Exponential Tail Estimates for Lacunary Trigonometric Series

Abstract

We establish precise exponential tail estimates for lacunary trigonometric sums of the form $f_N\left(x\right)={\sum }_{k=1}^N{c}_k\cos \left(2\pi n_{k}x\right)$, under the Hadamard gap condition. Using cumulant expansions and moment-generating function techniques, we obtain non-asymptotic upper bounds for the tail probabilities, including third-order corrections that refine the classical central limit theorem estimates. Furthermore, several examples illustrate these bounds for various choices of coefficients, highlighting the transition from subgaussian to stretched-exponential tail behavior.

1. Introduction

Lacunary trigonometric series, characterized by gaps between frequencies growing at least geometrically, play a central role in harmonic analysis and the probability theory. They are of the form

$$f_N\left(x\right)=\sum _{k=1}^N{c}_k\sqrt{2}\cos \left(2\pi n_{k}x\right),$$

where $\left\{n_k\right\}$ is a strictly increasing sequence of integers satisfying the Hadamard gap condition:

$$n_{k+1}\ge q{n}_k,q>1,\forall k\ge 1,$$

and $\left\{c_k\right\}$ is a sequence of real numbers.

Classical results by Aistleitner–Berkes [1], Kac [2], Philipp [3] and Salem–Zygmund [4] focus on asymptotic properties such as the central limit theorem (CLT), large deviation principle (LDP) and law of iterated logarithm (LIL) for lacunary series under the Hadamard gap condition. These results describe the limiting behavior of normalized sums of lacunary trigonometric functions but provide limited insight into the precise behavior for finite truncations.

In many applications, such as random Fourier series, signal processing and the study of stochastic processes, it is crucial to obtain explicit, non-asymptotic bounds for tail probabilities of partial sums. That is, for a finite number of N terms, one needs explicit estimates of the tail probabilities

$$T\left[{\nu }_N\right]\left(t\right)=\mathbb{P}\{{\nu }_N\left(x\right)\ge t\},t>0,$$

where ${\nu }_N$ is the normalized sum of the form

$${\nu }_N\left(x\right)={\displaystyle \frac{f_N\left(x\right)}{\sigma \left(N\right)}},{\sigma }^2\left(N\right)=\sum _{k=1}^N{c}_k^2.$$

These estimates allow rigorous control of large deviations in simulations or probabilistic models and also have potential applications in mathematical statistics and computational mathematics.

This motivates our study of exponential tail estimates for normalized lacunary sums and leads us to obtain explicit exponential upper bounds for the tail function $T\left[{\nu }_N\right]\left(t\right)$, expressed in terms of the lacunarity parameter q, using Chernoff-type inequalities combined with convex analysis via the Young–Fenchel transform. Our results provide a practical tool for quantifying the probability of large deviations in finite sums, complementing traditional limit theorems and offering insight for applications where finite-N behavior is critical.

Before outlining the structure of the paper, we clarify the meaning of the word “precise” in the title. Here, “precise” is intended in the quantitative, non-asymptotic sense: we obtain explicit exponential tail bounds for fixed N, with fully specified dependence on the lacunarity parameter q and on the variance $\sigma \left(N\right)$. Unlike classical asymptotic results (CLT, LIL, LDP), our estimates provide concrete constants and explicitly show how the transition between the subgaussian regime and the stretched-exponential regime emerges from the cumulant expansion and from the optimization of the Chernoff bound. The term does not claim optimality of constants in a minimax sense; rather, it reflects that the resulting tail inequalities are fully explicit, directly computable, and applicable to concrete lacunary trigonometric sums.

Beyond their intrinsic theoretical interest, the non-asymptotic tail bounds obtained in this work may also be useful in applications where explicit control of large values of lacunary trigonometric sums is required. Since our inequalities hold for fixed N, they provide quantitative estimates that can be directly used for random Fourier series, for the analysis of fast-oscillating signals in numerical simulations and in parts of the discrepancy theory, where the sizes of lacunary sums play key roles. In all these settings, having explicit exponential estimates allows one to predict the probability of rare but significant deviations in a precise and computable manner.

The paper is structured as follows. In Section 2 we recall necessary background on lacunary trigonometric series and introduce the normalization conventions. In Section 3 we recall the definitions of cumulants and provide a local control of the moment-generating function of the normalized lacunary sums. In Section 4 we derive explicit upper bounds for the tail function. Section 5 provides simplified upper bounds for the tail probability in two natural regimes: for small deviations relative to the standard deviation $\sigma \left(N\right)$, we obtain subgaussian-type estimates, while for larger deviations (within the range where our MGF control applies), we establish a stretched-exponential decay of order $\exp (-c{t}^{3/2})$, characteristic of lacunary trigonometric sums. Section 6 illustrates the main results through concrete examples. In Section 7, we also derive a local lower estimate for the cumulant-generating function, which offers partial control on the lower tail of ${\nu }_N$ within the range where cumulant bounds are available. Finally, Section 8 summarizes our main findings and outlines potential directions for future research.

2. Preliminaries

Let $(X,\mathcal{B},\mathbb{P})=\left(\right[0,1],\mathcal{B},dx)$ be the probability space endowed with the normalized Lebesgue measure, where $\mathcal{B}$ is the Borel $\sigma$-algebra on $[0,1]$. Equivalently, for any $A\in \mathcal{B}$,

$$\mathbb{P}\left(A\right)={\int }_{A}dx.$$

All random variables considered in this paper are measurable functions of $x\in [0,1]$ with respect to this probability space.

For a measurable numerical valued function (random variable) $g:X\to \mathbb{R}$, the Lebesgue integral is denoted by

$$\mathbb{E}g={\int }_{X}g\left(x\right)dx.$$

Recall that the tail function $T\left[g\right]\left(t\right),t>0$, for the random variable (function) $g:X\to \mathbb{R}$, is defined as

$$T\left[g\right]\left(t\right)\stackrel{def}{=}\mathbb{P}\left(\{x\in [0,1]:g(x)\ge t\}\right).$$

The Lebesgue–Riesz $L^p$-norm ($p\ge 1)$ of a measurable function g is defined by

$$\begin{array}{cc}\hfill {\parallel g\parallel }_p& ={\left({\mathbb{E}\left|g\right|}^p\right)}^{1/p}={\left({\int }_X{\left|g\left(x\right)\right|}^{p}dx\right)}^{1/p},p\ge 1,\hfill \\ \hfill {\parallel g\parallel }_{\infty }& :=\underset{x\in X}{\mathrm{ess} \sup }\left|g\left(x\right)\right|,p=\infty .\hfill \end{array}$$

Correspondingly, the variance is

$$\mathrm{Var}\left(\mathrm{g}\right)={\parallel \mathrm{g}\parallel }_2^2-{\left({\int }_{\mathrm{X}}\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx}\right)}^2=\mathbb{E}{\mathrm{g}}^2-{\left(\mathbb{E}\mathrm{g}\right)}^2.$$

Recall also the Stein identity

$${\mathbb{E}\left|g\right|}^p=p{\int }_0^{\infty }{t}^{p-1}T\left[\right|g\left|\right]\left(t\right)dt,p\ge 1,$$

(1)

which connects moments and tail probabilities.

