A Note on the Large Deviation Principle for Discrete Associated Random Variables

N(0, 1). SLLN asserts that the arithmetic mean S n /n converges to μ almost surely when n → ∞, whereas the CLT specifies the probability that S n differs fromμn by the quantity of order√n. Such deviation is called normal. However, in the case of S n and μn being distant by the quantity of order n, we deal with so-called large deviation events. It turns out that under certain condition on the tail distribution function of X i , for a > 0, the probability P(S n ≥


Introduction and the Notation
Let {  } ≥1 be a strictly stationary sequence of squareintegrable random variables (r.v.'s).Denote   = ∑  =1   ,  =  1 , and  2 = Var  1 + 2 ∑ ∞ =2 Cov( 1 ,   ).Consider a sequence satisfying the strong law of large numbers (SLLN) and the central limit theorem (CLT), that is, (  −)/√   → N(0, 1).SLLN asserts that the arithmetic mean   / converges to  almost surely when  → ∞, whereas the CLT specifies the probability that   differs from  by the quantity of order √.Such deviation is called normal.However, in the case of   and  being distant by the quantity of order , we deal with so-called large deviation events.
It turns out that under certain condition on the tail distribution function of   , for  > 0, the probability (  ≥ ( + )) tends to zero with exponential rate; that is, lim where () is called the rate function.
In the i.i.d.case, the well-known Cramer theorem gives the explicit formula for the rate function (see, for example, [1]).Let us recall that if the moment generating function () =   1 is finite for all  ∈ R, then lim  → ∞ (1/) ln (  ≥ ) = −() for all  > , where which means that () is the Fenchel-Legendre transform of the function ln ().
In general setting, we follow [1] to give the definition of the rate function as the function defined on a Polish space X, taking values in closed half line [0, ∞] and satisfying three conditions: Let us further recall that it is said that the sequence of probability measures {  } ≥1 satisfies the large deviation principle (LDP) with rate  and the rate function , if (D1)  is the rate function in the sense of the invoked definition; It is also well known that the rate function corresponding to the sequence {  } ≥1 is uniquely determined.
The role of open and closed sets in the above definition is similar to the one they play in Portmanteau theorem stating equivalent conditions for the weak convergence of probability measures.In fact, LDP may be thought of as the analogue of the weak convergence but in the exponential scale.Bryc [2] proved that if the sequence of probability measures {  } ≥1 is exponentially tight (see Appendix) and for all continuous and bounded functions  defined on the Polish space X (denoted by (X)) the limit exists, then {  } ≥1 satisfies LDP with rate  and the rate function This result explains the relation between the LDP and the concept of weak convergence of probability measures.Moreover, it appears very useful in proving the LDP, since it suffices to show existence of Λ() for sufficiently large subfamily of continuous and bounded functions.
There is only a few results on LDP for dependent r.v.'s other than Markov processes.Bryc [3] and Bryc and Dembo [4] studied LDP for strongly mixing sequences of r.v.'s; Henriques and Oliveira [5,6] dealt with stationary sequences of associated absolutely continuous r.v.'s.They assumed that the probability density function    / of the random variable   / satisfies the following condition:    / () ≤  ⋅   , for some  > 0,  > 1. ( Our goal is to extend the results of Henriques and Oliveira [5,6] by proving the LDP for a stationary sequence {  } ≥1 of integer-valued r.v.'s satisfying the following additional conditions: (A1)   ,  ≥ 1, are associated; (A2)   ,  ≥ 1, are uniformly bounded; that is, there exists For the definition of associated r.v.'s and their properties we refer the reader to the monographs of Bulinski and Shashkin [7], Oliveira [6], and Rao [8].
The paper is organized as follows.Section 2 presents the LDP in question, its proof, and an example demonstrating the applicability of the result.For convenience while reading, we place the technical lemmas used in the proof in Section 3. At the end, in the Appendix, we recall the Gärtner-Ellis theorem and essential results of Varadhan [9] and Bryc [2] as well as two lemmas on convergence of real sequences.In the proofs we will follow the ideas of Henriques and Oliveira [5,6].

