1, . . . , xn) ∈ Rn and h ∈ G, we have
F
(X)
(g1,...,gn)(x1, . . . , xn)) =
= µ
({
(ωg)g∈G : (ωg)g∈G ∈ RG & (ωg1 , . . . , ωgn) ∈
n∏
k=1
(−∞, xk]
})
=
=
(∑
k∈N
αkp
G
k
)({
(ωg)g∈G : (ωg)g∈G ∈ RG & (ωg1 , . . . , ωgn) ∈
n∏
k=1
(−∞, xk]
})
=
=
∑
k∈N
αkp
G
k
({
(ωg)g∈G : (ωg)g∈G ∈ RG & (ωg1 , . . . , ωgn) ∈
n∏
k=1
(−∞, xk]
})
=
=
∑
k∈N
αkp
G
k
({
(ωg)g∈G : (ωg)g∈G ∈ RG & (ωg1+h, . . . , ωgn+h) ∈
n∏
k=1
(−∞, xk]
})
=
=
(∑
k∈N
αkp
G
k
)({
(ωg)g∈G : (ωg)g∈G ∈ RG & (ωg1+h, . . . , ωgn+h) ∈
n∏
k=1
(−∞, xk]
})
=
= F
(X)
(g1+h,...,gn+h)(x1, . . . , xn)).
Lemma 2.9 is proved.
3. Formulations and proofs of the main results.
Theorem 3.1. Let F be a strictly increasing continuous distribution function on R and p be a
Borel probability measure on R defined by F. Let G be an infinite additive group. Then the family
of Borel G-powers (pGθ )θ∈R of Borel shift measures (pθ)θ∈R is strongly separated and the family
of all coordinate projections X = (Prg)g∈G defined on a probability space (RG,B(RG), pGθ ) is a
G-process for every θ ∈ R.
Proof. By Lemma 2.2, there is a unique Borel extension of the Baire measure pGθ . We preserve
the notation pGθ for its Borel extension. Let G0 be a subset of G with card(G0) = ℵ0. By Lemma 2.1
we can establish that pGθ = pG0
θ × p
G\G0
θ . By Lemma 2.6, (pG0
θ )θ∈R is strongly separated, which
implies that there exists a family (Zθ)θ∈R of elements of the σ-algebra ∩θ∈Rdom(pG0
θ ) such that:
(i) pG0
θ (Zθ) = 1 for θ ∈ R;
(ii) Zθ1 ∩ Zθ2 = ∅ for all different parameters θ1 and θ2 from R;
(iii) ∪θ∈RZθ = RG0 .
Now setting Dθ = Zθ × RG\G0 , we get that (Dθ)θ∈R is a family of elements of the σ-algebra
S = ∩θ∈Rdom(pGθ ) such that the following three conditions are fulfilled:
(i∗) pGθ (Dθ) = 1 for θ ∈ R;
(ii∗) Dθ1 ∩Dθ2 = ∅ for all different parameters θ1 and θ2 from R;
(iii∗) ∪θ∈RDθ = RG.
Theorem 3.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
ON THE SEPARATION PROBLEM FOR A FAMILY OF BOREL AND BAIRE G-POWERS . . . 479
Remark 3.1. Let us equip Θ = R with a usual metric. Under the conditions of Theorem 3.1, by
Lemmas 2.5 and 2.6 we deduce that the mapping θ̃ : RG → Θ defined by θ̃(x) = θ for x ∈ Dθ is a
consistent estimator of the parameter θ.
Theorem 3.2. Let F be a distribution function on R such that the integral
∫
R
xdF (x) is finite.
Suppose that p is a Borel probability measure on R defined by F. Let G be an infinite additive
group. Then the family of Baire G-powers (pGθ )θ∈R of Borel shift measures (pθ)θ∈R on R is strongly
separated and the family of all coordinate projections X = (Prg)g∈G defined on a probability space
(RG, Ba(RG), pGθ ) is a G-process for every θ ∈ R.
Remark 3.2. The proof of Theorem 3.2 can be obtained by using Lemmas 2.1 and 2.8. Let us
equip Θ = R with a usual metric. Under the conditions of Theorem 3.2, by Lemma 2.5 we claim
that there exists a consistent estimator of the parameter θ.