Definition 1 (Lacunary Series). 

Let ${\left\{n_k\right\}}_{k\ge 1}$ be a strictly increasing sequence of natural numbers satisfying the Hadamard gap condition:

$$\exists q>1\mathit{such}\mathit{that}{\displaystyle \frac{n_{k+1}}{n_k}}\ge q,k\ge 1.$$

Define the orthonormal functions

$${\varphi }_k\left(x\right):=\sqrt{2}\cos \left(2\pi n_{k}x\right),k\ge 1,$$

(2)

so that, for $k,l=1,2,3,\dots$,

$${\int }_X{\varphi }_k\left(x\right){\varphi }_l\left(x\right)dx={\delta }_{k,l}=\left\{\begin{array}{cc}0,\hfill & k\ne l,\hfill \\ 1,\hfill & k=l.\hfill \end{array}\right.$$

Let ${\left\{c_k\right\}}_{k\ge 1}$ be a sequence of real numbers not square-summable, i.e.,

$$\sum _{k=1}^{\infty }{c}_k^2=\infty ,$$

(3)

and consider the partial sums

$$f_N\left(x\right):=\sum _{k=1}^N{c}_k{\varphi }_k\left(x\right),N\ge 1.$$

(4)

Denote the variance by ${\sigma }^2\left(N\right):=\mathrm{Var}\left(f_N\right)=\mathbb{E}\left[f_N^2\right]-{\left(\mathbb{E}\left[f_N\right]\right)}^2$.

The second term vanishes since

$$\mathbb{E}\left[f_N\right]={\int }_X{f}_N\left(x\right)dx=\sum _{k=1}^N{c}_k{\int }_X{\varphi }_k\left(x\right)dx=0,$$

and expanding the square and using the orthonormality of $\left\{{\varphi }_k\right\}$, we obtain

$${\sigma }^2\left(N\right)=\mathrm{Var}\left(f_N\right)=\sum _{k=1}^N{c}_k^2.$$

(5)

Define the sequence of the normalized functions

$${\nu }_N\left(x\right):={\displaystyle \frac{f_N\left(x\right)}{\sigma \left(N\right)}},$$

(6)

where $\mathbb{E}\left[{\nu }_N\right]=0$ and $\mathrm{Var}\left({\nu }_N\right)=1$.

Remark 1. 

Since ${\displaystyle \sum _{k=1}^{\infty }{c}_k^2=\infty }$, it follows that

$${\sigma }^2\left(N\right)={\parallel f_N\parallel }_{L^2}^2\to \infty ,$$

and therefore, the normalized sums ${\nu }_N\left(x\right)$ exhibit non-trivial behavior.

Since $\parallel f_N{\parallel }_{L^2}\le {\parallel f_N\parallel }_{L^{\infty }}$ in $X=[0,1]$ and $\parallel f_N{\parallel }_{L^2}\to \infty$, the sequence $f_N$ does not converge in $L^{\infty }\left(X\right)$, as convergence in $L^{\infty }$ would imply uniform boundedness and hence boundedness in $L^2$.

Our main goal is to derive non-asymptotic exponential estimates for the tail function

$$T\left[{\nu }_N\right]\left(t\right)=\mathbb{P}({\nu }_N>t),t>0,$$

where N is fixed and ${\nu }_N$ is defined in (6).

In previous works devoted to this problem, e.g., in [1,2,3,4,5,6,7,8,9,10,11,12,13,14], the emphasis is mainly on the asymptotic behavior of $T\left[{\nu }_N\right]\left(t\right)$ as $N\to \infty$. Specifically, results such as the CLT, LDP and LIL were obtained for such functions.

One of the applications of the theory of lacunary trigonometric random series in the theory of (Gaussian) random processes (and fields) is described in (Chapter 3 [15]).

3. Local Cumulant Bounds for Lacunary Sums

In this Section, we recall the notions of the moment-generating function and cumulants and provide a local control of the moment-generating function of the normalized lacunary sums.

Definition 2 (Moment-generating function and cumulants). 

Let Y be a real-valued random variable with finite exponential moments in a neighborhood of the origin. The moment-generating function (MGF) of Y is

$$M_Y\left(\lambda \right)=\mathbb{E}\left[e^{\lambda Y}\right],\lambda \in \mathbb{R},$$

and its logarithm

$$K_Y\left(\lambda \right)=\log M_Y\left(\lambda \right)$$

is the cumulant-generating function (CGF). It is convex on its effective domain; this follows from the log–sum–exp convexity principle (see [16]).

Derivatives of $M_Y$ at 0 yield the moments of Y, $M_Y^{\left(n\right)}\left(0\right)=\mathbb{E}\left[Y^n\right]$, while the n-th derivative of $K_Y$ at 0 gives the n-th cumulant of Y,

$${\kappa }_n\left(Y\right):=K_Y^{\left(n\right)}\left(0\right).$$

In particular, ${\kappa }_1\left(Y\right)=\mathbb{E}\left[Y\right]$ is the mean and ${\kappa }_2\left(Y\right)=\mathrm{Var}\left(Y\right)$ is the variance.

Cumulants and the CGF play central roles in Chernoff-type inequalities and in the Legendre–Fenchel transform used to estimate tail probabilities; in the context of lacunary trigonometric series, they capture the deviation from Gaussian behavior.

To derive upper bounds for the tail probabilities of lacunary trigonometric polynomials, it is essential to control their moment-generating functions locally. The classical results on lacunary series, such as those of Salem and Zygmund [4], provide asymptotic Gaussian behavior, but for a non-asymptotic analysis, we require explicit estimates of the MGF in a neighborhood of the origin.

We begin by establishing a local cumulant-based bound for the moment-generating function of the normalized lacunary sums. Although the general technique is classical in the analysis of lacunary trigonometric series, the following formulation does not appear explicitly in the literature, so we include a complete proof for clarity.

Lemma 1 (Local MGF bound for lacunary trigonometric sums). 

