Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

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A Law of the Iterated Logarithm for the Signum Function 

 

 Santosh Ghimire 

Department of Applied Sciences and Chemical Engineering, Pulchowk Campus, Institute of Engineering, Tribhuvan 

University, Kathmandu, Nepal. Email: santoshghimire@ioe.edu.np 

 

Article History: 

Received: 02-08-2024 

Revised: 12-09-2024 

Accepted: 20-09-2024 

Abstract 

In 2023, S. Ghimire established a one-sided version of a law of the iterated logarithm, 

denoted as LIL, for summations of signum functions analogous to the LIL proposed by 

Salem and Zygmund for trigonometric series. In this article, we complete the LIL for 

these signum functions by establishing the complimentary version of the LIL. 

Keywords: Signum functions, q-lacunary series, law of the iterated logarithm, Borel-

Cantelli lemma. 

 

1. Introduction 

The LIL is a widely recognized theorem in probability that complements two other fundamental 

theorems: the central limit theorem (CLT) and the law of large numbers (LLN). While the LLN 

describes the tendencies of the average of independent random variables with increasing sample size, 

and the CLT outlines the distribution of these sums, the LIL provides insights into the fluctuations of 

these sums, especially concerning their bounds. The LIL emerged from Khintchine's [1] 

investigations, in which he sought to ascertain the precise rate of convergence of normal numbers. 

Kolmogorov [5] later generalized this result to include independent random variables. Since its 

inception, the LIL has developed into a fundamental theorem with extensive applications spanning 

various areas of mathematics and statistics. A similar LIL has been developed across different fields, 

including harmonic functions [7], martingales [10], [12], q-lacunary series [8], random walks, 

stochastic processes, and more. 

In the realm of mathematics, Salem and Zygmund [8] were the first to introduce a LIL for the sums 

of q-lacunary trigonometric series. Erdos and Gal [6] subsequently obtained a comparable outcome 

for a particular category of q-lacunary series. In this LIL, only the sum of the first n-terms of the 

lacunary series was considered as in Kolmogorov's LIL. Subsequently, M. Weiss [4] obtained an LIL 

for q-lacunary series analogous to Kolmogorov's LIL. Additionally, in the same paper, Salem and 

Zygmund [8] introduced another LIL for q-lacunary series, as stated below: 

Theorem 1 (Salem and Zygmund) Let �̃�𝑁 = ∑ (𝑎𝑖 cos 𝑛𝑖𝜃 + 𝑏𝑖 sin 𝑛𝑖𝜃)
∞
𝑖=𝑁  with 

𝑛𝑖+1

𝑛𝑖
> 𝑞 > 1 and 

𝑐𝑖
2 = 𝑎𝑖

2 + 𝑏𝑖
2 satisfy ∑ 𝑐𝑖

2∞
𝑖=1 < ∞. Define �̃�𝑀 = ∑ 𝑐𝑖

2∞
𝑖=𝑀  and  �̃�𝑀 = max

i≥M
|𝑐𝑖|. Assume that �̃�1 < ∞ 

and �̃�𝑀
2 ≤ 𝐾𝑀 (

�̃�𝑀
2

ln ln
1

�̃�𝑀

) where 𝐾𝑀 approaches to 0 and M approaches to infinity. Then  



Communications on Applied Nonlinear Analysis 

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limsup
M→∞

�̃�𝑀(𝜃)

√2�̃�𝑀
2 ln ln

1

�̃�𝑀
 

≤ 1 

for a.e. 𝜃 in the unit circle. 

In this LIL variant, the focus lies on the sums beyond the initial n-terms, specifically on the tail sums 

of the series. Consequently, this version is often referred to as  the ``tail LIL" because of its emphasis 

on the tail sum aspect. Salem and Zygmund only derived the upper bound in this tail LIL. Under 

similar conditions, S. Ghimire and C.N. Moore [9] established the converse of the aforementioned 

result. Their result is: 

Theorem 2. Assuming the same notation and hypotheses as stated in the preceding theorem, we have 

limsup
N→∞

�̃�𝑁(𝜃)

√2�̃�𝑁
2 ln ln

1

�̃�𝑁
 

≥ 1 

for a.e. 𝜃 in [0, 2𝜋]. 