Example 3.1. Let p be a Gaussian Borel measure on R. Then by Theorem 3.1 (or by Theo-
rem 3.2) we deduce that the family of Borel (or Baire) G-powers (pGθ )θ∈R of shift measures (pθ)θ∈R
on R is strong separated for an arbitrary additive group G, and that the family of all coordinate
projections X = (Prg)g∈G defined on a probability space (RG,B(RG), pGθ ) (or (RG, Ba(RG), pGθ ))
is a G-process for every θ ∈ R.
Example 3.2. Let p be a Poisson Borel probability measure on R. Then by Theorem 3.2 we
deduce that the family of Baire G-powers (pGθ )θ∈R of shift measures (pθ)θ∈R on R is strongly
separated for an arbitrary additive group G, and that the family of all coordinate projections X =
= (Prg)g∈G defined on a probability space (RG, Ba(RG), pGθ ) is a G-process for every θ ∈ R.
We cannot apply Theorem 3.1 in order to establish the validity of this fact since the family of shift
measures (pθ)θ∈R does not satisfy the conditions of this theorem.
Example 3.3. Let p be a Cauchy Borel probability measure on R. Then by Theorem 3.1,
we deduce that for an arbitrary additive group G, the family of Borel G-powers (pGθ )θ∈R of shift
measures (pθ)θ∈R on R is strongly separated and the family of all coordinate projections X =
= (Prg)g∈G defined on a probability space (RG,B(RG), pGθ ) is a G-process for every θ ∈ R. We
cannot apply Theorem 3.2 in order to establish the validity of this fact since the integral
∫
R
xdF (x)
does not converge.
Theorem 3.3. Let (Θi)i∈I be a partition of the real axis R, such that card(Θi) ≤ ℵ0, where
ℵ0 is the cardinality of the set of all natural numbers. Let (α
(i)
θ )θ∈Θi
be a sequence of positive real
numbers such that
∑
θ∈Θi
α
(i)
θ = 1 for i ∈ I. Let µθ be a θ-shift of the Borel probability measure
µ on R with a strictly increasing continuous distribution function. Let G be an infinite additive
group. For i ∈ I, we define a Borel probability measure λi on RG by λi =
∑
θ∈Θi
α
(i)
θ µ
G
θ . Then
(λi)i∈I is strongly separated and the family of all coordinate projections X = (Prg)g∈G defined on
a probability space (RG,B(RG), λi) is a G-process for every i ∈ I.
Proof. By Lemma 2.2, the Baire measure µGθ admits a unique Borel extension for θ ∈ R for
which we preserve the same notation. Note that λi =
∑
θ∈Θi
α
(i)
θ µ
G
θ will be a Borel probability
measure on RG for i ∈ I.
By Lemma 2.9, the family of all coordinate projections X = (Prg)g∈G defined on a probability
space (RG,B(RG), λi) is a G-process for every i ∈ I.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
480 Z. ZERAKIDZE, G. PANTSULAIA, G. SAATASHVILI
By Theorem 3.1, the family of Borel probability measures (µGθ )θ∈R is strongly separated, i.e.,
there exists a family (Dθ)θ∈R of elements of the σ-algebra ∩θ∈Rdom(µGθ ) such that:
(i) µGθ (Dθ) = 1 for θ ∈ R;
(ii) Dθ1 ∩Dθ2 = ∅ for all different parameters θ1 and θ2 from R;
(iii) ∪θ∈RDθ = RG.
We set Ei = ∪θ∈Θi
Dθ for i ∈ I. Now it is clear that (Ei)i∈I is a pairwise disjoint family of
elements of the σ-algebra ∩i∈Idom(λi) such that λi(Ei) = 1 for i ∈ I.
Theorem 3.3 is proved.
Remark 3.3. Let us equip a set I with a discrete metric. Under the conditions of Theorem 3.3,
by Lemma 2.5 we claim that a mapping θ̃ : RG → I, defined by θ̃(x) = i for x ∈ Ei, is a consistent
estimator of the parameter i.
The following theorem is a simple consequence of Theorem 3.2.