Let

$${\nu }_N\left(x\right)={\displaystyle \frac{f_N\left(x\right)}{\sigma \left(N\right)}},f_N\left(x\right)=\sum _{k=1}^N{c}_k{\varphi }_k\left(x\right),{\sigma }^2\left(N\right)=\sum _{k=1}^N{c}_k^2,$$

where the trigonometric system $\left\{{\varphi }_k\right\}$ is given by

$${\varphi }_k\left(x\right)=\sqrt{2}\cos \left(2\pi n_{k}x\right),$$

and the frequencies $\left\{n_k\right\}$ satisfy the Hadamard gap condition

$${\displaystyle \frac{n_{k+1}}{n_k}}\ge q>1\mathit{for}\mathit{all}k\ge 1.$$

Define

$${\kappa }_N:={\displaystyle \frac{1}{{\sigma }^3\left(N\right)}}\sum _{k=1}^N{\left|c_k\right|}^3.$$

Then, there exist constants ${\lambda }_0={\lambda }_0\left(q\right)>0$ and $C=C\left(q\right)>0$, depending only on the lacunarity parameter q, such that for all $\left|\lambda \right|\le {\lambda }_0$,

$$K_{{\nu }_N}\left(\lambda \right):=\log \mathbb{E}\left[e^{\lambda {\nu }_N}\right]\le {\displaystyle \frac{{\lambda }^2}{2}}+C{\kappa }_N{\left|\lambda \right|}^3.$$

(7)

In particular, after renormalizing the constant C if necessary, we may write

$$\mathbb{E}\left[e^{\lambda {\nu }_N}\right]\le \exp \left({\displaystyle \frac{{\lambda }^2}{2}}+{\displaystyle \frac{{\kappa }_N}{3}}{\left|\lambda \right|}^3\right),\left|\lambda \right|\le {\lambda }_0\left(q\right).$$

Proof. 

The cumulant-generating function (CGF) of ${\nu }_N$ is

$$K_{{\nu }_N}\left(\lambda \right)=\log \mathbb{E}\left[e^{\lambda {\nu }_N}\right]=\sum _{r=1}^{\infty }{\displaystyle \frac{{\kappa }_r\left({\nu }_N\right)}{r!}}{\lambda }^r,$$

where ${\kappa }_r\left({\nu }_N\right)$ denotes the r-th cumulant of ${\nu }_N$.

Since ${\nu }_N$ is centered and normalized, we have

$${\kappa }_1\left({\nu }_N\right)=0,{\kappa }_2\left({\nu }_N\right)=1,$$

and therefore,

$$K_{{\nu }_N}\left(\lambda \right)={\displaystyle \frac{{\lambda }^2}{2}}+{\displaystyle \frac{{\kappa }_3\left({\nu }_N\right)}{3!}}{\lambda }^3+\sum _{r\ge 4}{\displaystyle \frac{{\kappa }_r\left({\nu }_N\right)}{r!}}{\lambda }^r.$$

Step 1: Third cumulant. We write

$${\kappa }_3\left({\nu }_N\right)=\mathbb{E}\left[{\nu }_N^3\right]={\displaystyle \frac{1}{{\sigma }^3\left(N\right)}}{\int }_0^1{\left[f_N\left(x\right)\right]}^{3}dx.$$

Expanding $f_N^3$ and using the orthogonality of $\left\{{\varphi }_k\right\}$ together with the Hadamard gap condition, many frequency combinations lead to non-vanishing integrals only finitely. Classical computations (Kac [2], Salem–Zygmund [4], Gaposhkin [17]) give

$$|{\kappa }_3\left({\nu }_N\right)|\le C_3\left(q\right){\displaystyle \frac{1}{{\sigma }^3\left(N\right)}}\sum _{k=1}^N{\left|c_k\right|}^3=C_3\left(q\right){\kappa }_N.$$

Step 2: Higher-order cumulants. For $r\ge 4$, ${\kappa }_r\left({\nu }_N\right)$ is a linear combination of integrals of products of r trigonometric functions. Lacunarity implies that almost all mixed-frequency interactions average out, and one obtains uniform bounds

$$|{\kappa }_r\left({\nu }_N\right)|\le C_r\left(q\right){\kappa }_N,r\ge 3,$$

(8)

where $C_r\left(q\right)$ depends only on q. Such bounds appear in the works of Aistleitner–Berkes [1] and Aistleitner et al. [5] for more general lacunary systems.

• Step 3: Local control of the cumulant series. Let

$$R\left(\lambda \right):=\sum _{r\ge 4}{\displaystyle \frac{{\kappa }_r\left({\nu }_N\right)}{r!}}{\lambda }^r.$$

Using (8),

$$\left|R\left(\lambda \right)\right|\le {\kappa }_N\sum _{r\ge 4}{\displaystyle \frac{C_r\left(q\right)}{r!}}{\left|\lambda \right|}^r={\kappa }_N{\left|\lambda \right|}^3\sum _{r\ge 4}{\displaystyle \frac{C_r\left(q\right)}{r!}}{\left|\lambda \right|}^{r-3}.$$

Since the power series ${\sum }_{r\ge 4}{C}_r\left(q\right){\lambda }^{r-3}/r!$ converges near the origin, there exist constants $C\left(q\right)>0$ and ${\lambda }_0\left(q\right)>0$ such that

$$\left|R\left(\lambda \right)\right|\le C\left(q\right){\kappa }_N{\left|\lambda \right|}^3,\left|\lambda \right|\le {\lambda }_0\left(q\right).$$

Step 4: Conclusion. For $\left|\lambda \right|\le {\lambda }_0\left(q\right)$, we combine the three parts:

$$K_{{\nu }_N}\left(\lambda \right)\le {\displaystyle \frac{{\lambda }^2}{2}}+{\displaystyle \frac{C_3\left(q\right)}{6}}{\kappa }_N{\left|\lambda \right|}^3+C\left(q\right){\kappa }_N{\left|\lambda \right|}^3.$$

Absorbing constants into a new $C\left(q\right)$ yields (7). Finally, rescaling ${\kappa }_N$ to absorb $3C\left(q\right)$ gives

$$\mathbb{E}\left[e^{\lambda {\nu }_N}\right]\le \exp \left({\displaystyle \frac{{\lambda }^2}{2}}+{\displaystyle \frac{{\kappa }_N}{3}}{\left|\lambda \right|}^3\right),\left|\lambda \right|\le {\lambda }_0\left(q\right).$$

(Here, as is standard in cumulant-based estimates, constants depending only on q are absorbed into the definition of ${\kappa }_N$ without changing its qualitative role). □

For the tail estimates, we will use a slightly more explicit bound, where the dependence on the lacunarity parameter q is made explicit and the third-order term is normalized in a simpler way. This relies on more detailed harmonic analyses of lacunary sums; a sketch of the computation is provided in Appendix A.

Lemma 2 (MGF bound for lacunary trigonometric sums, explicit constant). 

Let $f_N$ be the lacunary trigonometric sum

$$f_N\left(x\right)=\sum _{k=1}^N{c}_k{\varphi }_k\left(x\right),{\varphi }_k\left(x\right)=\sqrt{2}\cos \left(2\pi n_{k}x\right),$$

where ${\left\{n_k\right\}}_{k\ge 1}$ satisfies the Hadamard gap condition

$${\displaystyle \frac{n_{k+1}}{n_k}}\ge q>1,k\ge 1,$$

and ${\left\{c_k\right\}}_{k\ge 1}$ are real coefficients. Let

$${\sigma }^2\left(N\right)=\mathrm{Var}\left(f_N\right)=\sum _{k=1}^N{c}_k^2,{\nu }_N\left(x\right)={\displaystyle \frac{f_N\left(x\right)}{\sigma \left(N\right)}},$$

and define

$${\tilde{\kappa }}_N={\displaystyle \frac{2\sqrt{2q}}{q-1}}·{\displaystyle \frac{1}{\sigma \left(N\right)}}.$$

(9)

Then, there exists ${\lambda }_0={\lambda }_0\left(q\right)>0$ such that, for all real λ with $\left|\lambda \right|\le {\lambda }_0$,

$${\int }_0^1\exp \left(\lambda {\nu }_N\left(x\right)\right)dx\le \exp \left({\displaystyle \frac{{\lambda }^2}{2}}+{\displaystyle \frac{{\tilde{\kappa }}_N}{3}}{\left|\lambda \right|}^3\right).$$

(10)

Proof. 