Now Theorem 2, when combined with Theorem 1 give the conclusion 

limsup
N→∞

�̃�𝑁(𝜃)

√2�̃�𝑁
2 ln ln

1

�̃�𝑁
 

= 1 

for a.e. 𝜃 in [0, 2𝜋]. A similar LIL for summation of signum functions was recently obtained by S. 

Ghimire [11] showing that the convergence rate of summation of the functions is controlled by the 

tail sums of the square function as in the LIL introduced by Salem and Zygmund. The one-sided 

version of S. Ghimire's LIL is as follows: 

Theorem 3. Suppose {𝑢𝑖} is a sequence of signum functions defined by 𝑢𝑖(𝑡) = 𝑠𝑔𝑛 (sin 2
𝑖𝜋𝑡) and 

{𝑏𝑖}𝑖=1
∞  where 𝑏𝑖 ∈ ℝ satisfies ∑ 𝑏𝑖

2 < ∞.∞
𝑖=1   Then  

limsup
n→∞

| ∑ 𝑏𝑖𝑢𝑖(𝑡)|
∞
𝑖=𝑛+1

√2∑ 𝑏𝑖
2 ∞

𝑖=𝑛+1 ln ln (
1

∑ 𝑏𝑖
2 ∞

𝑖=𝑛+1

) 

≤ 1 

for a.e. 𝑡 ∈ [0, 1). 

Here, we obtain the lower limit version of the LIL for the summation of signum functions. Our main 

result is: 

Theorem 4. Suppose {𝑢𝑖} is a sequence of signum functions defined by 𝑢𝑖(𝑡) = 𝑠𝑔𝑛 (sin 2
𝑖𝜋𝑡) and 

{𝑏𝑖}𝑖=1
∞  where {𝑏𝑖} is a square integrable real-valued  sequence with 𝐵𝑛 = ∑ 𝑏𝑖

2∞
𝑖=𝑛  and assume 

lim
n→∞

𝑏𝑛
2

𝐵𝑛
= 0. Then 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

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limsup
n→∞

| ∑ 𝑏𝑖𝑢𝑖(𝑡)|
∞
𝑖=𝑛+1

√2𝐵𝑛 ln ln
1

𝐵𝑛
 

≥ 1 

for a.e. 𝑡 ∈ [0, 1). 

Note that if the sequence {|𝑏𝑖|} is non increasing, or more generally, if lim
i→∞

|
𝑏𝑖

𝑏𝑖+1
| ≥ 1,  then for large 

𝑛, we have  

𝑏𝑛
2

𝐵𝑛2
≤

𝑏𝑛
2

∑ 𝑏𝑖
2∞

𝑖=𝑚

≤
1

𝑛 + 1
 

and assumption in the theorem is satisfied. The proof consists of stopping time argument with 

application of certain estimates. To prove our main result, we first utilize sub-Gaussian type 

estimates for the summation of signum functions followed by the application of both versions of 

Borelli lemma. In what follows, we use measure space (𝐼 = (0,1), ℬ, 𝜇) and |. | stands for probability 

measure 𝜇 restricted on I. To establish our main result, we begin by introducing some definitions and 

obtaining estimates. 

2. Preliminaries  

 Let's revisit the definition of a general signum function: 

𝑠𝑔𝑛(𝑡) = {
1 𝑖𝑓 𝑡 ≥ 0;
−1 𝑖𝑓 𝑡 < 0.

 

In constructing a sequence, we define 𝑢𝑖(𝑡) = 𝑠𝑔𝑛(sin 2𝑖𝜋𝑡 )  on the interval (0, 1).  We say 𝐴𝑛 

happens infinitely often, abbreviated as 𝐴𝑛 𝑖. 𝑜., if for all 𝑛 there is 𝑚 ≥ 𝑛 such that 𝐴𝑚 is true. We 

now state Borelli lemma of both versions. Please see [3] for the proof. 

Lemma 5 (Borel-Cantelli 1) If {𝐴𝑘} satisfies ∑ |𝐴𝑘| < ∞,
∞
𝑘=1   then |{𝐴𝑘 𝑖. 𝑜. }| = 0. 