Theorem 3.4. Let (Θi)i∈I be a partition of the real axis R such that card(Θi) ≤ ℵ0, where
ℵ0 is the cardinality of the set of all natural numbers. Let (α
(i)
θ )θ∈Θi
be a sequence of positive real
numbers such that
∑
θ∈Θi
α
(i)
θ = 1 for i ∈ I. Let µθ be a θ-shift of the Borel probability measure
µ on R such that the integral
∫
R
xdF (x) is finite, where F is the distribution function defined by
µ. Let G be an infinite additive group. For i ∈ I, we define a Baire probability measure on RG by
λi =
∑
θ∈Θi
α
(i)
θ µ
G
θ . Then (λi)i∈I is strongly separated and the family of all coordinate projections
X = (Prg)g∈G defined on a probability space (RG, Ba(RG), λi) is a G-process for every i ∈ I.
Remark 3.4. Let us equip the set I with a discrete metric. Under the conditions of Theorem 3.4,
by Lemma 2.5 we deduce that there exists a consistent estimator of the parameter i.
4. On consistent estimators of a useful signal in the linear one-dimensional stochastic model
when the expectation of the transformed signal is not defined. Suppose that Θ is a vector subspace
of the infinite-dimensional topological vector space of all real-valued sequences RN equipped with
the product topology.
In the information transmission theory we consider the linear one-dimensional stochastic system
(ξk)k∈N = (θk)k∈N + (∆k)k∈N, (4.1)
where (θk)k∈N ∈ Θ is a sequence of useful signals, (∆k)k∈N is sequence of independent identically
distributed random variables (the so-called generalized “white noise” ) defined on the some probabil-
ity space (Ω,F , P ) and (ξk)k∈N is a sequence of transformed signals. Let µ be a Borel probability
measure on R defined by a random variable ∆1. Then the N-power of the measure µ denoted by µN
coincides with the Borel probability measure on RN defined by the generalized “white noise”, i.e.,
(∀X)(X ∈ B(RN)→ µN(X) = P ({ω : ω ∈ Ω & (∆k(ω))k∈N ∈ X})),
where B(RN) is the Borel σ-algebra of subsets of RN.
In the information transmission theory, the general decision is that the Borel probability measure
λ, defined by the sequence of transformed signals (ξk)k∈N coincides with
(
µN
)
θ0
for some θ0 ∈ Θ
provided that
(∃θ0)(θ0 ∈ Θ→ (∀X)(X ∈ B(RN)→ λ(X) =
(
µN
)
θ0
(X))),
where
(
µN
)
θ0
(X) = µN (X − θ0) for X ∈ B(RN).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
ON THE SEPARATION PROBLEM FOR A FAMILY OF BOREL AND BAIRE G-POWERS . . . 481
Here we consider a particular case of the above model when a vector space of useful signals Θ
has the form
Θ = {(θ, θ, . . .) : θ ∈ R}.
For θ ∈ R, a measure µNθ , defined by
µNθ = µθ × µθ × . . . ,
where µθ is a θ-shift of µ (i.e., µθ(X) = µ(X − θ) for X ∈ B(R)), is called the N-power of the
θ-shift of µ on R. It is obvious that µNθ =
(
µN
)
(θ,θ,...)
.
Following Lemma 2.7, the sample mean is a consistent estimator of a parameter θ ∈ R (in the
sense of almost everywhere convergence) for the family (µNθ )θ∈R if the first order absolute moment of
µ is finite. We have a different picture when the first order absolute moment of µ is not defined. In that
case, Lemma 2.7 cannot be used. Unfortunately, we could not find in the literature any method that
would allow us to estimate a useful signal for model (4.1) when the expectation of the transformed
signal is not defined. In the remaining part of the paper we resolve this problem.
Definition 4.1. A Borel measurable function Tn : Rn → R, n ∈ N, is called a consistent
estimator of a parameter θ (in the sense of everywhere convergence) for the family (µNθ )θ∈R if the
condition
µNθ ({(xk)k∈N : (xk)k∈N ∈ RN & lim
n→∞
Tn(x1, . . . , xn) = θ}) = 1
holds for each θ ∈ R.
