One-Sided Version of Law of the Iterated Logarithm for Summations of Signum Functions

. Te law of the iterated logarithm (LIL), which describes the rate of convergence for a convergent lacunary series, was established by R. Salem and A. Zygmund. Tis rate is determined based on the variance-like term of the remainder after n terms of the series. In this article, we investigate a comparable one-sided LIL for sums of signum functions, which also relies on the remainder after n terms.


Introduction
LIL is a fundamental theorem in probability theory that characterizes the properties displayed by the sums of independent random variables and can be viewed as a refnement of two famous theorems, namely, central limit theorem (CLT) and law of large numbers (LLN).Te LIL provides a more accurate estimation as compared to the CLT and LLN in the cases where the fuctuations of the sample average are anticipated to be small, yet the occurrence of signifcant deviations remains possible.Te LIL was frst introduced by Russian mathematician Khintchine [1] in 1924 while describing the size of deviation from the expected mean for Bernoulli's random variables.In 1929, Kolmogorov [2] extended this result to apply to the sums of independent random variables.Since then, the LIL has gained signifcance as a fundamental theorem and has found applications in diverse felds of mathematics and statistics.After the introduction of Kolmogorov's LIL, various extensions and generalizations of the LIL have been introduced in the diferent felds such as harmonic functions [3], martingales [4,5], q-lacunary series, random walks, stochastic processes, etc. Salem and Zygmund [6] introduced an LIL for the sums of q-lacunary trigonometric series.We recall that a sequence n i   is called Hadamard gap condition if it satisfes q � n i+1 /n i > 1.A trigonometric series of the form S(x) �  ∞ i�1 (a i cos n i x + b i sin n i x) in which n i satisfes the Hadamard gap condition is known as q-lacunary trigonometric series.Te LIL formulated by Salem and Zygmund is stated below.Theorem 1. Suppose that S(θ) is q-lacunary series and n k are positive integers.Set for almost every θ in a unit circle T.
We note that the abovementioned theorem is not entirely analogous to the result introduced by Kolmogorov.In this direction, Erdös and Gál [7] derived a similar result for a particular form of q-lacunary series.Eventually, Weiss [8] succeeded in deriving a similar LIL for a general q-lacunary series analogous to the LIL introduced by Kolmogorov.
Theorem 2 (M.Weiss).Let S m (θ) �  m i�1 (a i cos n i θ + b i sin n i θ) be a q-lacunary series and n i be the integers.Set for almost every θ in the unit circle.
Salem and Zygmund [6] also introduced another LIL for q-lacunary series, stated as follows: Theorem 3 (Salem and Zygmund).
) for K M approaches to 0 as M approaches to infnity.Ten, for almost every θ in the unit circle.
Hence, it is evident that the LIL introduced by Salem and Zygmund is applicable to both divergent and convergent qlacunary series.For the divergent series, they analyzed the partial sums and established the extent of their deviation, which is infuenced by the term resembling variance, denoted as B 2 m .Meanwhile, in the convergent series, they analyzed the tail sums and derived another LIL that estimates the convergence rate of q-lacunary series, which also depends on the tail sums of variance-like term B 2 m .Tis LIL is commonly referred to as the "tail" LIL because of the tail sum component.Te abovementioned result represents the one-sided version of the LIL.Te other direction of the above theorem was estimated by Ghimire and Moore [9], who obtained the following outcome under the similar assumptions to those in the previous theorem.Theorem 4. Assuming the same notation and hypotheses as stated in the preceding theorem, we have for a.e.θ ∈ [0, 2π].
In this article, we derive a similar LIL for the sums of signum functions.We recall that a general signum function sgn(t) is defned as follows: In order to form a sequence, we consider signum function as u i (t) � sgn(sin 2 i πt) on [0, 1).Note that for i � 1, u 1 (t) � sgn(sin 2πt) will give value 1 on [0, 1/2] and − 1 on (1/2, 1) and similarly the rest of the signum functions in the sequence will fuctuate between − 1 and 1.Consider a sequence of real number b i   ∞ i�1 and defne g n (t) �  n i�1 b i u i (t).Burkholder and Gundy in [10] proved that t : where the sets are almost everywhere equal.Note that Here, we derive a LIL for which is similar to the LIL introduced by Salem and Zygmund.Our result estimates the rate at which g n converges to g and the rate at which it converges is governed by the tail sums of the square function  ∞ i�1 b 2 i .We only obtain the one-sided version of LIL and our main result is as follows: for almost every t ∈ [0, 1).

Preliminaries
During the course of proving our main result, we will require certain estimates that will be utilized in the proof of the fnal outcome.We now prove these estimates.
Ten, we have the following estimate: where c ∈ R.
As n increases, we show that h(n) decreases.For this, 2 Journal of Mathematics where we used the fact of function g n is constant on n th generation interval Applying cosh t ≤ e t 2 /2 , we obtain Tus we have h(n + 1) ≤ h(n).We next show h(1) ≤ 1.For this, Hence, h(1) ≤ 1. Tis with h(n + 1) ≤ h(n) gives h(n) ≤ 1. Tus, we have Ten, c scaling gives ∞ i�1 be a sequence of signum function defned by u i (t) � sgn(sin 2 i πt) where Ten, for all η > 0 and for a fxed n, we have where I � [0, 1).
Proof.Defne g n (t) �  n i�1 b i u i (t).Let n be fxed and Q l be the interval in [0, 1) of length 1/2 l .Ten where M(e Using Lemma 6, we obtain i.e.
We note that For a fxed n, defne For this function, the above estimate becomes Here, Ten for all α > 0 and a fxed number n, we have where I � [0, 1).

Journal of Mathematics
Proof.Let Here, each C i is independent and symmetric with mean 0 and variance 1. Invoking Levy's inequality, we have Let M ≫ n.Ten, we obtain Tus, We have (41) □

Proof of Our Main Result
Consider θ > 1 and defne stopping times Using Lemma 8 for a fxed m, we obtain For stopping time n i , this becomes So, by Borel-Cantelli Lemma [11] for a.e.
for a large i, with i ≥ N, N being fxed.Clearly, the value of N is based on t.We fx t and then consider n ≥ n N .We can fndi ≥ N satisfying n i ≤ n < n i+1 .Clearly  ∞ i�n i+1 +1 b 2 i < 1/θ i+1 .But n i+1 > n.So  ∞ for a.e.t ∈ I.