Lemma 6 (Borel-Cantelli 2) If independent events {𝐴𝑘} satisfies ∑ |𝐴𝑘| = ∞,
∞
𝑘=1   then |{𝐴𝑘 𝑖. 𝑜. }| =

1. 

Next, we state a result on exponential estimate for independent random variables which will be used 

in the proof of our main result. For the proof, please see [2]. 

Theorem 7. Suppose {𝑌𝑘} is a sequence of random variables on sample spac (𝐼 = (0,1), ℬ, 𝜇), with 

zero mean and variance 𝜎𝑘
2. Let  

𝑆𝑛 =∑ 𝑌𝑘
𝑛

𝑘=1
,    𝑠𝑛

2 =∑ 𝜎𝑘
2

𝑛

𝑘=1
 𝑎𝑛𝑑 𝑍𝑛 =

max
k≤n

|𝑌𝑘|

𝑠𝑛
 

Then, for given 𝛽 > 0, if 𝑍𝑛(𝛽) is very small and 𝛾 = 𝛾(𝛽) is very large, then  

|{𝑡 ∈ 𝐼
|𝑆𝑛(𝑡)|

𝑠𝑛
> 𝛾} | > exp(−

𝛾2

2
(1 + 𝛽) . 

Following, we present a sub-Gaussian type estimate crucial to proving our main result. For a detailed 

proof, refer to [11]. We sketch the proof. 



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Lemma 8. Let {𝑏𝑖} where 𝑏𝑖 ∈ ℝ and {𝑢𝑖} be a sequence of signum functions defined by 𝑢𝑖(𝑡) =

𝑠𝑔𝑛 (sin 2𝑖𝜋𝑡). Then for all 𝛼 > 0 and for a fixed number 𝑛, we have  

|{𝑡 ∈ 𝐼: sup
m≥n

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞

𝑖=𝑚+1
| > 𝛼  }| ≤ 12 exp(

−𝛼2

2 ∑ 𝑏𝑖
2 ∞

𝑖=𝑛+1

). 

Proof: Let 𝑀 ≫ 𝑛 and we write 𝑔𝑖(𝑡) = ∑ 𝑏𝑘𝑢𝑘(𝑡).
𝑖
𝑘=1  Then using Levy's inequality, we get 

(2.1)                       |{𝑡 ∈ 𝐼: max
M≥m≥n

|𝑔𝑚(𝑡) − 𝑔𝑛(𝑡)| > 𝛼  }| ≤ |{𝑡 ∈ 𝐼: |𝑔𝑀(𝑡) − 𝑔𝑛(𝑡)| > 𝛼}| 

Using Lemma 1 in [11], we get 

(2.2)                                 |{𝑡 ∈ 𝐼:max
m≥n

|𝑔𝑚(𝑡) − 𝑔𝑛(𝑡)| > 𝛼  }| ≤ 6 exp (
−𝛼2

2 ∑ 𝑏𝑖
2 ∞

𝑖=𝑛+1

) 

Using (2.1) in (2.2), we get 

|{𝑡 ∈ 𝐼: sup
M≥m≥n

|𝑔𝑀(𝑡) − 𝑔𝑚(𝑡)| > 𝛼  }| ≤ 12 exp (
−𝛼2

2 ∑ 𝑏𝑖
2 ∞

𝑖=𝑛+1

). 

Then continuity property gives 

|{𝑡 ∈ 𝐼: sup
m≥n

|𝑔(𝑡) − 𝑔𝑚(𝑡)| > 𝛼  }| ≤ lim
M→∞

|{𝑡 ∈ 𝐼: sup
M≥m≥n

|𝑔𝑀(𝑡) − 𝑔𝑚(𝑡)| > 𝛼  }|

≤ 12 exp(
−𝛼2

2 ∑ 𝑏𝑖
2 ∞

𝑖=𝑛+1

). 

Thus we have  

|{𝑡 ∈ 𝐼: sup
m≥n

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞

𝑖=𝑚+1
| > 𝛼  }| ≤ 12 exp(

−𝛼2

2 ∑ 𝑏𝑖
2 ∞

𝑖=𝑛+1

). 