We sketch the argument. The cumulant-generating function of ${\nu }_N$ is

$$K_{{\nu }_N}\left(\lambda \right)=\log \mathbb{E}\left[e^{\lambda {\nu }_N}\right]=\sum _{m=1}^{\infty }{\displaystyle \frac{{\lambda }^m}{m!}}{\kappa }_m\left({\nu }_N\right),$$

with ${\kappa }_1\left({\nu }_N\right)=0$ and ${\kappa }_2\left({\nu }_N\right)=1$. The third cumulant satisfies

$${\kappa }_3\left({\nu }_N\right)=\mathbb{E}\left[{\nu }_N^3\right]={\displaystyle \frac{1}{{\sigma }^3\left(N\right)}}{\int }_0^1{\left[f_N\left(x\right)\right]}^{3}dx.$$

To estimate the integral, expand

$${\left[f_N\left(x\right)\right]}^3=\sum _{i,j,k=1}^N{c}_i{c}_j{c}_k{\varphi }_i\left(x\right){\varphi }_j\left(x\right){\varphi }_k\left(x\right).$$

Since each ${\varphi }_k$ is a linear combination of exponentials, the product ${\varphi }_i{\varphi }_j{\varphi }_k$ is a finite sum of exponentials of the form $e^{2\pi i(\pm n_i\pm n_j\pm n_k)x}$. The integral over $[0,1]$ vanishes unless the frequency is zero, i.e.,

$$\pm n_i\pm n_j\pm n_k=0.$$

Under the Hadamard gap condition $n_{k+1}/n_k\ge q>1$, the number of such “resonant” triples $(i,j,k)$ is uniformly bounded in terms of q. A detailed harmonic analysis (see, e.g., [2,3,4,17]) then yields the explicit estimate

$$\left|{\int }_0^1{\left[f_N\left(x\right)\right]}^{3}dx\right|\le {\displaystyle \frac{2\sqrt{2q}}{q-1}}{\sigma }^2\left(N\right),$$

which implies

$$|{\kappa }_3\left({\nu }_N\right)|={\displaystyle \frac{1}{{\sigma }^3\left(N\right)}}\left|{\int }_0^1{\left[f_N\left(x\right)\right]}^{3}dx\right|\le {\displaystyle \frac{2\sqrt{2q}}{q-1}}·{\displaystyle \frac{1}{\sigma \left(N\right)}}={\tilde{\kappa }}_N.$$

(11)

Higher-order cumulants satisfy

$$|{\kappa }_m\left({\nu }_N\right)|\le C_m\left(q\right){\tilde{\kappa }}_N,m\ge 3,$$

so the remainder of the cumulant expansion obeys

$$\left|\sum _{m\ge 4}{\displaystyle \frac{{\lambda }^m}{m!}}{\kappa }_m\left({\nu }_N\right)\right|\le C\left(q\right){\tilde{\kappa }}_N{\left|\lambda \right|}^4.$$

Thus, for $\left|\lambda \right|\le {\lambda }_0\left(q\right)$ (chosen so that $C\left(q\right){\lambda }_0\le 1/6$),

$$K_{{\nu }_N}\left(\lambda \right)\le {\displaystyle \frac{{\lambda }^2}{2}}+{\displaystyle \frac{{\tilde{\kappa }}_N}{3}}{\left|\lambda \right|}^3.$$

(12)

Exponentiating yields (10). □

Remark 2. 

The exact value of ${\lambda }_0\left(q\right)$ is not important for our purposes; it is enough to know that such a positive radius of analyticity and control exists, depending only on the lacunarity parameter q.

Remark 3. 

For the detailed computation of the constant $\widetilde{{\kappa }_N}$, see Appendix A.

4. Upper Tail Bound

In this section, we derive an upper estimate for the tail function

$$T\left[{\nu }_N\right]\left(t\right):=\mathbb{P}({\nu }_N>t),t>0,$$

for the normalized lacunary trigonometric sum

$${\nu }_N\left(x\right)={\displaystyle \frac{f_N\left(x\right)}{\sigma \left(N\right)}},$$

where $f_N$ is defined in (4) and ${\sigma }^2\left(N\right)=\mathrm{Var}\left(f_N\right)$ is its variance.

We will combine the MGF estimate of Lemma 2 with a Chernoff-type bound based on the Legendre transform of a suitable convex function. A standard exposition of these concentration techniques and convex-analytic tools may be found in [18].

Theorem 1 (Upper tail bound). 

Let $f_N$ be the lacunary trigonometric sum defined in (4), and let ${\nu }_N=f_N/\sigma \left(N\right)$. Let ${\lambda }_0={\lambda }_0\left(q\right)>0$ be the constant from Lemma 2, and define

$${\tilde{\kappa }}_N={\displaystyle \frac{2\sqrt{2q}}{q-1}}{\displaystyle \frac{1}{\sigma \left(N\right)}}.$$

For every $t>0$ and every $0<\lambda \le {\lambda }_0$, one has

$$T\left[{\nu }_N\right]\left(t\right)\le \exp \left(-\lambda t+{\displaystyle \frac{{\lambda }^2}{2}}+{\displaystyle \frac{{\tilde{\kappa }}_N}{3}}{\lambda }^3\right).$$

(13)

Moreover, for each $t>0$, let

$$\widehat{\lambda }\left(t\right):={\displaystyle \frac{-1+\sqrt{1+4{\tilde{\kappa }}_{N}t}}{2{\tilde{\kappa }}_N}},$$

(14)

which is the unique positive solution of the equation

$$\lambda +{\tilde{\kappa }}_N{\lambda }^2=t.$$

If $\widehat{\lambda }\left(t\right)\le {\lambda }_0$, then inserting $\lambda =\widehat{\lambda }\left(t\right)$ into (13) yields the sharpened bound

$$T\left[{\nu }_N\right]\left(t\right)\le \exp \left(-{\displaystyle \frac{{\left(\widehat{\lambda }\left(t\right)\right)}^2}{2}}-{\displaystyle \frac{2}{3}}{\tilde{\kappa }}_N{\left(\widehat{\lambda }\left(t\right)\right)}^3\right).$$

(15)

Proof. 

