On the Large Deviations for Engel's, Sylvester's Series and Cantor's Products

In this paper, we study the large deviations for Engel's series, Sylvester's series and Cantor's products from number theory. For a real number, there are various ways to represent it as an expansion of digits. For example, perhaps the most well-known expansion is continued fractions. Statistical properties of the digits in the representations of real numbers have been well studied in the literature, see e. Erd˝ os et al. [4] and for a brief summary see e.g. Rényi [10]. One particular class of representations of a real number is Engel's series, and it is related to Sylvester's series and Cantor's products, see e.g. Erd˝ os et al. [4] and Rényi [8]. The statistical properties, including the central limit theorem, law of iterated logarithms have been well understood for Engel's series, Sylvester's series and Can-tor's products. In this article, we are interested to study the large deviations for these representations.

One particular class of representations of a real number is Engel's series, and it is related to Sylvester's series and Cantor's products, see e.g.Erdős et al. [4] and Rényi [8].The statistical properties, including the central limit theorem, law of iterated logarithms have been well understood for Engel's series, Sylvester's series and Cantor's products.In this article, we are interested to study the large deviations for these representations.
A sequence of probability measures (P n ) n∈N on a topological space X satisfies a large deviation principle with speed n and rate function I : X → R if I is non-negative, lower semicontinuous and for any measurable set A, we have For example, one can study the probability that the empirical mean of a sequence of random variables deviates away from its ergodic mean.Those probabilities are exponentially small in general and follow a large deviation principle.For an introduction to the theory of large deviations, we refer to Dembo and Zeitouni [3] and Varadhan [14].
We will obtain the large deviation principles for Engel's series in Section 1, Sylvester's series in Section 2 and Cantor's products in Section 3.

Engel's Series
For any real number 0 < x < 1, it can be represented in the form of Engel's series, where q n ∈ N and q n+1 ≥ q n ≥ 2 for any n ∈ N.
Borel [1] announced, without proof, that for a.e.x, (1.2) Lévy [6] proved this and also pointed out that log(q n+1 /q n ) are in a certain sense asymptotically independent and identially distributed as exponential random variables.Moreover, Lévy [6] showed that where P is the Lebesgue measure and Φ(t 2π e −u 2 /2 du.The proof was simplified in Erdős et al. [4] by observing that the sequence q n follows a homogeneous Markov chain with transition probabilities given by , k ≥ j. ( Rényi and Révész [9] showed that (1.3) holds for any probability measure Q, if Q is absolutely continuous with respect to the Lebesgue measure.Now, we can ask the question, what is the probability of the rate event that log qn n deviates away from its ergodic mean?Informally speaking, we are interested to estimate the small probability Q( log qn n x), where x = 1.That leads to the studies of large deviations.
Before we go to the statement and proof of the large deviations result for Engel's series, let us first state and prove the following two lemmas.Lemma 1.1.Assume x is a uniformly distributed random variable on (0, 1) and let (q n ) ∞ n=1 be the Engel's series defined in (1.1).Then, Proof.Recall that q n is a homogeneous Markov chain with transition probability P(q n = k|q n−1 = j) = j−1 k(k−1) , for k ≥ j.Large deviations theory for homogeneous Markov chain for the compact state space is well known, see e.g.Dembo and Zeitouni [3].However, in our case, (q n ) ∞ n=1 is supported on N\{1} and the general results do not apply directly.Some careful analysis is needed.
Next, let us show that for any −1 < θ < 1, (1.8) For any θ < 1, for j ≥ N , where N is a sufficiently large positive integer depending on > 0. Therefore, we have This implies that there exists some constant M > 0, so that On the other hand, ECP 19 (2014), paper 2.
Let Q be any probability measure that is equivalent to P so that dP dQ ∈ L p (Q) and dQ dP ∈ L p (P) for any 1 < p < ∞.Then, for any Borel set A and positive numbers p, q so ECP 19 (2014), paper 2. that 1 p + 1 q = 1, by Hölder's inequality, Since it holds for any p > 1, we get (1.32) Similarly, one can show that lim sup (1.33) Therefore the large deviation principle also holds for Q( log qn n ∈ •) with the same rate function I(x).
Remark 1.3.Prof. S. R. S. Varadhan pointed out to the author that if we consider the conditional probability measures P(•|q 1 = q), q ≥ 2, q ∈ N, then, one can show that P( log qn n ∈ •|q 1 = q) satisfies a large deviation principle with rate function (1.34) Remark 1.4.Lévy [6] pointed out that for uniformly distributed x on (0, 1), as n → ∞, log qn(x) qn−1(x) are asymptotically close to i.i.d.exponential random variables with mean 1.However, as we have seen in Theorem 1.2, the rate function I(x) coincides with the rate function for the empirical mean of i.i.d.exponentially distributed random variables only when x ≥ 1 2 .That indicates that in the context of tail probabilities and large deviations, the approximation of qn(x) qn−1(x) by i.i.d.exponential random variables is not enough.What is interesting though, is that we will see later in Section 2 and Section 3 that the rate functions of the large deviations for Sylvester's series and Cantor's products coincide with the rate function of the exponentials.

Sylvester's Series
For a real number x ∈ (0, 1), Sylverster's series, also known as Engel's series of the second kind, is defined via the expansion where Let P be the probability measure so that x is a uniformly distributed random variable on (0, 1).
For any > 0, there exists a sufficiently large N ∈ N so that for any j ≥ N ,  Therefore, for any n ≥ N , we have
Assume x is a uniformly distributed random variable on (0, 1) and let (Q n ) ∞ n=1 be the Sylvester's series defined in (2.1).Then, Lemma 2.1.