 

3. Main Result: Proof of Theorem 4 

Let 𝜃 be very large and 0 < < 𝜖 < 1. We next choose 0 < 𝛼 < 2 in such a way that (1 − 𝜖2)(1 +

𝛼) > 1. Define stopping times by  

𝑛𝑗 = min (𝑛:∑ 𝑏𝑖
2 

∞

𝑖=𝑛+1
<
1

𝜃𝑗
 ) . 

We have 

  

∑ 𝑏𝑖
2 

∞

𝑖=𝑛𝑗

= 𝑏𝑛𝑗
2 +∑ 𝑏𝑖

2 
∞

𝑖=𝑛𝑗+1
 

Then for sufficiently large 𝑛𝑗 , we have  

(3.1)                                     (1 − 𝜖2)∑ 𝑏𝑖
2 

∞

𝑖=𝑛𝑗

<∑ 𝑏𝑖
2 

∞

𝑖=𝑛𝑗+1
<
1

𝜃𝑗
 



Communications on Applied Nonlinear Analysis 

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By definition of 𝑛𝑗 ,  

(3.2)                                              (1 − 𝜖2)
1

𝜃𝑗
 < (1 − 𝜖2)∑ 𝑏𝑖

2 
∞

𝑖=𝑛𝑗

             

So from (3.1) and (3.2), we get, 

(1 − 𝜖2)
1

𝜃𝑗
 < ∑ 𝑏𝑖

2 
∞

𝑖=𝑛𝑗+1
<
1

𝜃𝑗
 

Thus we have  

(3.3)                                                                         
∑ 𝑏𝑖

2 ∞
𝑖=𝑛𝑗+1

∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1+1

≥ (1 − 𝜖2)𝜃 

This gives 

|{𝑡 ∈ 𝐼: sup
n≥nj+1

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞

𝑖=𝑛+1
| > √ 

2(1 + 𝛼)

𝜃
 ∑ 𝑏𝑖

2 
∞

𝑖=𝑛𝑗+1
ln ln (

1

∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

)}| 

= ||

{
 

 

𝑡 ∈ 𝐼: sup
n≥nj+1

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛+1 |

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1+1

> √
2(1 + 𝛼)∑ 𝑏𝑖

2 ∞
𝑖=𝑛𝑗+1

𝜃        ∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1+1

ln ln (
1

∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

)

}
 

 

|| 

≤ ||

{
 

 

𝑡 ∈ 𝐼: sup
n≥nj+1

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛+1 |

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1+1

 √
2(1 + 𝛼)

𝜃
 𝜃(1 − 𝜖2) ln ln (

1

∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

)

}
 

 

|| 

= |{𝑡 ∈ 𝐼: sup
n≥nj+1

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞

𝑖=𝑛+1
| > √  2(1 + 𝛼)(1 − 𝜖2) ∑ 𝑏𝑖

2 
∞

𝑖=𝑛𝑗+1+1
ln ln (

1

∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

)}| 

= |{𝑡 ∈ 𝐼: sup
n≥nj+1

|∑ 𝑏𝑖𝑢𝑖(𝑡) −∑ 𝑏𝑖𝑢𝑖(𝑡)
𝑛

𝑖=1

∞

𝑖=1
|

> √  2(1 + 𝛼)(1 − 𝜖2) ∑ 𝑏𝑖
2 

∞

𝑖=𝑛𝑗+1+1
ln ln (

1

∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

)}| 

Using Lemma 8, we have  



Communications on Applied Nonlinear Analysis 

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|{𝑡 ∈ 𝐼: sup
n≥nj+1

|∑ 𝑏𝑖𝑢𝑖(𝑡) −∑ 𝑏𝑖𝑢𝑖(𝑡)
𝑛

𝑖=1

∞

𝑖=1
|

> √  2(1 + 𝛼)(1 − 𝜖2) ∑ 𝑏𝑖
2 

∞

𝑖=𝑛𝑗+1+1
ln ln (

1

∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

)}| 

≤ 24 exp

(

 
 
2(1 + 𝛼)(1 − 𝜖2) ∑ 𝑏𝑖

2 ∞
𝑖=𝑛𝑗+1+1

ln ln (
1

∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

)

2 ∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1+1

)

 
 

 

= 24 (ln(
1

∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

))