By Lemma 2, for all $\left|\lambda \right|\le {\lambda }_0\left(q\right)$, we have

$$\mathbb{E}{e}^{\lambda {\nu }_N}\le \exp \left({\displaystyle \frac{{\lambda }^2}{2}}+{\displaystyle \frac{{\tilde{\kappa }}_N}{3}}{\left|\lambda \right|}^3\right).$$

In particular, for every $\lambda \in (0,{\lambda }_0]$, we may apply Markov’s inequality to the non-negative random variable $e^{\lambda {\nu }_N}$ and obtain

$$\mathbb{P}({\nu }_N>t)=\mathbb{P}\left(e^{\lambda {\nu }_N}>e^{\lambda t}\right)\le e^{-\lambda t}\mathbb{E}\left[e^{\lambda {\nu }_N}\right].$$

Combining this with the MGF bound from Lemma 2 gives

$$T\left[{\nu }_N\right]\left(t\right)=\mathbb{P}({\nu }_N>t)\le \exp \left(-\lambda t+{\displaystyle \frac{{\lambda }^2}{2}}+{\displaystyle \frac{{\tilde{\kappa }}_N}{3}}{\lambda }^3\right),0<\lambda \le {\lambda }_0,$$

which proves (13). This is precisely the standard Chernoff method: one first applies Markov’s inequality to $e^{\lambda {\nu }_N}$ and then optimizes the resulting bound with respect to $\lambda$.

To describe this optimization, define the convex function

$$b\left(\lambda \right)={\displaystyle \frac{{\lambda }^2}{2}}+{\displaystyle \frac{{\tilde{\kappa }}_N}{3}}{\lambda }^3$$

and

$${\Phi }_t\left(\lambda \right)=\lambda t-b\left(\lambda \right),\lambda \ge 0.$$

The Young–Fenchel (Legendre) transform of b is

$$b*\left(t\right):=\underset{\lambda >0}{sup}\left(\lambda t-b\left(\lambda \right)\right)=\underset{\lambda >0}{sup}{\Phi }_t\left(\lambda \right),t>0.$$

Then, (13) can be written as

$$T\left[{\nu }_N\right]\left(t\right)\le \exp \left(-{\Phi }_t\left(\lambda \right)\right),0<\lambda \le {\lambda }_0.$$

A direct computation shows that

$${\Phi }_t^{\prime }\left(\lambda \right)=t-b^{\prime }\left(\lambda \right)=t-\left(\lambda +{\tilde{\kappa }}_N{\lambda }^2\right),$$

so the critical points of ${\Phi }_t$ satisfy

$$\lambda +{\tilde{\kappa }}_N{\lambda }^2=t.$$

For each fixed $t>0$, this quadratic equation has exactly one positive solution $\widehat{\lambda }\left(t\right)$, given by (14), and this point is the unique maximizer of ${\Phi }_t$ on $(0,\infty )$, that is, $b*\left(t\right)={\Phi }_t\left(\widehat{\lambda }\left(t\right)\right)$.

If the corresponding optimizer satisfies $0<\widehat{\lambda }\left(t\right)\le {\lambda }_0$, then it is admissible in (13), and we obtain

$$T\left[{\nu }_N\right]\left(t\right)\le \exp \left(-{\Phi }_t\left(\widehat{\lambda }\left(t\right)\right)\right).$$

Since $\widehat{\lambda }\left(t\right)$ satisfies the optimality condition $t=\widehat{\lambda }\left(t\right)+{\tilde{\kappa }}_N{\left(\widehat{\lambda }\left(t\right)\right)}^2$, we have

$$\widehat{\lambda }\left(t\right)t={\left(\widehat{\lambda }\left(t\right)\right)}^2+{\tilde{\kappa }}_N{\left(\widehat{\lambda }\left(t\right)\right)}^3.$$

Substituting this identity into ${\Phi }_t\left(\lambda \right)=\lambda t-b\left(\lambda \right)$ and simplifying, we obtain

$${\Phi }_t\left(\widehat{\lambda }\left(t\right)\right)={\displaystyle \frac{\widehat{\lambda }{\left(t\right)}^2}{2}}+{\displaystyle \frac{2}{3}}{\tilde{\kappa }}_N\widehat{\lambda }{\left(t\right)}^3.$$

So that

$$T\left[{\nu }_N\right]\left(t\right)\le \exp \left(-{\displaystyle \frac{\widehat{\lambda }{\left(t\right)}^2}{2}}-{\displaystyle \frac{2}{3}}{\tilde{\kappa }}_N\widehat{\lambda }{\left(t\right)}^3\right),$$

which is exactly (15). □

Remark 4 (On Fenchel duality and inverse Chernoff bounds). 

The Fenchel–Legendre transform reflects the standard duality between moment-generating function bounds and tail estimates. Indeed, if a random variable η satisfies

$$\mathbb{E}{e}^{\lambda \eta }\le \exp \left(b\left(\lambda \right)\right),\lambda >0,$$

then Chernoff’s variational formula yields the upper tail bound

$$\mathbb{P}(\eta >t)\le \exp (-b*\left(t\right)).$$

Conversely, under additional Orlicz–regularity assumptions on b (convexity, superlinearity, differentiability) and a mild regularity condition on the tail behavior, one also obtains an inverse estimate:

$$\mathbb{E}{e}^{\lambda \eta }\le \exp \left(Cb\left(\lambda \right)\right),\left|\lambda \right|\le {\lambda }_0,$$

for suitable constants $C,{\lambda }_0>0$. Such two-sided equivalences between exponential tails and MGF control are classical (see [19] (Ch. 2–3)). Chernoff inequality is often used in the theory of the so-called Grand Lebesgue Spaces (see, e.g., [15,19,20,21]).

A very important example is the subgaussian random variable

$$\psi \left(\lambda \right)={\displaystyle \frac{{\lambda }^2}{2}},\lambda >0,$$

with its Young–Fenchel transform $\psi *\left(t\right)=t^2/2$.

Remark 5 (Range of validity of the optimized bound). 

Since the function $\widehat{\lambda }\left(t\right)$ defined in (14) is strictly increasing in t, there exists a unique value $t_{max}>0$ such that

$$\widehat{\lambda }\left(t_{max}\right)={\lambda }_0.$$

Solving ${\lambda }_0+{\tilde{\kappa }}_N{\lambda }_0^2=t_{max}$ gives the explicit expression

$$t_{max}={\lambda }_0+{\tilde{\kappa }}_N{\lambda }_0^2.$$

(16)

Hence,

• For all $0<t\le t_{max}$, one has $0<\widehat{\lambda }\left(t\right)\le {\lambda }_0$, and therefore the optimized bound (15) applies.
• For $t>t_{max}$, the optimizer $\widehat{\lambda }\left(t\right)$ lies outside the admissible range of Lemma 2, and the best admissible choice is $\lambda ={\lambda }_0$ in (13), yielding

$$T\left[{\nu }_N\right]\left(t\right)\le \exp \left(-{\lambda }_{0}t+{\displaystyle \frac{{\lambda }_0^2}{2}}+{\displaystyle \frac{{\tilde{\kappa }}_N}{3}}{\lambda }_0^3\right).$$

Thus, the tail behavior splits naturally into the “Chernoff-optimized” regime $0<t\le t_{max}$ and the “linear-exponential” regime $t>t_{max}$.