Definition 4.2. A Borel measurable function Tn : Rn → R, n ∈ N, is called a consistent
estimator of a parameter θ (in the sense of convergence in probability) for the family (µNθ )θ∈R if the
condition
lim
n→∞
µNθ
({
(xk)k∈N : (xk)k∈N ∈ RN &
∣∣∣Tn(x1, . . . , xn)− θ| > ε
})
= 0
holds for every ε > 0 and θ ∈ R.
Definition 4.3. A Borel measurable function Tn : Rn → R, n ∈ N, is called a consistent
estimator of a parameter θ (in the sense of convergence in distribution) for the family (µNθ )θ∈R if the
condition
lim
n→∞
∫
RN
f(Tn(x1, . . . , xn))dµNθ ((xk)k∈N) = f(θ)
holds for every continuous bounded real-valued function f on R.
Remark 4.1. Based on [6, p. 272] (see Theorem 2), for the family (µNθ )θ∈R we make the
following conclusions:
(a) the existence of a consistent estimator of a parameter θ in the sense of everywhere conver-
gence implies the existence of a consistent estimator of a parameter θ in the sense of convergence in
probability;
(b) the existence of a consistent estimator of a parameter θ in the sense of convergence in
probability implies the existence of a consistent estimator of a parameter θ in the sense of convergence
in distribution.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
482 Z. ZERAKIDZE, G. PANTSULAIA, G. SAATASHVILI
Theorem 4.1. Let F be a strictly increasing continuous distribution function on R and µ be
a Borel probability measure on R defined by F. For θ ∈ R, we set Fθ(x) = F (x − θ), x ∈ R,
and denote by µθ the Borel probability measure on R defined by Fθ (obviously, this is an equivalent
definition of the θ-shift of µ). Then a function Tn : Rn → R, defined by
Tn(x1, . . . , xn) = −F−1(n−1#
(
{x1, . . . , xn} ∩ (−∞; 0])
)
(4.2)
for (x1, . . . , xn) ∈ Rn, n ∈ N, is a consistent estimator of a parameter θ for the family (µNθ )θ∈R in
the sense of almost everywhere convergence.
Proof. It is clear that Tn is a Borel measurable function for n ∈ N. For θ ∈ R, we set
Aθ =
{
(xk)k∈N : (xk)k∈N is µθ-uniformly distributed on R
}
.
By Corollary 2.4, we obtain µNθ (Aθ) = 1 for θ ∈ R.
For θ ∈ R, we have
µNθ
({
(xk)k∈N : (xk)k∈N ∈ RN & lim
n→∞
Tn(x1, . . . , xn) = θ
})
=
= µNθ
({
(xk)k∈N : (xk)k∈N ∈ RN & lim
n→∞
F−1(n−1#({x1, . . . , xn} ∩ (−∞; 0])) = −θ
})
=
= µNθ
({
(xk)k∈N : (xk)k∈N ∈ RN & lim
n→∞
n−1#({x1, . . . , xn} ∩ (−∞; 0]) = F (−θ)
})
=
= µNθ
({
(xk)k∈N : (xk)k∈N ∈ RN & lim
n→∞
n−1#({x1, . . . , xn} ∩ (−∞; 0]) = Fθ(0)
})
≥
≥ µNθ (Aθ) = 1.
Theorem 4.1 is proved.
The following corollaries are simple consequences of Theorem 4.1 and Remark 4.1.
Corollary 4.1. An estimator Tn defined by (4.2) is a consistent estimator of a parameter θ for
the family (µNθ )θ∈R in the sense of convergence in probability.
Corollary 4.2. An estimator Tn defined by (4.2) is a consistent estimator of a parameter θ for
the family (µNθ )θ∈R in the sense of convergence in distribution.
Remark 4.2. Combining Lemma 2.7 and Theorem 4.1,we get the validity of the condition
µNθ
({
(xk)k∈N : (xk)k∈N ∈ RN & − lim
n→∞
F−1
(
n−1#({x1, . . . , xn} ∩ (−∞; 0])
)
=
= lim
n→∞
n−1
n∑
k=1
xk = θ
})
= 1
for θ ∈ R when µ is equivalent to the linear standard Gaussian measure on R, the first order absolute
moment of µ is finite and the first order moment of µ is equal to zero.