−(1+𝛼)(1−𝜖2)

 

< 24  (
1

ln 𝜃𝑗
)
(1+𝛼)(1−𝜖2)

 

Thus, 

|{𝑡 ∈ 𝐼: sup
n≥nj+1

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞

𝑖=𝑛+1
| > √ 

2(1 + 𝛼)

𝜃
 ∑ 𝑏𝑖

2 
∞

𝑖=𝑛𝑗+1
ln ln (

1

∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

)}|

< 24  (
1

ln 𝜃𝑗
)
(1+𝛼)(1−𝜖2)

 

Define 𝐴 = {𝑡 ∈ 𝐼: sup
n≥nj+1

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛+1 | > √ 

2(1+𝛼)

𝜃
 ∑ 𝑏𝑖

2 ∞
𝑖=𝑛𝑗+1

ln ln (
1

∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

)} 

Hence 

|𝐴| <
24

(ln 𝜃)(1+𝛼)(1−𝜖
2)

1

𝑗(1+𝛼)(1−𝜖
2)
. 

Set 

𝑆𝑛 =∑ 𝑏𝑖𝑢𝑖 ,   
∞

𝑖=𝑛+1
   𝑠𝑛

2 =∑ 𝑏𝑖
2 

𝑛

𝑖=𝑚
  𝑎𝑛𝑑   𝑍𝑛 =

max
k≤n

|𝑏𝑘𝑢𝑘|

𝑠𝑛
 

Fix 𝛽 > 0 and choose 𝑍𝑛(𝛽) and 𝛾(𝛽) accordingly. Suppose 𝑛𝑗  is sufficiently large. Then for this 𝑛𝑗 , 

Theorem 7 gives  

||

{
 

 

𝑡 ∈ 𝐼 ∶
|∑ 𝑏𝑖𝑢𝑖(𝑡)

𝑛
𝑖=𝑛+1 |

√∑ 𝑏𝑖
2 𝑛

𝑖=𝑛𝑗+1

> 𝛾

}
 

 

|| > exp (
−𝛾2(1 + 𝛽)

2
) . 



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Choose 𝛾 = √
(2−𝛼)

(1+𝛽)
ln ln(

1

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

 )  where 𝛼 > 0. Clearly for large 𝑛𝑗 , 𝛾 is large as needed in 

Theorem 7. Thus,  

|
|

{
 
 

 
 

𝑡 ∈ 𝐼 ∶
|∑ 𝑏𝑖𝑢𝑖(𝑡)

𝑛
𝑖=𝑛𝑗+1

|

√∑ 𝑏𝑖
2 𝑛

𝑖=𝑛𝑗+1

> √
(2 − 𝛼)

(1 + 𝛽)
ln ln

(

 
1

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

 

)

  

}
 
 

 
 

|
|

> exp

(

 
−(2 − 𝛼)

(1 + 𝛽)
ln ln

(

 
1

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

 

)

  
(1 + 𝛽)

2

)

  

                          ≥
1

(𝑗 ln 𝜃 + ln(1 − 𝜖2))
2−𝛼

2

 

Therefore for large 𝑛𝑗 , we have  

|

|

|

{
 
 
 

 
 
 

𝑡 ∈ 𝐼 ∶
|∑ 𝑏𝑖𝑢𝑖(𝑡)

𝑛
𝑖=𝑛𝑗+1

|

√∑ 𝑏𝑖
2  
(2−𝛼)

(1+𝛽)
ln ln (

1

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

 ) 𝑛
𝑖=𝑛𝑗+1

> 1

}
 
 
 

 
 
 

|

|

|

>
1

2

1

(𝑗 ln 𝜃)
2−𝛼

2

 

This gives  

|

|

|

{
 
 
 

 
 
 

𝑡 ∈ 𝐼 ∶
|∑ 𝑏𝑖𝑢𝑖(𝑡)

∞
𝑖=𝑛𝑗+1

− ∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛+1 |

√2∑ 𝑏𝑖
2  ln ln(

1

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

 ) 𝑛
𝑖=𝑛𝑗+1

> √
(2 − 𝛼)

2(1 + 𝛽)

}
 
 
 

 
 