Remark 6. 

For t such that $\widehat{\lambda }\left(t\right)>{\lambda }_0$, the bound (15) need not hold, but (13) remains valid for any fixed choice of $\lambda \in (0,{\lambda }_0]$. In particular, one can choose $\lambda ={\lambda }_0$ to obtain a linear-exponential bound of the form

$$T\left[{\nu }_N\right]\left(t\right)\le \exp \left(-{\lambda }_{0}t+\frac{{\lambda }_0^2}{2}+\frac{{\tilde{\kappa }}_N}{3}{\lambda }_0^3\right)$$

for all sufficiently large t.

Remark 7. 

The bound for negative deviations follows by applying the same argument to the random variable $-{\nu }_N$ since

$$\mathbb{E}{e}^{\lambda (-{\nu }_N)}=\mathbb{E}{e}^{-\lambda {\nu }_N}.$$

Thus,

$$\mathbb{P}({\nu }_N<-t)=\mathbb{P}(-{\nu }_N>t),t>0,$$

satisfies the same estimate as in Theorem 1, with the same constants.

5. Simplified Tail Estimates

In this Section, we derive more explicit upper bounds for $T\left[{\nu }_N\right]\left(t\right)$ in two natural regimes of the parameter t, defined with respect to the standard deviation $\sigma \left(N\right)$. Recall that Theorem 1 provides the estimate

$$T\left[{\nu }_N\right]\left(t\right)\le \exp \left(-{\displaystyle \frac{\widehat{\lambda }{\left(t\right)}^2}{2}}-{\displaystyle \frac{2}{3}}{\tilde{\kappa }}_N\widehat{\lambda }{\left(t\right)}^3\right)$$

whenever $0<\widehat{\lambda }\left(t\right)\le {\lambda }_0\left(q\right)$, where

$$\widehat{\lambda }\left(t\right)={\displaystyle \frac{-1+\sqrt{1+4{\tilde{\kappa }}_{N}t}}{2{\tilde{\kappa }}_N}},{\tilde{\kappa }}_N={\displaystyle \frac{2\sqrt{2q}}{q-1}}·{\displaystyle \frac{1}{\sigma \left(N\right)}}.$$

We consider two regimes:

$$Z_1:=\{t:0\le t\le \sigma \left(N\right)\},Z_2:=\{t:t\ge \sigma \left(N\right)\},$$

corresponding to small and moderate/large deviations, respectively. All estimates below are to be understood for those t such that $\widehat{\lambda }\left(t\right)\le {\lambda }_0\left(q\right)$, i.e., within the range where the MGF bound is available.

5.1. Small Deviations: Subgaussian Behavior for t in $Z_1$ ($t\le \sigma \left(N\right)$)

We derive here a simplified, non-asymptotic upper bound for the tail probability $T\left[{\nu }_N\right]\left(t\right)$ valid for small deviations. Throughout this subsection, no asymptotic limit is taken: N is fixed.

Recall that, whenever $\widehat{\lambda }\left(t\right)\le {\lambda }_0\left(q\right)$, Theorem 1 yields

$$T\left[{\nu }_N\right]\left(t\right)\le \exp \left(-{\displaystyle \frac{{\left(\widehat{\lambda }\left(t\right)\right)}^2}{2}}-{\displaystyle \frac{2}{3}}{\tilde{\kappa }}_N{\left(\widehat{\lambda }\left(t\right)\right)}^3\right),\widehat{\lambda }\left(t\right)={\displaystyle \frac{-1+\sqrt{1+4{\tilde{\kappa }}_{N}t}}{2{\tilde{\kappa }}_N}}.$$

The cubic term in the exponent is negative; discarding it gives the weaker but valid bound

$$T\left[{\nu }_N\right]\left(t\right)\le \exp \left(-{\displaystyle \frac{{\left(\widehat{\lambda }\left(t\right)\right)}^2}{2}}\right).$$

(17)

To make (17) explicit, we obtain a lower bound for $\widehat{\lambda }\left(t\right)$. Using the elementary inequality (valid for all $x\ge 0$)

$$\sqrt{1+x}\ge 1+{\displaystyle \frac{x}{2}}-{\displaystyle \frac{x^2}{8}},$$

and applying it with $x=4{\tilde{\kappa }}_{N}t$, we obtain

$$\sqrt{1+4{\tilde{\kappa }}_{N}t}\ge 1+2{\tilde{\kappa }}_{N}t-2{\tilde{\kappa }}_N^2{t}^2.$$

Substituting this into the definition of $\widehat{\lambda }\left(t\right)$ gives the non-asymptotic bound

$$\widehat{\lambda }\left(t\right)\ge t-{\tilde{\kappa }}_N{t}^2.$$

(18)

This inequality is valid for all $t\ge 0$, for which $\widehat{\lambda }\left(t\right)\le {\lambda }_0\left(q\right)$.

Plugging (18) into (17) yields the explicit estimate

$$T\left[{\nu }_N\right]\left(t\right)\le \exp \left(-{\displaystyle \frac{{(t-{\tilde{\kappa }}_N{t}^2)}^2}{2}}\right),0\le t\le \sigma \left(N\right),\widehat{\lambda }\left(t\right)\le {\lambda }_0\left(q\right).$$

(19)

Since

$${\tilde{\kappa }}_N={\displaystyle \frac{2\sqrt{2q}}{q-1}}{\displaystyle \frac{1}{\sigma \left(N\right)}},$$

we have ${\tilde{\kappa }}_{N}t=O(t/\sigma \left(N\right))$. Thus, for all $0\le t\le \sigma \left(N\right)$ in the admissible MGF range, (19) shows that

$$T\left[{\nu }_N\right]\left(t\right)\le \exp \left(-{\displaystyle \frac{t^2}{2}}\left(1-O(t/\sigma (N\left)\right)\right)\right),$$

i.e., the normalized lacunary sum ${\nu }_N$ exhibits a subgaussian-type tail behavior for small deviations.

Remark 8. 

Using

$${(t-{\tilde{\kappa }}_N{t}^2)}^2=t^2\left(1-2{\tilde{\kappa }}_{N}t+{\tilde{\kappa }}_N^2{t}^2\right),$$

and ${\tilde{\kappa }}_N={\displaystyle \frac{2\sqrt{2q}}{q-1}}{\displaystyle \frac{1}{\sigma \left(N\right)}}$, we can rewrite the exponent in (19) as

$$-{\displaystyle \frac{{(t-{\tilde{\kappa }}_N{t}^2)}^2}{2}}=-{\displaystyle \frac{t^2}{2}}\left(1-O\left(t/\sigma \left(N\right)\right)\right),$$

for $0\le t\le \sigma \left(N\right)$. In particular, the leading term is $-t^2/2$ and the correction is of order $t^3/\sigma \left(N\right)$, so ${\nu }_N$ exhibits a subgaussian-type decay for small deviations.

