Natural boundary of random Dirichlet series
For the random Dirichlet series $$\sum\limits_{n = 0}^\infty {X_n (\omega )e^{ - s\lambda _n } } (s = \sigma + it \in \mathbb{C}, 0 = \lambda _0 < \lambda _n \uparrow \infty )$$ whose coefficients are uniformly nondegenerate independent random variables, we provide some explicit con...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509611330633728 |
|---|---|
| author | Ding, Xiaoqing Xiao, Yimin Дін, Сяоцин Сяо, Імінь |
| author_facet | Ding, Xiaoqing Xiao, Yimin Дін, Сяоцин Сяо, Імінь |
| author_sort | Ding, Xiaoqing |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:56:18Z |
| description | For the random Dirichlet series
$$\sum\limits_{n = 0}^\infty {X_n (\omega )e^{ - s\lambda _n } } (s = \sigma + it \in \mathbb{C}, 0 = \lambda _0 < \lambda _n \uparrow \infty )$$
whose coefficients are uniformly nondegenerate independent random variables, we provide some explicit conditions for the line of convergence to be its natural boundary a.s. |
| first_indexed | 2026-03-24T02:43:51Z |
| format | Article |
| fulltext |
UDC 519.21
Xiaoqing Ding∗ (Northwestern Polytechn. Univ., China),
Yimin Xiao∗∗ (Michigan State Univ., USA)
NATURAL BOUNDARY OF RANDOM DIRICHLET SERIES
NATURAL\NA HRANYCQ VYPADKOVOHO RQDU DIRIXLE
For the random Dirichlet series
∞∑
n=0
Xn(ω) e−sλn (s = σ + it ∈ C, 0 = λ0 < λn ↑ ∞),
whose coefficients are uniformly nondegenerate independent random variables, we provide some explicit con-
ditions for the line of convergence to be its natural boundary a.s.
Dlq vypadkovoho rqdu Dirixle
∞∑
n=0
Xn(ω) e−sλn (s = σ + it ∈ C, 0 = λ0 < λn ↑ ∞),
koefici[nty qkoho — rivnomirno nevyrodΩeni nezaleΩni vypadkovi zminni, vvedeno deqki qvni umovy,
za qkyx liniq zbiΩnosti [ joho natural\nog hranyceg majΩe napevno.
1. Introduction. Consider the random Dirichlet series
∞∑
n=0
Xn e
−sλn , (1.1)
where the coefficients {Xn} are complex-valued random variables, 0 = λ0 < λ1 <
< λ2 < . . . is an increasing sequence of real numbers such that λn → ∞ and s = σ + it
is the complex variable. Let
σc(ω) = inf
{
σ ∈ R :
∞∑
n=0
Xn(ω) e−σλn converges
}
, (1.2)
then it is well known that (1.1) converges at s = σ + it if σ > σc(ω) and diverges at s if
σ < σc(ω). Hence σc(ω) is called the abscissa of convergence and the line s = σc(ω)+it,
t ∈ R, is called the convergence line of the random Dirichlet series (1.1). It is known that
the sum fω(s) of (1.1) is analytic in the half-plane {s = σ + it: σ > σc(ω)}.
When the coefficients {Xn, n ≥ 0} are independent, the Kolmogorov zero-one law
implies that σc(ω) = σc a.s. for some constant −∞ ≤ σc ≤ ∞. It has been of importance
to answer the following two natural questions: (a) how do we determine σc? (b) when is
the line of convergence σ = σc the natural boundary of the series (1.1) itself?
In the special case of λn = n for all n ≥ 0, (1.1) is reduced to a random Taylor series
in z = es. The natural boundary problem for random Taylor seriesF (z) =
∑∞
n=0
Xnz
n,
z ∈ C, has been investigated by several authors. We refer to Kahane [1] and the ref-
erences therein for more information. A fundamental theorem in this area is due to
Ryll-Nardzewski [2] who proved the following conjecture of Blackwell: If the coef-
ficients {Xn, n ≥ 0} are independent, then there exists a deterministic Taylor series
f(z) =
∑∞
n=0
an z
n such that
(i) the circle of convergence of
∑∞
n=0
(Xn − an) zn is a.s. a natural boundary for
F (z) − f(z);
∗ Research supported by the Science Foundation of the Northwestern Polytechnical University, China.