 

|

|

|

>
1

2

1

(𝑗 ln 𝜃)
2−𝛼

2

 

Using (3.1) for 𝑛 ≥ 𝑛𝑗+1, we have  

∑ 𝑏𝑖
2 

𝑛

𝑖=𝑛𝑗+1
=∑ 𝑏𝑖

2 
∞

𝑖=𝑛𝑗+1
−∑ 𝑏𝑖

2 
∞

𝑖=𝑛+1
≥ (1 − 𝜖2)

1

𝜃𝑗
−

1

𝜃𝑗+1
≥∑ 𝑏𝑖

2 
∞

𝑖=𝑛𝑗+1
(1 − 𝜖2 −

1

𝜃
)  

Then this gives  



Communications on Applied Nonlinear Analysis 

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|

|

|

{
 
 
 

 
 
 

𝑡 ∈ 𝐼 ∶
|∑ 𝑏𝑖𝑢𝑖(𝑡)

∞
𝑖=𝑛𝑗+1

− ∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛+1 |

√2∑ 𝑏𝑖
2 (1 − 𝜖2 −

1

𝜃
) ln ln(

1

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

 ) ∞
𝑖=𝑛𝑗+1

> √
(2 − 𝛼)

2(1 + 𝛽)

}
 
 
 

 
 
 

|

|

|

>
1

2

1

(𝑗 ln 𝜃)
2−𝛼

2

 

i.e. 

|

|

|

{
 
 
 

 
 
 

𝑡 ∈ 𝐼 ∶
|∑ 𝑏𝑖𝑢𝑖(𝑡)

∞
𝑖=𝑛𝑗+1

− ∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛+1 |

√2∑ 𝑏𝑖
2  ln ln(

1

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

 ) ∞
𝑖=𝑛𝑗+1

> √
(2 − 𝛼) (1 − 𝜖2 −

1

𝜃
)

2(1 + 𝛽)
 

}
 
 
 

 
 
 

|

|

|

>
1

2

1

(𝑗 ln 𝜃)
2−𝛼

2

 

Define 

𝐵 ≔

{
 
 
 

 
 
 

𝑡 ∈ 𝐼 ∶
|∑ 𝑏𝑖𝑢𝑖(𝑡)

∞
𝑖=𝑛𝑗+1

− ∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛+1 |

√2∑ 𝑏𝑖
2  ln ln(

1

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

 ) ∞
𝑖=𝑛𝑗+1

> √
(2 − 𝛼) (1 − 𝜖2 −

1

𝜃
)

2(1 + 𝛽)
 

}
 
 
 

 
 
 

 

 

Consequently,   |𝐵| ≥
1

2

1

(𝑗 ln𝜃)
2−𝛼
2

. Next define  

𝐶 ≔

{
 
 
 

 
 
 

𝑡 ∈ 𝐼 ∶
|∑ 𝑏𝑖𝑢𝑖(𝑡)

∞
𝑖=𝑛𝑗+1+1

− ∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛𝑗+1

|

√2∑ 𝑏𝑖
2 ln ln(

1

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

 ) ∞
𝑖=𝑛𝑗+1

> √
(2 − 𝛼) (1 − 𝜖2 −

1

𝜃
)

2(1 + 𝛽)
− 2√

(1 − 𝜖2)

𝜃
(1 + 𝛼)

}
 
 
 

 
 
 

 

Using triangle inequality, we have 𝐵⋂𝐴𝑐 ⊂ 𝐶. So we have |𝐵 − 𝐴| ≤ |𝐶|. Thus, we have  

|𝐶| ≥
1

2(𝑗 ln 𝜃)
2−𝛼

2

−
24

(𝑗 ln 𝜃)(1−𝜖
2)(1+𝛼)

. 

Since 𝛼 ∈ (0, 2) and (1 − 𝜖2)(1 + 𝛼) > 1,   for large 𝑗, 



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1

3(𝑗 ln 𝜃)
(2−𝛼)

2

≥
24

(𝑗 ln 𝜃)(1−𝜖
2)(1+𝛼)

 

This gives 

|𝐶| ≥
1

6(𝑗 ln 𝜃)
(2−𝛼)

2

. 