5.2. Moderate/Large Deviations: Stretched-Exponential Decay in $Z_2$ ($t\ge \sigma \left(N\right)$)

We now consider t such that $\widehat{\lambda }\left(t\right)\le {\lambda }_0$, but t is comparable to or larger than $\sigma \left(N\right)$. Using the inequality $\sqrt{1+x}\ge \sqrt{x}$ for $x\ge 0$, we obtain

$$\widehat{\lambda }\left(t\right)={\displaystyle \frac{-1+\sqrt{1+4{\tilde{\kappa }}_{N}t}}{2{\tilde{\kappa }}_N}}\ge {\displaystyle \frac{-1+\sqrt{4{\tilde{\kappa }}_{N}t}}{2{\tilde{\kappa }}_N}}=\sqrt{{\displaystyle \frac{t}{{\tilde{\kappa }}_N}}}-{\displaystyle \frac{1}{2{\tilde{\kappa }}_N}}.$$

If $t\ge 1/{\tilde{\kappa }}_N$, then

$$\sqrt{{\displaystyle \frac{t}{{\tilde{\kappa }}_N}}}\ge \sqrt{{\displaystyle \frac{1}{{\tilde{\kappa }}_N^2}}}={\displaystyle \frac{1}{{\tilde{\kappa }}_N}},$$

and hence,

$$\sqrt{{\displaystyle \frac{t}{{\tilde{\kappa }}_N}}}-{\displaystyle \frac{1}{2{\tilde{\kappa }}_N}}\ge {\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{t}{{\tilde{\kappa }}_N}}}.$$

Thus, for all $t\ge 1/{\tilde{\kappa }}_N$, we have the lower bound

$$\widehat{\lambda }\left(t\right)\ge {\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{t}{{\tilde{\kappa }}_N}}}.$$

(20)

Substituting (20) into the exponent in (15) and discarding the quadratic term, which is negative and therefore only strengthens the bound, we obtain

$$T\left[{\nu }_N\right]\left(t\right)\le \exp \left(-{\displaystyle \frac{2}{3}}{\tilde{\kappa }}_N\widehat{\lambda }{\left(t\right)}^3\right),$$

and hence, for all t such that $t\ge 1/{\tilde{\kappa }}_N$ and $\widehat{\lambda }\left(t\right)\le {\lambda }_0$,

$$\begin{array}{cc}\hfill T\left[{\nu }_N\right]\left(t\right)& \le \exp \left(-{\displaystyle \frac{2}{3}}{\tilde{\kappa }}_N{\left({\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{t}{{\tilde{\kappa }}_N}}}\right)}^3\right)\hfill \\ \hfill & =\exp \left(-{\displaystyle \frac{1}{12}}{\tilde{\kappa }}_N^{-1/2}{t}^{3/2}\right).\hfill \end{array}$$

Using ${\tilde{\kappa }}_N={\displaystyle \frac{2\sqrt{2q}}{q-1}}·{\displaystyle \frac{1}{\sigma \left(N\right)}}$, we can rewrite this as

$${\tilde{\kappa }}_N^{-1/2}={\left({\displaystyle \frac{q-1}{2\sqrt{2q}}}\right)}^{1/2}{\left(\sigma \left(N\right)\right)}^{1/2}=:C_{*}\left(q\right)\sqrt{\sigma \left(N\right)},$$

and therefore,

$$T\left[{\nu }_N\right]\left(t\right)\le \exp \left(-C\left(q\right)\sqrt{\sigma \left(N\right)}{t}^{3/2}\right),$$

for all t in the range where both $\widehat{\lambda }\left(t\right)\le {\lambda }_0$ and $t\ge 1/{\tilde{\kappa }}_N$, with $C\left(q\right)>0$ depending only on q.

Remark 9. 

Since

$${\displaystyle \frac{1}{{\tilde{\kappa }}_N}}={\displaystyle \frac{q-1}{2\sqrt{2q}}}\sigma \left(N\right)=:C^{\#}\left(q\right)\sigma \left(N\right),$$

the condition $t\ge 1/{\tilde{\kappa }}_N$ corresponds to deviations of size that are at least a constant multiple of $\sigma \left(N\right)$. Thus, the stretched-exponential regime described above covers moderate and large deviations, as long as $\widehat{\lambda }\left(t\right)$ remains in the admissible range $[0,{\lambda }_0\left(q\right)]$ where the MGF bound of Lemma 2 applies.

6. Examples

We now illustrate the tail estimates derived in the previous sections through several representative families of lacunary trigonometric sums with different coefficient sequences.

Recall the variance

$${\sigma }^2\left(N\right)=\sum _{k=1}^N{c}_k^2,$$

and the parameter

$${\tilde{\kappa }}_N={\displaystyle \frac{2\sqrt{2q}}{q-1}}{\displaystyle \frac{1}{\sigma \left(N\right)}},$$

which governs the third-order correction in the MGF bound. In each case, we comment on the regimes where the subgaussian and the stretched-exponential estimates apply.

6.1. Example: Bounded Coefficients, $c_k=1$, $n_k=2^k$

Let $c_k=1$ and $n_k=2^k$. Then,

$${\sigma }^2\left(N\right)=N,\sigma \left(N\right)=\sqrt{N},{\tilde{\kappa }}_N={\displaystyle \frac{4}{\sqrt{N}}}.$$

The frequencies satisfy the Hadamard gap condition with $q=2$.

• Small deviations. For $0\le t\le \sigma \left(N\right)=\sqrt{N}$, Theorem 1 yields

$$T\left[{\nu }_N\right]\left(t\right)\le \exp \left(-\frac{1}{2}{t}^2\left(1+O(t/\sqrt{N})\right)\right),$$

which is subgaussian up to a negligible $(t/\sqrt{N})$ correction coming from the cubic term in the MGF bound.

• Moderate deviations. For $t\ge \sqrt{N}$, but such that $\widehat{\lambda }\left(t\right)\le {\lambda }_0\left(2\right)$, the stretched-exponential inequality applies:

  $$T\left[{\nu }_N\right]\left(t\right)\le \exp \left(-C{N}^{1/4}{t}^{3/2}\right),$$

  for some constant $C>0$ depending only on q. This behavior reflects the cubic correction in the MGF and is specific to lacunary trigonometric sums.

6.2. Example: Polynomially Growing Coefficients, $c_k=k^{\alpha }$,$n_k=2^k$

Let $c_k=k^{\alpha }$, $\alpha >0$. Then,

$${\sigma }^2\left(N\right)=\sum _{k=1}^N{k}^{2\alpha }.$$

Using standard integral comparison,

$$\sum _{k=1}^N{k}^{2\alpha }\sim {\displaystyle \frac{N^{2\alpha +1}}{2\alpha +1}},$$

and therefore,

$${\sigma }^2\left(N\right)\asymp N^{2\alpha +1},\sigma \left(N\right)\asymp N^{\alpha +1/2}.$$

Therefore,

$${\tilde{\kappa }}_N={\displaystyle \frac{2\sqrt{2q}}{q-1}}·{\displaystyle \frac{1}{\sigma \left(N\right)}}\asymp N^{-(\alpha +1/2)}.$$

Small deviations. The subgaussian regime corresponds to deviations in size

$$t=O\left(\sigma \left(N\right)\right)=O\left(N^{\alpha +1/2}\right),$$

where the cubic correction satisfies ${\tilde{\kappa }}_{N}t=O\left(1\right)$.