∗∗ Research partially supported by the NSF grant DMS-0404729.
c© XIAOQING DING, YIMIN XIAO, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7 997
998 XIAOQING DING, YIMIN XIAO
(ii) the radius of convergence r(F − f) of
∑∞
n=0
(Xn − an) zn is maximal with
respect to the choice of deterministic Taylor series;
(iii) if g(z) is another deterministic Taylor series such that r(F − g) = r(F − f) a.s.
then the circle of convergence of F (z) − g(z) is a.s. its natural boundary.
Kahane [1, p. 40, 41] gives a simplified alternative proof of Ryll-Nardzewski’s result
and, moreover, he proves that if the coefficients {Xn, n ≥ 0} are symmetric, then the
circle of convergence is a natural boundary for F (z). As a related result, we also mention
that Holgate [3] has proved that Ryll-Nardzewski’s result still holds for certain random
Taylor series with dependent coefficients.
The question (b) for random Dirichlet series (1.1) with independent coefficients has
been addressed by Kahane [1] (Section 6 of Chapter 4). In particular, he has extended the
above result of Ryll-Nardzewski to random Dirichlet series and has shown that the line of
convergence is the natural boundary for (1.1) provided the coefficients {Xn, n ≥ 0} are
independent and symmetric random variables.
However, if the coefficients {Xn, n ≥ 0} are not assumed to be symmetric, Kahane’s
result does not give any information about the location of the natural boundary of random
Dirichlet series (1.1). Even in the special case of random Taylor series, very little has
been known. Sun [4] has considered the natural boundary problem for random Taylor
series with nonsymmetric independent coefficients, but his condition is rather restrictive;
see Remark 3.2.
The objective of this paper is to investigate the questions (a) and (b) above for random
Dirichlet series (1.1) with independent and uniformly nondegenerate coefficients {Xn,
n ≥ 0} (see Section 2 for definition). We will provide some explicit conditions for (1.1)
to have a.s. the convergence line as its natural boundary. When applied to random Taylor
series, our result gives a more convenient condition for the convergence circle to be the
natural boundary a.s.
We remark that the existence of singular points on the convergence line for random
Dirichlet series (1.1) whose coefficients {Xn, n ≥ 0} form a martingale difference se-
quence has been considered by Ding [5]. However, as far as we know, no results on the
natural boundary of such series have been established. Similarly, the problem of finding
the location of natural boundaries for the random Taylor series considered by Holgate [3]
is also open. For general references on other aspects of deterministic and random Dirich-
let series such as convergence, growth and value distributions, we refer to Mandelbrojt
[6] and Yu [7].
2. Main result and its proof. Consider the random Dirichlet series
∞∑
n=0
Xn(ω) e−sλn , s = σ + it, 0 = λ0 < λn ↑ ∞, (2.1)
where {Xn, n ≥ 0} is a sequence of independent, complex-valued random variables
defined on a complete probability space (Ω,F , P) such that
sup
n≥0
sup
a∈C
P{Xn = a} < 1. (2.2)
Any sequence {Xn, n ≥ 0} of random variables satisfying (2.2) is said to be uni-
formly nondegenerate. Clearly, the Rademacher sequence, Steinhaus sequence and, more
generally, any sequence of continuous random variables are uniformly nondegenerate in
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
NATURAL BOUNDARY OF RANDOM DIRICHLET SERIES 999
the above sense. It is verified in Lemma 2.1 that the condition (2.2) holds if and only if
there is a sequence {Rn} of positive numbers such that
sup
n≥0
sup
a∈C
P
{
|Xn − a| ≤ Rn
}
< 1. (2.3)
This property is connected with the Stein property of {Xn}; see [8].
Let σ(1)
c be the abscissa of convergence of the series (2.1), which is a constant or
infinity a.s. by Kolmogorov’s zero-one law. Denote by σ
(2)
c the abscissa of convergence
of the Dirichlet series
∞∑
n=0
R2
n e
−2sλn . (2.4)
The main result of the paper is as follows.