Main Result
Theorem 1.Let {  } ≥1 be a strictly stationary sequence of integer-valued r.v.'s satisfying conditions (A1), (A2), and (A3).Then the sequence of probability measures {  (⋅) = (  / ∈ ⋅)} ≥1 satisfies the large deviation principle with rate  and the rate function Λ * () being Fenchel-Legendre transform of the function which means that Proof.The main tool in proving LDP for dependent r.v.'s is the Gärtner-Ellis theorem (see Appendix) which states that it is enough to verify the existence of limit ( 6) together with the differentiability of Λ to have LDP.
To be more precise, in order to obtain the upper bound (A.1), we need to prove that limit (6) exists, which is shown in Lemma 3 (see Section 3).Actually, we prove even morethe finiteness of ( 6) for all  ∈ R.
With a view to getting the lower bound (A.2), we first need to verify the existence of a more general limit for any real continuous, concave, and bounded from above function  (the class of such functions will be denoted by   (R)).It is presented in Lemma 4, Section 3. In fact, these are continuous and bounded functions ((R)) which is required in limit (A.5).Nevertheless, in order to claim the sole existence of this limit, it suffices to consider the subfamily   (R), since it is well separated (see Definition B.7 and Theorem B.8 in [6]).Further, from (A2), it is easy to see that the distributions of   / are exponentially tight (see Appendix).
As a result, by Lemma A.4, we can claim that the LDP holds with rate  and the rate function However, to make sure that  is the Fenchel-Legendre transform of Λ() defined by ( 6), we are still in need of showing convexity of  (see Lemma A.2). To this end, somewhat unobvious implication is inevitable.
It is shown in Lemma 6.We are already in a position to prove the convexity of , proceeding exactly like in the proof of Theorem 3.20 in [6].
According to Theorem B.2 in [6], the rate function  may be presented in the following form: Hence, which means that  is midconvex (called by some authors Jensen-convex or J-convex).The function  is measurable; thus according to Sierpiński Theorem (see [10], Theorem 9.4.2. and Theorem 5.3.5), it is convex and the proof is completed.
Finally, let us present an example of a sequence satisfying the assumptions of our theorem; for this sequence obviously the results of [5] do not apply.
Example 2. Let {  } ≥1 be a Gaussian sequence with the squared exponential covariance function; that is, This sequence is stationary and associated (positively correlated Gaussian).Define a sequence {  } ≥1 as follows: where  ∈ R is an arbitrary number.The r.v.'s {  } ≥1 inherit the properties of association and stationarity from the sequence {  } ≥1 (we applied the same nonincreasing indicator function to the r.v.'s   which are associated).Furthermore, for  ≥ 1, from the covariance inequality for Gaussian r.v.'s (see [11]) we have Therefore and the binary sequence {  } ≥1 is stationary and fulfills assumptions (A1), (A2), and (A3).

Auxiliary Results
Lemma 3. Let {  } ≥1 be the sequence as in Theorem 1. Then limit (6) exists and is finite for all  ∈ R.
Proof.Except for the steps where discreteness of r.v.'s in question is involved, the proof goes along exactly like in Theorem 3.18 in [6].
Let us now define, for  ∈ [−, ] and  ∈ N, the sequence of continuous functions   : R → R in the following way: so   is absolutely continuous (since it is Lipschitz continuous) and almost everywhere differentiable with |   ()| ≤ .We will now make use of the Newman identity (see [12]) which allows us to express the covariance of two absolutely continuous functions of arbitrary r.v.'s via the covariance of the indicators of these r.v.'s.Let us recall that if  1 and  2 are absolutely continuous functions and ,  are random variables, such that  1 () and  2 () are square-integrable, then Cov ( In light of the above identity, we can write where the last inequality is a consequence of nonpositivity of  and application of the well-known Hoeffding identity stating that By the assumption of stationarity of {  } ∈N , we can bound the above expression in the following way: Hence, we get which, on the basis of uniform boundedness of   ,  ∈ N, yields  (  /)  (( ++ − + )/)  (  /)  (( ++ − + )/) ≥ 1 −  2 ( + )  ()  (+) .

𝜃 ( 𝑛
Therefore it is easy to show that putting  = ( + )/ln Proof.Again, apart from the calculations we conduct with remark that the r.v.'s are integer-valued, the proof goes like in Theorem 3.19 in [6].