TY - JOUR
T1 - Large deviation principles for lacunary sums
AU - Aistleitner, Christoph
AU - Gantert, Nina
AU - Kabluchko, Zakhar
AU - Prochno, Joscha
AU - Ramanan, Kavita
PY - 2023/1/1
Y1 - 2023/1/1
N2 - Let (a
k)
k∈N be an increasing sequence of positive integers satisfying the Hadamard gap condition a
k
+1/a
k > q > 1 for all k ∈ N, and let S
n(ω) = n ∑ cos(2πa
kω), k=1 n ∈ N, ω ∈ [0, 1]. Then Sn is called a lacunary trigonometric sum, and can be viewed as a random variable defined on the probability space Ω = [0, 1] endowed with Lebesgue measure. Lacunary sums are known to exhibit several properties that are typical for sums of independent random variables. For example, a central limit theorem for (Sn)
n∈
N has been obtained by Salem and Zygmund, while a law of the iterated logarithm is due to Erdős and Gál. In this paper we study large deviation principles for lacunary sums. Specifically, under the large gap condition a
k
+1/a
k → ∞, we prove that the sequence (Sn/n)
n∈
N does indeed satisfy a large deviation principle with speed n and the same rate function I
~ as for sums of independent random variables with the arcsine distribution. On the other hand, we show that the large deviation principle may fail to hold when we only assume the Hadamard gap condition. However, we show that in the special case when a
k = q
k for some q ∈ {2, 3, . . .}, (Sn/n)
n∈
N satisfies a large deviation principle (with speed n) and a rate function I
q that is different from I
~, and describe an algorithm to compute an arbitrary number of terms in the Taylor expansion of Iq. In addition, we also prove that I
q converges pointwise to I
~ as q → ∞. Furthermore, we construct a random perturbation (a
k)
k∈
N of the sequence (2
k)
k∈
N for which a
k
+1/a
k → 2 as k → ∞, but for which at the same time (Sn/n)
n∈
N satisfies a large deviation principle with the same rate function I
~ as in the independent case, which is surprisingly different from the rate function I
2 one might naïvely expect. We relate this fact to the number of solutions of certain Diophantine equations. Together, these results show that large deviation principles for lacunary trigonometric sums are very sensitive to the arithmetic properties of the sequence (a
k)
k∈N.
AB - Let (a
k)
k∈N be an increasing sequence of positive integers satisfying the Hadamard gap condition a
k
+1/a
k > q > 1 for all k ∈ N, and let S
n(ω) = n ∑ cos(2πa
kω), k=1 n ∈ N, ω ∈ [0, 1]. Then Sn is called a lacunary trigonometric sum, and can be viewed as a random variable defined on the probability space Ω = [0, 1] endowed with Lebesgue measure. Lacunary sums are known to exhibit several properties that are typical for sums of independent random variables. For example, a central limit theorem for (Sn)
n∈
N has been obtained by Salem and Zygmund, while a law of the iterated logarithm is due to Erdős and Gál. In this paper we study large deviation principles for lacunary sums. Specifically, under the large gap condition a
k
+1/a
k → ∞, we prove that the sequence (Sn/n)
n∈
N does indeed satisfy a large deviation principle with speed n and the same rate function I
~ as for sums of independent random variables with the arcsine distribution. On the other hand, we show that the large deviation principle may fail to hold when we only assume the Hadamard gap condition. However, we show that in the special case when a
k = q
k for some q ∈ {2, 3, . . .}, (Sn/n)
n∈
N satisfies a large deviation principle (with speed n) and a rate function I
q that is different from I
~, and describe an algorithm to compute an arbitrary number of terms in the Taylor expansion of Iq. In addition, we also prove that I
q converges pointwise to I
~ as q → ∞. Furthermore, we construct a random perturbation (a
k)
k∈
N of the sequence (2
k)
k∈
N for which a
k
+1/a
k → 2 as k → ∞, but for which at the same time (Sn/n)
n∈
N satisfies a large deviation principle with the same rate function I
~ as in the independent case, which is surprisingly different from the rate function I
2 one might naïvely expect. We relate this fact to the number of solutions of certain Diophantine equations. Together, these results show that large deviation principles for lacunary trigonometric sums are very sensitive to the arithmetic properties of the sequence (a
k)
k∈N.
KW - Diophantine equations
KW - Hadamard gap condition
KW - hyperbolic dynamics
KW - Lacunary series
KW - lacunary trigonometric sums
KW - large deviations
KW - large gap condition
KW - normal numbers
UR - http://www.scopus.com/inward/record.url?scp=85132782034&partnerID=8YFLogxK
U2 - 10.1090/tran/8788
DO - 10.1090/tran/8788
M3 - Article
SN - 0002-9947
VL - 376
SP - 507
EP - 553
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 1
ER -