Definition 4.4. Following [1], the family (µNθ )θ∈R is called strongly separated in the usual
sense if there exists a family (Zθ)θ∈R of Borel subsets of RN such that
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
ON THE SEPARATION PROBLEM FOR A FAMILY OF BOREL AND BAIRE G-POWERS . . . 483
(i) µNθ (Zθ) = 1 for θ ∈ R;
(ii) Zθ1 ∩ Zθ2 = ∅ for all different parameters θ1 and θ2 from R;
(iii) ∪θ∈RZθ = RN.
Definition 4.5. Following [1], a Borel measurable function T : RN → R is called an infinite
sample consistent estimator of a parameter θ for the family (µNθ )θ∈R if the following condition:
(∀θ)
(
θ ∈ R→ µNθ
({
(xk)k∈N : (xk)k∈N ∈ RN & T ((xk)k∈N) = θ
})
= 1
)
ia fulfilled.
Remark 4.3. The existence of an infinite sample consistent estimator of a parameter θ for the
family (µNθ )θ∈R implies that the family (µNθ )θ∈R is strongly separated in a usual sense. Indeed, if
we set Zθ =
{
(xk)k∈N : (xk)k∈N ∈ RN & T ((xk)k∈N) = θ
}
for θ ∈ R, then all the conditions of
Definition 2.5 will be satisfied.
Theorem 4.2. Let F be a strictly increasing continuous distribution function on R and µ
be the Borel probability measure on R defined by F. For θ ∈ R, we set Fθ(x) = F (x − θ),
x ∈ R, and denote by µθ the Borel probability measure on R defined by Fθ. Then the estimators
lim T̃n := infn supm≥n T̃m and lim T̃n := supn infm≥n T̃m are infinite sample consistent estimators
of a parameter θ for the family (µNθ )θ∈R, where T̃n : RN → R is defined by
(∀(xk)k∈N)
(
(xk)k∈N ∈ RN → T̃n((xk)k∈N) = −F−1(n−1#({x1, . . . , xn} ∩ (−∞; 0]))
)
. (4.3)
Proof. Following [6, p. 189], the functions lim T̃n and lim T̃n are Borel measurable. By Corol-
lary 2.4, we have µNθ (Aθ) = 1 for θ ∈ R, which implies
µNθ
({
(xk)k∈N : (xk)k∈N ∈ RN & lim T̃n(xk)k∈N = θ
})
≥
≥ µNθ
({
(xk)k∈N : (xk)k∈N ∈ RN & lim T̃n(xk)k∈N = lim T̃n(xk)k∈N = θ
})
≥
≥ µNθ (Aθ) = 1,
where
Aθ =
{
(xk)k∈N : (xk)k∈N is µθ-uniformly distributed on R
}
for θ ∈ R.
From the last relation it follows that lim T̃n is the infinite sample consistent estimator of a
parameter θ for the family (µNθ )θ∈R.
Using the above scheme, we can established the validity of an analogous fact for the estimator
lim T̃n.
Theorem 4.2 is proved.
Remark 4.4. By Remark 4.3 and Theorem 4.2, we deduce that the family (µNθ )θ∈R is strongly
separated in the usual sense. Since each Borel subset of RN is an element of the σ-algebra S :=
:= ∩θ∈Rdom(µθ), we claim that Theorem 4.2 extends the result of Lemma 2.6.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
484 Z. ZERAKIDZE, G. PANTSULAIA, G. SAATASHVILI
Example 4.1. Let µ(θ,1) be a linear Gaussian measure on R with parameters (θ, 1). Let [·]
denote the integer part of a real number. Since a sequence of real numbers (π × k − [π × k])k∈N is
uniformly distributed on (0, 1) (see [2, p. 17], Example 2.1), by Lemma 2.4 we claim that a sequence
(xk)k∈N defined by
xk = F−1(π × k − [π × k]) + θ
is a µ(θ,1)-equidistributed sequence on R, where F denotes a linear standard Gaussian distribution
function on R with parameters (θ, 1).