• Moderate deviations. Whenever $\widehat{\lambda }\left(t\right)\le {\lambda }_0\left(q\right)$, the stretched-exponential bound

  $$T\left[{\nu }_N\right]\left(t\right)\le \exp \left(-C\left(q\right)\sigma {\left(N\right)}^{1/2}{t}^{3/2}\right)$$

  applies up to the admissible range of $\lambda$. Since $\sigma \left(N\right)$ diverges polynomially, the stretched-exponential regime is visible for a significant range of t.

6.3. Example: Slowly Decaying Coefficients, $c_k=k^{-1/2}$

Let $c_k=k^{-1/2}$ and $n_k=3^k$. Then,

$${\sigma }^2\left(N\right)=\sum _{k=1}^N{\displaystyle \frac{1}{k}}\sim \log N,\sigma \left(N\right)\sim {\left(\log N\right)}^{1/2}.$$

Thus,

$${\tilde{\kappa }}_N\sim C\left(q\right){\left(\log N\right)}^{-1/2}.$$

Small deviations. For $t=O\left({\left(\log N\right)}^{1/2}\right)$, the behavior of ${\nu }_N$ is nearly Gaussian:

$$T\left[{\nu }_N\right]\left(t\right)\le \exp \left(-\frac{1}{2}{t}^2(1+o\left(1\right))\right).$$

Moderate deviations. Since ${\tilde{\kappa }}_N$ decays only logarithmically, the stretched-exponential correction becomes relevant only for relatively large t, but still within the MGF–admissible region.

7. Remarks on Lower Tail Bounds

The main results of this work provide non-asymptotic upper bounds for the tail probabilities $T\left[{\nu }_N\right]\left(t\right)=\mathbb{P}({\nu }_N\ge t)$. Obtaining matching lower bounds is considerably more delicate and typically requires additional tools.

In this section, we record a local two-sided control of the cumulant-generating function, which is a natural ingredient in such analyses. We do not pursue a full lower-tail theory here, as it would require assumptions extending beyond those used for our upper bounds.

Local Lower Bound for the Cumulant-Generating Function

Recall that the cumulant-generating function (CGF) of ${\nu }_N$ is

$$K_{{\nu }_N}\left(\lambda \right)=\log \mathbb{E}\left[e^{\lambda {\nu }_N}\right],\lambda \in \mathbb{R}.$$

Since ${\nu }_N$ is centered and normalized,

$${\kappa }_1\left({\nu }_N\right)=0,{\kappa }_2\left({\nu }_N\right)=1,$$

and the cumulant expansion yields

$$K_{{\nu }_N}\left(\lambda \right)={\displaystyle \frac{{\lambda }^2}{2}}+{\displaystyle \frac{{\kappa }_3\left({\nu }_N\right)}{6}}{\lambda }^3+R_4\left(\lambda \right),R_4\left(\lambda \right)=\sum _{m\ge 4}{\displaystyle \frac{{\kappa }_m\left({\nu }_N\right)}{m!}}{\lambda }^m.$$

From Lemma 2,

$$|{\kappa }_3\left({\nu }_N\right)|\le {\tilde{\kappa }}_N={\displaystyle \frac{2\sqrt{2q}}{q-1}}{\displaystyle \frac{1}{\sigma \left(N\right)}}.$$

Moreover, standard lacunary estimates (see, e.g., [2,3,4,17]) imply that there exists ${\lambda }_0={\lambda }_0\left(q\right)>0$ and $C\left(q\right)>0$ such that

$$|R_4\left(\lambda \right)|\le C\left(q\right){\tilde{\kappa }}_N{\left|\lambda \right|}^4,\left|\lambda \right|\le {\lambda }_0.$$

Combining these bounds, for $\left|\lambda \right|\le {\lambda }_0$, we obtain

$$\begin{array}{cc}\hfill K_{{\nu }_N}\left(\lambda \right)& \ge {\displaystyle \frac{{\lambda }^2}{2}}-{\displaystyle \frac{{\tilde{\kappa }}_N}{6}}{\left|\lambda \right|}^3-C\left(q\right){\tilde{\kappa }}_N{\left|\lambda \right|}^4\hfill \\ \hfill & \ge {\displaystyle \frac{{\lambda }^2}{2}}-C\left(q\right){\tilde{\kappa }}_N{\left|\lambda \right|}^3,\hfill \end{array}$$

after absorbing the quartic term into the cubic one (which is allowed because ${\lambda }_0$ is fixed and depends only on q). Thus,

$$K_{{\nu }_N}\left(\lambda \right)\ge {\displaystyle \frac{{\lambda }^2}{2}}-C\left(q\right){\tilde{\kappa }}_N{\left|\lambda \right|}^3,\left|\lambda \right|\le {\lambda }_0.$$

(21)

Discussion. Inequality (21) is a local counterpart to the upper bound provided in (12) by Lemma 2. It shows that in a fixed neighborhood of the origin (depending only on q) the deviation in $K_{{\nu }_N}\left(\lambda \right)$ from the Gaussian CGF $\frac{1}{2}{\lambda }^2$ is controlled by the same parameter ${\tilde{\kappa }}_N$ that appears in our upper tail analysis.

A full non-asymptotic lower-tail theory for lacunary trigonometric sums would require additional bounds on cumulants of order $m\ge 4$ or access to a suitable local limit theorem. As these tools fall outside the scope of the present work, we restrict ourselves to recording the local CGF control (21), which is the portion that can be established under our current assumptions.

8. Conclusions

We have obtained explicit non-asymptotic upper bounds for the tail probabilities of normalized lacunary sums under the Hadamard gap condition, combining local control of the moment-generating function with Chernoff-type bounds and Legendre transforms.

The resulting estimates exhibit two main regimes:

• a subgaussian regime for small deviations, where the tail behaves essentially like that of a standard normal random variable;
• a stretched-exponential regime for larger deviations (within the MGF control range), where the tail decays approximately as $\exp (-c\sigma {\left(N\right)}^{1/2}{t}^{3/2})$ with an explicit dependence on the lacunarity parameter q.

The examples highlight how the interplay between the choice of coefficients $c_k$ and the lacunarity parameter affects the scale of deviations and the strength of the exponential decay.

Possible directions for future research include the derivation of sharp non-asymptotic lower bounds matching our upper estimates, extensions to random signs and vector-valued lacunary sums and applications to quantitative bounds in random Fourier analysis and stochastic models driven by lacunary structures.