Theorem 2.1. Let {Xn, n ≥ 0} be a sequence of independent complex-valued
random variables satisfying the condition (2.2). Then:
(i) σ
(1)
c ≥ σ
(2)
c a.s.;
(ii) if σ(1)
c < ∞, then with probability one for every rational number t the disc{
s ∈ C :
∣∣s− (σ(1)
c + 1 + it)
∣∣ ≤ σ(1)
c + 1 − σ(2)
c
}
contains at least one singularity of the series (2.1);
(iii) if σ(1)
c = σ
(2)
c < ∞, then with probability one the line Re s = σ
(1)
c is the natural
boundary of the series (2.1).
In Section 3, we will give some sufficient conditions on the sequences {Xn, n ≥ 0}
and {λn, n ≥ 0} so that σ(1)
c = σ
(2)
c a.s.
For the proof of Theorem 2.1, we need the following two lemmas.
Lemma 2.1. The condition (2.2) holds if and only if there is a sequence {Rn} of
positive numbers such that (2.3) holds.
Proof. The condition (2.3) implies clearly the condition (2.2). The reverse follows
immediately from the following proposition1 : If a random variable X has the property
that for some constant ε ∈ (0, 1),
sup
a∈C
P
{
X = a
}
≤ 1 − ε, (2.5)
then there exists a constant R > 0 such that
sup
a∈C
P
{
|X − a| ≤ R
}
≤ 1 − ε
2
. (2.6)
Assume the above proposition does not hold. Then for every integer n ≥ 1, there
exists an ∈ C such that
P
{
|X − an| ≤
1
n
}
> 1 − ε
2
. (2.7)
We may choose {an, n ≥ 1} such that either lim
n→∞
|an| = ∞ or lim
n→∞
an = a∞ ∈ C.
In the first case, (2.7) implies that for every positive number C, P
{
|X| ≥ C
}
≥ 1 − ε
2
,
which is impossible. In the second case, we note that for an arbitrary δ > 0, we can
choose n >
2
δ
such that |a∞ − an| <
δ
2
. Hence by (2.7) we derive
The proof of (2.6) is suggested by the referee. It is simpler than our original proof.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
1000 XIAOQING DING, YIMIN XIAO
P
{
|X − a∞| ≤ δ
}
≥ P
{
|X − an| ≤
1
n
}
> 1 − ε
2
.
Letting δ → 0+, we obtain that P
{
X = a∞
}
≥ 1 − ε
2
, which contradicts (2.5). The
proof of Lemma 2.1 is completed.
Lemma 2.2 below is a consequence of Lemma 2.1 and Corollary 2 of Burkholder [8].
It extends Theorem 1 in Chapter 2 of Kahane [1].
Lemma 2.2. Let {Xn, n ≥ 0} be a sequence of independent complex-valued ran-
dom variables satisfying the condition (2.2). Suppose {bmn, m, n ≥ 0} is a double
sequence of complex numbers such that for each m ≥ 0 the series
∑∞
n=0
bmnXn con-
verges a.s. to a random variable Ym. If the sequence {Ym, m ≥ 0} is convergent a.s.,
then there is a sequence {bn, n ≥ 0} of complex numbers such that
lim
m→∞
∞∑
n=0
|bmn − bn|2 R2
n = 0 and
∞∑
n=0
|bn|2 R2
n < ∞. (2.8)
Proof. Put Zn = Xn/Rn. Then Lemma 2.1 implies that
sup
n≥0
sup
a∈C
P {|Zn − a| < 1} < 1.
Now by applying Corollary 2 of Burkholder [8] to the series
∑∞
n=0
(bmnRn)Zn we com-
plete the proof.
Proof of Theorem 2.1. In order to prove (i), we assume σ(1)
c < ∞; otherwise there
is nothing to prove. For any σ > σ
(1)
c the random Dirichlet series (2.1) converges a.s. for
s = σ. By applying Lemma 2.2 to bmn = e−σλn , we see that the series (2.4) converges
at s = σ. Hence σ(2)
c ≤ σ so that (i) holds.