Now summing over all 𝑗 we have, 

∑

|

|

|

{
 
 
 

 
 
 

𝑡 ∈ 𝐼 ∶
|∑ 𝑏𝑖𝑢𝑖(𝑡)

∞
𝑖=𝑛𝑗+1+1

− ∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛𝑗+1

|

√2∑ 𝑏𝑖
2 ln ln(

1

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

 ) ∞
𝑖=𝑛𝑗+1

∞

𝑗=1

> √
(2 − 𝛼) (1 − 𝜖2 −

1

𝜃
)

2(1 + 𝛽)
− 2√

(1 − 𝜖2)

𝜃
(1 + 𝛼)

}
 
 
 

 
 
 

|

|

|

 

≥∑
1

6(𝑗 ln 𝜃)
(2−𝛼)

2

∞

𝑗=1
=

1

6(ln 𝜃)
(2−𝛼)

2

∑
1

𝑗
(2−𝛼)

2

= ∞.
∞

𝑗=1
 

Here we note that {∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛𝑗+1

− ∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛𝑗+1+1

}
𝑖=1

∞

is a sequence of independent random 

variables. Apply Lemma 5 for a.e. 𝑡, there exists an infinite sequence 𝑛1 < 𝑛2 < 𝑛3 < ⋯ such that, 

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛𝑗+1+1

− ∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛𝑗+1

|

√2∑ 𝑏𝑖
2 ln ln (

1

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

 ) ∞
𝑖=𝑛𝑗+1

> √
(2 − 𝛼) (1 − 𝜖2 −

1

𝜃
)

2(1 + 𝛽)
− 2√

(1 − 𝜖2)

𝜃
(1 + 𝛼) 

By triangle inequality, we have  

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛𝑗+1+1

|

√2∑ 𝑏𝑖
2 ln ln (

1

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

 ) ∞
𝑖=𝑛𝑗+1

+
|∑ 𝑏𝑖𝑢𝑖(𝑡)

∞
𝑖=𝑛𝑗+1

|

√2∑ 𝑏𝑖
2 ln ln(

1

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

 ) ∞
𝑖=𝑛𝑗+1

 



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(3.4)                                   > √
(2 − 𝛼) (1 − 𝜖2 −

1

𝜃
)

2(1 + 𝛽)
− 2√

(1 − 𝜖2)

𝜃
(1 + 𝛼). 

We have  

|𝐴| = |{𝑡 ∈ 𝐼: sup
n≥nj+1

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞

𝑖=𝑛+1
| > √ 

2(1 + 𝛼)

𝜃
 ∑ 𝑏𝑖

2 (1 − 𝜖2) 
∞

𝑖=𝑛𝑗+1
ln ln (

1

∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

)}|

<
24

(ln 𝜃)(1+𝛼)(1−𝜖
2)

1

𝑗(1+𝛼)(1−𝜖
2)
.   

So 

 ∑ |{𝑡 ∈ 𝐼: sup
n≥nj+1

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛+1 | > √ 

2(1+𝛼)

𝜃
 ∑ 𝑏𝑖

2(1 − 𝜖2) ∞
𝑖=𝑛𝑗+1

ln ln (
1

∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

)}|∞
𝑗=1  

< ∑
24

(ln 𝜃)(1+𝛼)(1−𝜖
2)

1

𝑗(1+𝛼)(1−𝜖
2)

∞

𝑗=1
=

24

(ln 𝜃)(1+𝛼)(1−𝜖
2)
∑

1

𝑗(1+𝛼)(1−𝜖
2)

∞

𝑗=1
< ∞. 

Applying Lemma 6, for a.e. 𝑡, we get 

sup
n≥nj+1

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞

𝑖=𝑛+1
| ≤ √ 

2(1 + 𝛼)

𝜃
 ∑ 𝑏𝑖

2(1 − 𝜖2) 
∞

𝑖=𝑛𝑗+1
ln ln (

1

∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

) 

i.e.  