It is obvious that (xk)k∈N is a certain realization of model (4.1), where (∆k)k∈N is a sequence of
independent Gaussian random variables on R.
In the sequel we use the notation introduced above:
(i) n is a number of trials;
(ii) Tn is the estimator defined by formula (4.2);
(iii) Xn is the sample average;
(iv) θ is a “useful signal”.
We have considered the construction of the one-dimensional linear stochastic model (4.1) for
θ = 1. Below we present the numerical results obtained by Microsoft Excel.
Table 4.1
n Tn Xn θ n Tn Xn θ
50 0.994457883 1.146952654 1 550 1.04034032 1.034899747 1
100 1.036433389 1.010190601 1 600 1.036433389 1.043940988 1
150 1.022241387 1.064790041 1 650 1.03313984 1.036321771 1
200 1.036433389 1.037987511 1 700 1.030325691 1.037905202 1
250 1.027893346 1.045296447 1 750 1.033578332 1.03728633 1
300 1.036433389 1.044049728 1 800 1.03108705 1.032630945 1
350 1.030325691 1.034339407 1 850 1.033913784 1.037321098 1
400 1.036433389 1.045181911 1 900 1.031679632 1.026202323 1
450 1.031679632 1.023083495 1 950 1.034178696 1.036669278 1
500 1.036433389 1.044635371 1 1000 1.036433389 1.031131694 1
Note that the results of computations presented in Table 4.1 do not contradict Remark 4.2, which
asserts that under the conditions of Theorem 4.1, the estimators Tn and Xn coincide and both are
consistent estimators of the “useful signal” θ.
We have also considered similar model constructions when F is a Cauchy distribution function
on R. The results of relevant numerical computations are given in the next table.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
ON THE SEPARATION PROBLEM FOR A FAMILY OF BOREL AND BAIRE G-POWERS . . . 485
Table 4.2
n Tn Xn θ n Tn Xn θ
50 1.20879235 2.555449288 1 550 1.017284476 41.08688757 1
100 0.939062506 1.331789564 1 600 1.042790358 41.30221291 1
150 1.06489184 71.87525566 1 650 1.014605804 38.1800532 1
200 1.00000000 54.09578271 1 700 1.027297114 38.03399768 1
250 1.06489184 64.59240343 1 750 1.012645994 35.57956117 1
300 1.021166379 54.03265563 1 800 1.015832638 35.25149408 1
350 1.027297114 56.39846672 1 850 1.018652839 33.28723503 1
400 1.031919949 49.58316089 1 900 1.0070058 31.4036155 1
450 1.0070058 44.00842613 1 950 1.023420701 31.27321466 1
500 1.038428014 45.14322051 1 1000 1.012645994 29.73405416 1
We see that the results of numerical computations in Table 4.2 do not contradict Theorem 4.2,
which asserts that Tn is a consistent estimator of the parameter θ = 1. These results also show
that our attempt to estimate a useful signal by the sample average in our model is not successful.
These computational results seem natural because the mean and the variance do not exist for Cauchy
random variables.
1. Ibramkhallilov I. Sh., Skorokhod A. V. On well-off estimates of parameters of stochastic processes (in Russian). –
Kiev, 1980.
2. Kuipers L., Niederreiter H. Uniform distribution of sequences. – New York etc.: John Wiley & Sons, 1974.
3. Pantsulaia G. R. On orthogonal families of probability measures (in Russian) // Trans. GPI. – 1989. – 8(350). –
P. 106 – 112.
4. Pantsulaia G. R. On separation properties for families of probability measures // Georg. Math. J. – 2003. – 10(2). –
P. 335 – 342.
5. Pantsulaia G. R. Invariant and quasiinvariant measures in infinite-dimensional topological vector spaces. – New
York: Nova Sci. Publ., Inc., 2007.
6. Shiryaev A. N. Probability (in Russian). – Moscow: Nauka, 1980.
7. Zerakidze Z. On weakly separated and separated families of probability measures (in Russian) // Bull. Acad. Sci.
Georg.SSR. – 1984. – 113(2). – P. 273 – 275.
Received 20.05.11,
after revision — 16.01.13
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
|