Next we prove (ii). Let fω(s) be the sum function of the random Dirichlet series (2.1)
for Re s > σ
(1)
c . For t ∈ R, put
R(t, ω) =
lim sup
n→∞
∣∣∣∣∣f
(n)
ω (σ(1)
c + 1 + it)
n!
∣∣∣∣∣
1
n
−1
, (2.9)
where f (n) denotes the n-th derivative of f . Then R(t, ω) is the radius of convergence of
the Taylor expansion of fω(s) around s0 = σ
(1)
c + 1 + it. Suppose (ii) is not true, then
there is a rational number t0 such that, with positive probability, the disc{
s ∈ C :
∣∣s− (
σ(1)
c + 1 + it0
)∣∣ ≤ σ(1)
c + 1 − σ(2)
c
}
does not contain any singularities of the series (2.1). This implies that
P
{
R(t0, ω) > σ(1)
c + 1 − σ(2)
c
}
> 0. (2.10)
However, by Kolmogorov’s zero-one law and (2.10), R(t0, ω) is a.s. a constant R(t0) so
that
R(t0, ω) = R(t0) > σ(1)
c + 1 − σ(2)
c a.s. (2.11)
Now we choose σ0 < σ
(2)
c such that R(t0) > σ
(1)
c + 1 − σ0 and let
Ω0 =
{
ω : R(t0, ω) > σ(1)
c + 1 − σ0
}
.
Then (2.11) implies P {Ω0} = 1.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
NATURAL BOUNDARY OF RANDOM DIRICHLET SERIES 1001
Put ζ0 = σ
(1)
c +1+it0. Since σ0 < σ
(2)
c we can choose s1
(
e.g. s1 =
σ0 + σ
(2)
c
2
+it0
)
such that
|s1 − ζ0| < σ(1)
c + 1 − σ0, σ0 < Re s1 = σ1 < σ(2)
c . (2.12)
For each ω ∈ Ω0, we consider the Taylor series
Y (ω) =
∞∑
n=0
f
(n)
ω (ζ0)
n!
(s1 − ζ0)n. (2.13)
It follows from (2.11) and R(t0, ω) > σ
(1)
c +1−σ0 > |s1 − ζ0| that (2.13) is convergent
for every ω ∈ Ω0. Now put
bmn =
m∑
k=0
φ
(k)
n (ζ0)
k!
(s1 − ζ0)k,
where φn(s) = e−sλn . Then for every m ≥ 0, the series
∑∞
n=0
bmnXn converges a.s.
to the series
m∑
k=0
(s1 − ζ0)k
k!
∞∑
n=0
φ(k)
n (ζ0)Xn(ω) =
m∑
k=0
(s1 − ζ0)k
k!
f (k)
ω (ζ0) =̂Ym(ω).
Moreover, (2.13) implies that the sequence {Ym(ω), m ≥ 0} converges a.s. to the random
variable Y (ω). Therefore it follows from Lemma 2.2 and limm→∞ bmn = φn(s1) =
= e−s1λn that
∞∑
n=0
R2
n e
−2λnRe s1 < ∞
so that the series (2.4) converges for Re s1 < σ
(2)
c (see (2.12)). This is a contradiction,
which proves (ii).
Finally we prove (iii). Since the series (2.1) converges a.s. in the half plane: Re s >
> σ
(1)
c , we have inft∈Q R(t, ω) ≥ 1 a.s., where Q is the set of rational numbers. On the
other hand, by (ii) and the hypothesis that σ(1)
c = σ
(2)
c , we have supt∈Q R(t, ω) ≤ 1 a.s.
Therefore R(t, ω) = 1 a.s. for all t ∈ Q, that is, σ(1)
c + it is a singularity of the function
fω(s) with probability one. Therefore, (iii) follows readily from the facts that Q is dense
in R and the set of the singularities of an analytic function is closed. This finishes the
proof of Theorem 2.1.
3. Three corollaries. In this section, we apply Theorem 2.1 to random Dirichlet series
and random Taylor series with different conditions on their coefficients and to derive more
explicit results on the line of convergence and natural boundary.