   (3.5)                            sup
n≥nj+1

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛+1 |

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1
2 ln ln (

1

∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

) 

≤ √ 
(1 − 𝜖2)(1 + 𝛼)

𝜃
  

for sufficiently large 𝑗 ≥ 𝑁 (say). Thus from (3.4) and (3.5), for a.e. 𝑡 we get 𝑛1 < 𝑛2 < 𝑛3 < ⋯ 

such that, 

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛𝑗+1

|

√2∑ 𝑏𝑖
2 ln ln(

1

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛𝑗+1

 ) ∞
𝑖=𝑛𝑗+1

 

> √
(2 − 𝛼) (1 − 𝜖2 −

1

𝜃
)

2(1 + 𝛽)
− 3√

(1 − 𝜖2)

𝜃
(1 + 𝛼)                                . 

Consequently, for a.e. 𝑡 



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|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛+1 |

√2∑ 𝑏𝑖
2 ln ln (

1

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛+1

 ) ∞
𝑖=𝑛+1

 > √
(2 − 𝛼) (1 − 𝜖2 −

1

𝜃
)

2(1 + 𝛽)
− 3√

(1 − 𝜖2)

𝜃
(1 + 𝛼)   . 

Letting 𝜃 ↗ ∞, 𝜖, 𝛼, 𝛽 ↘ 0, we get 

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛+1 |

√2∑ 𝑏𝑖
2 ln ln (

1

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛+1

 ) ∞
𝑖=𝑛+1

≥ 1. 

Thus, 

limsup
n→∞

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛+1 |

√2∑ 𝑏𝑖
2 ln ln (

1

√∑ 𝑏𝑖
2 ∞

𝑖=𝑛+1

 ) ∞
𝑖=𝑛+1

≥ 1. 

This gives 

limsup
n→∞

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛+1 |

√2 𝐵𝑛 ln ln
1

𝐵𝑛

≥ 1 

for a.e. 𝑡 ∈ (0, 1). This completes the proof of the main theorem. 

Conclusion  

When we combine the result of Theorem 3 with result obtained in Theorem 4, we have  

limsup
n→∞

|∑ 𝑏𝑖𝑢𝑖(𝑡)
∞
𝑖=𝑛+1 |

√2 𝐵𝑛 ln ln
1

𝐵𝑛

= 1 

for a. e. 𝑡 ∈ (0, 1). This completes the law of the iterated logarithm for the summation of the signum 

functions. 

References 

[1] A. Khintchine, Uber einen Sat: der Wahrscheinlichkeitsrechnung, Fundamenta Mathematica, 6(1924), 9-20. 

[2] A. Zygmund, Trigonometrical Series, Cambridge University Press, Cambridge, 1959.  

[3] H.L. Royden and P.M. Fitzpatrick, Real Analysis, Fourth Edition, Prentice Hall of India, 2010. 

[4]  M. Weiss, The Law of the Iterated Logarithm for Lacunary Trigonometric series, Transaction of American 

Mathematical Society, 91 (1959), 444-469. 

[5] N. Kolmogorov, Uber des Geset: des iterierten Logarithmus , Mathematische Annalen 101 (1929), 136-139. 

[6] P. Erdos and I.S. Gal, On the law of the iterated logarithm, Proceedings of the Koninklijke Nederlandse Akademie 

van Wetenschappen A, 58 (1955), 65-84. 

[7] R. Banelos and C. N. Moore, Probabilistic Behaviour of Harmonic Functions, Birkhasaur Verlag, 1991. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

366 
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[8] R. Salem and A. Zygmund, La loi du logarithme itre pour les series trigonometriques lacunaires, Bulletin des 

Sciences Mathématiques, 74 (1950), 209-224. 

[9] S. Ghimire and C.N. Moore, A lower bound in the tail law of the iterated logarithm for lacunary trigonometric 

series, Proceeding of American Mathematical Society, 142(2014), 3207-3216. 

[10] S. Ghimire, One-sided law of the iterated logarithm for dyadic martingale using sub-Gaussian estimates, Open 

Journal of Mathematical Analysis, 6(1) (2022), 1-6. 

[11] S. Ghimire, One-sided version of the law of the iterated logarithm for summations of signum functions, Journal of 

Mathematics, 2023 (2023), 1-7. 

[12] W. F. Stout, A martingale analogue of Kolmogorov's law of the iterated logarithm, Zeitschriftfr 

Wahrscheinlichkeitstheoire Und Verwandte Gebiete, 15(4)(1970), 279-290. 

 

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