3.1. First we consider the random Dirichlet series (2.1) with coefficients {Xn}
which are identically distributed, independent and uniformly nondegenerate random vari-
ables. We assume further that either
E{|X0|β} < ∞ for some β ∈ (0, 1) (3.1)
or
E(X0) = 0 and E{|X0|β} < ∞ for some β ∈ [1, 2]. (3.2)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
1002 XIAOQING DING, YIMIN XIAO
Theorem 3.1. Under the above assumptions, the following statements hold:
(i)
D
2
≤ σ(1)
c ≤ D
β
, where D = lim supn→∞
log n
λn
;
(ii) if D = 0, the line Re s = 0 is a.s. the natural boundary of the series (2.1).
Remark 3.1. Theorem 3.1 applies directly to random Dirichlet series (2.1) whose
coefficients are i.i.d. stable random variables.
Proof. (i) Since the random variables {Xn, n ≥ 0} are identically distributed and
uniformly nondegenerate, it follows from Lemma 2.1 that (2.3) is valid for Rn = R,
where R > 0 is some positive constant. Hence the series (2.4) reduces to the series∑∞
n=0
R2 e−2sλn . Clearly, its abscissa σ(2)
c of convergence is at least 0. When D > 0,
then it is known (see, e.g., [6, 7]) that σ(2)
c = D/2. Hence by (i) of Theorem 2.1 we
always have D/2 ≤ σ
(1)
c .
To prove σ(1)
c ≤ D/β we assume without loss of generality that D < ∞. Note that
for any ε > 0, the series
∞∑
n=0
E
{∣∣∣Xn e
− (D+ε)λn
β
∣∣∣β}
=
∞∑
n=0
E
{
|Xn|β
}
e−(D+ε)λn ,
is obviously convergent. It follows from (3.1), (3.2) and Corollary 3 in [9, p. 114] that
the series (2.1) converges a.s. for s = (D + ε)/β. Since ε > 0 is arbitrary, this implies
σ
(1)
c ≤ D
β
, as desired.
When D = 0, from the above we have σ(1)
c = σ
(2)
c = 0. Hence the statement (ii)
follows directly from (i) and Theorem 2.1 (iii).
3.2. Now we consider the random Dirichlet series (2.1) where {Xn} is a sequence
of independent random variables satisfying the following condition: there exists a con-
stant α > 0 such that
E {Xn} = 0 and 0 < αE1/2
{
|Xn|2
}
≤ E {|Xn|} (∀n ≥ 0). (3.3)
This condition was introduced by Marcinkiewicz and Zygmund [10] to establish max-
imal inequalities for the weighted partial sums of independent random variables {Xn,
n ≥ 0} with mean 0 and variance 1 and to study the convergence of the random series∑∞
n=0
anXn. A conditional version of (3.3) was formulated by Gundy [11] (who called
it Condition (MZ)∗) for extending the results of Marcinkiewicz and Zygmund [10] to se-
quences of martingale differences. See Stout [12] and the references therein for further
information.
The following lemma gives a sufficient condition for (3.3) to hold and it follows from
a remark of Marcinkiewicz and Zygmund [10, p. 70].
Lemma 3.1. Let {Xn, n ≥ 0} be a sequence of independent random variables with
mean 0. If there exist constants 0 < η < 1 and K > 0 such that
E
[
|Xn|21lAn
]
≤ η ∀n ≥ 0, (3.4)
where An = {|Xn| ≥ K E1/2(|Xn|2)}. Then (3.3) holds with α = (1 − η)/K.
Let σ(3)
c be the abscissa of convergence of the Dirichlet series
∞∑
n=0
E2 {|Xn|} e−2sλn . (3.5)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
NATURAL BOUNDARY OF RANDOM DIRICHLET SERIES 1003
Theorem 3.2. Let {Xn, n ≥ 0} be a sequence of independent random variables
satisfying the condition (3.3). Then the following statements hold:
(i) σ
(1)
c = σ
(3)
c a.s.;
(ii) if σ(3)
c is finite, then the line Re s = σ
(3)
c is a.s. the natural boundary of the
series (2.1).
Proof. For any σ > σ
(3)
c , the series
∑∞
n=0
E2 {|Xn|} e−2σ λn is convergent. It
follows from (3.3) that the series
∞∑
n=0
E
{
|Xn|2
}
e−2σ λn
also converges. Hence the random Dirichlet series (2.1) converges a.s. at s = σ. Conse-
quently, we have σ(1)
c ≤ σ
(3)
c .
To prove the reverse inequality, by (i) and (iii) of Theorem 2.1 it is sufficient to prove
sup
n≥0
sup
a∈C
P
{
|Xn − a| ≤ α
4
E1/2
{
|Xn|2
}}
< 1. (3.6)
That is, {Xn, n ≥ 0} satisfies (2.3) with Rn =
α
4
E1/2
{
|Xn|2
}
. In fact, the inequal-
ity (3.6) is a consequence of the following claim: If X is a random variable such that
E {X} = 0 and 0 < αE1/2{|X|2} ≤ E {|X|} , (3.7)
then
sup
a∈C
P
{
|X − a| ≤ α
4
E1/2
{
|X|2
}}
≤ 1 − 1
8
( α
2 + α
)2
. (3.8)
Now let us prove the above claim. Since E{X} = 0, Jensen’s inequality and the
triangle inequality imply that for all a ∈ C,
E {|X − a|} ≥ |a| and E {|X − a|} ≥ E {|X|} − |a|. (3.9)
Hence by (3.9) and the easily verifiable inequality
inf
y≥0
max {y, x− y} ≥ x
2
(∀x > 0),
we have
E {|X − a|} ≥ max
{
|a|, E{|X|} − |a|
}
≥ 1
2
E {|X|} . (3.10)
It follows from (3.10) and the Paley – Zygmund inequality [13] (cf. [1, p. 8]) that for any
λ ∈ (0, 1),
P1/2
{
|X − a| > λ
2
E {|X|}
}
≥ P1/2
{
|X − a| > λE {|X − a|}
}
≥
≥ (1 − λ)
E {|X − a|}
E1/2
{
|X − a|2
} ≥ 1 − λ√
2
max {|a|, E {|X|} − |a|}
E1/2
{
|X|2
}
+ |a| .
By taking λ = 1/2 and then using the equality
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
1004 XIAOQING DING, YIMIN XIAO
inf
y≥0
max {y, x1 − y}
x2 + y
≥ inf
y≥0
max
{
y
x2 + y
,
x1 − y
x2 + y
}
=
y
x2 + y
∣∣∣∣
y=
x1
2
=
=
x1
2x2 + x1
(∀x1 > 0, x2 > 0),
we have that for all a ∈ C,
P1/2
{
|X − a| > 1
4
E {|X|}
}
≥ 1
2
√
2
E {|X|}
2E1/2
{
|X|2
}
+ E {|X|} . (3.11)
It is easy to see that (3.8) follows from (3.7) and (3.11). Therefore the proof of Theo-
rem 3.2 is finished.
3.3. Finally, we apply Theorem 3.2 to the random Taylor series
∞∑
n=0
Xn(ω) zn, (3.12)
where the sequence {Xn} satisfies the conditions imposed in Theorem 3.2. Consider the
auxiliary series
∞∑
n=0
E2{Xn} z2n. (3.13)
Since the series (3.12) and (3.13) can be regarded as special cases of the series (2.1)
and (3.5), respectively, we have the following corollary of Theorem 3.2.
Theorem 3.3. Under the conditions of Theorem 3.2,
(i) the radius of convergence of the series (3.12) is a.s.
r =
1
lim sup
n→∞
(E{|Xn|})1/n
;
(ii) if the radius r is finite and positive, then the circle |z| = r is a.s. the natural
boundary of the random Taylor series (3.12).
Remark 3.2. Theorem 3.3 gives more concise information about the natural bound-
ary of random Taylor series than that of Sun [4], who has proved that the natural boundary
of the random Taylor series
∑∞
n=0
Yn(ω) zn is the circle
|z| =
1
lim sup
n→∞
(√
E{|Yn|2}
) 1/n
,
provided {Yn} is a sequence of independent random variables with mean 0 and finite
second moments satisfying the following condition: there exists a constant δ > 0 such
that for every event A with P{A} < δ,
inf
n≥0
∫
Ac
|Yn|2
E{|Yn|2}
dP >
1
2
, (3.14)
where Ac is the complement of A. Condition (3.14) is equivalent to assuming
E
(
|Yn|21lA) < E
(
|Yn|21lAc) holds uniformly for all events A with P{A} < δ and all
integers n ≥ 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
NATURAL BOUNDARY OF RANDOM DIRICHLET SERIES 1005
We now show that the condition (3.14) implies (3.3). Without loss of generality, we
will assume E(|Yn|2) = 1 for all n ≥ 0. Then Chebyshev’s inequality implies that for
any δ ∈ (0, 1), there exists a constant K > 0 such that
sup
n≥0
P
{
|Yn| ≥ K
}
< δ. (3.15)
Let An = {|Yn| ≥ K}. It follows from (3.14) and (3.15) that E(|Yn|21lAn
) ≤ 1/2 for all
integers n ≥ 0. Hence by Lemma 3.1 we see that (3.3) holds.
The following is an example of independent random variables {Yn, n ≥ 0} that veri-
fies the condition (3.3), but not the condition (3.14).
Example 3.1. Let {An, n ≥ 0} be a sequence of events with P(An) = (1 + n)−2.
We define a sequence of independent real-valued random variables {Yn, n ≥ 0} with
mean 0 such that |Yn| = 1 + n on Ac
n and |Yn| = (1 + n)2 on An. Then we have
E
(
|Yn|21lAn
) = (1 + n)2 and E
(
|Yn|21lAc
n
) = (1 + n)2(1 − (1 + n)−2), so (3.14) does
not hold. On the other hand, it is easy to see that the condition (3.3) is verified with
α = (
√
2)−1.
Acknowledgements. The work of this paper was done during the first author’s visit
to the Department of Statistics and Probability at Michigan State University. He thanks
the hospitality of the department.
The authors thank the referee for his/her comments that have led to several improve-
ments of this paper.
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– P. 190 – 205.
Received 18.11.2004,
after revision — 07.02.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 7
|
| id | umjimathkievua-article-3509 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:43:51Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/cb/c44b4db3dd69192d56221ee48f4927cb.pdf |
| spelling | umjimathkievua-article-35092020-03-18T19:56:18Z Natural boundary of random Dirichlet series Натуральна границя випадкового ряду Діріхле Ding, Xiaoqing Xiao, Yimin Дін, Сяоцин Сяо, Імінь For the random Dirichlet series $$\sum\limits_{n = 0}^\infty {X_n (\omega )e^{ - s\lambda _n } } (s = \sigma + it \in \mathbb{C}, 0 = \lambda _0 < \lambda _n \uparrow \infty )$$ whose coefficients are uniformly nondegenerate independent random variables, we provide some explicit conditions for the line of convergence to be its natural boundary a.s. Для випадкового ряду Діріхле $$\sum\limits_{n = 0}^\infty {X_n (\omega )e^{ - s\lambda _n } } (s = \sigma + it \in \mathbb{C}, 0 = \lambda _0 < \lambda _n \uparrow \infty )$$ коефіцієнти якого рівномірно невироджені незалежні випадкові змінні, введено деякі явні умови, за яких лінія збіжності с його натуральною границею майже напевно. Institute of Mathematics, NAS of Ukraine 2006-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3509 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 7 (2006); 997–1005 Український математичний журнал; Том 58 № 7 (2006); 997–1005 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3509/3755 https://umj.imath.kiev.ua/index.php/umj/article/view/3509/3756 Copyright (c) 2006 Ding Xiaoqing; Xiao Yimin |
| spellingShingle | Ding, Xiaoqing Xiao, Yimin Дін, Сяоцин Сяо, Імінь Natural boundary of random Dirichlet series |
| title | Natural boundary of random Dirichlet series |
| title_alt | Натуральна границя випадкового ряду Діріхле |
| title_full | Natural boundary of random Dirichlet series |
| title_fullStr | Natural boundary of random Dirichlet series |
| title_full_unstemmed | Natural boundary of random Dirichlet series |
| title_short | Natural boundary of random Dirichlet series |
| title_sort | natural boundary of random dirichlet series |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3509 |
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