Abstract

We present sufficient conditions under which the sequence of arithmetic means , where , is the partial sum built on a stationary sequence of associated integer-valued and uniformly bounded random variables, which satisfy the large deviation principle.

1. Introduction and the Notation

Let be a strictly stationary sequence of square-integrable random variables (r.v.’s). Denote , , and . Consider a sequence satisfying the strong law of large numbers (SLLN) and the central limit theorem (CLT), that is, . 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 , the probability tends to zero with exponential rate; that is, 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 is finite for all , then for all , where which means that is the Fenchel-Legendre transform of the function .

In general setting, we follow [1] to give the definition of the rate function as the function defined on a Polish space , taking values in closed half line and satisfying three conditions:(i) is not identically infinite;(ii) is lower semicontinuous;(iii) is compact for all .

Let us further recall that it is said that the sequence of probability measures 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;(D2), for all closed ;(D3), for all open .It is also well known that the rate function corresponding to the sequence 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 is exponentially tight (see Appendix) and for all continuous and bounded functions defined on the Polish space (denoted by ) the limit exists, then 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 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:

Our goal is to extend the results of Henriques and Oliveira [5, 6] by proving the LDP for a stationary sequence of integer-valued r.v.’s satisfying the following additional conditions:(A1), , are associated;(A2), , are uniformly bounded; that is, there exists such that ;(A3) for some .

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].

2. Main Result

Theorem 1. Let be a strictly stationary sequence of integer-valued r.v.’s satisfying conditions (A1), (A2), and (A3). Then the sequence of probability measures satisfies the large deviation principle with rate and the rate function 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 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 more—the finiteness of (6) for all .
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 ). It is presented in Lemma 4, Section 3. In fact, these are continuous and bounded functions which is required in limit (A.5). Nevertheless, in order to claim the sole existence of this limit, it suffices to consider the subfamily , 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 defined by (6), we are still in need of showing convexity of (see Lemma A.2). To this end, somewhat unobvious implication is inevitable. If are such that, for all , for , then 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: Since for there exists such that is immersed in , it is enough to write Let us now take such that . As the assumption of Lemma 6 is satisfied, we have 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 be a Gaussian sequence with the squared exponential covariance function; that is, This sequence is stationary and associated (positively correlated Gaussian). Define a sequence as follows: where is an arbitrary number. The r.v.’s inherit the properties of association and stationarity from the sequence (we applied the same nonincreasing indicator function to the r.v.’s which are associated). Furthermore, for , from the covariance inequality for Gaussian r.v.’s (see [11]) we have Therefore and the binary sequence is stationary and fulfills assumptions (A1), (A2), and (A3).

3. Auxiliary Results

Lemma 3. Let be the sequence as in Theorem 1. Then limit (6) exists and is finite for all .

Proof. The proof is nearly rewritten from Theorem 3.16 in [6]. For , , by assumption (A1), we have . Since the sequence of r.v.’s , , is stationary, we obtain . Denoting , by Lemma A.5 we get the existence of (6). It remains to verify its finiteness for all . By assumption (A2), we have Thus, for arbitrarily chosen , is finite.

Lemma 4. Let be the sequence as in Theorem 1. Then, for every real continuous, concave, and bounded from above function , the limit exists.

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].
Without loss of generality, we may and do assume that for and for some . Since is continuous and concave, it is a Lipschitz function with constant , for example. By assumption (A2), we get that for Thus, since is Lipschitz, By assumed concavity of , this implies that which is equivalent to When we denote , the above inequality takes the following form: We will now aim at finding the bound for the second term at the r.h.s. of inequality (24).
Let us now define, for and , the sequence of continuous functions in the following way: Each is thus a polyline with vertices having first coordinate of the form , , and second coordinate in the interval . Now, since is nonpositive and Lipschitz with constant , we have 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 and are absolutely continuous functions and , are random variables, such that and are square-integrable, then 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 , we can bound the above expression in the following way: Hence, we get which, on the basis of uniform boundedness of , , yields Let us now define the following quantity: , and restate (32): Going back to inequality (24), we obtain Choose , where is the constant taken from assumption (A3). By Lemma A.6, Therefore it is easy to show that putting , for sufficiently large , we have As a result, from inequality (34), we arrive at and by Lemma A.5 we conclude that the limit exists and the proof is completed.

Lemma 5. Let and be integer-valued, associated, square-integrable r.v.’s; then for any and .

Proof. The proof follows from the covariance inequality obtained in [13] for integer-valued associated r.v.’s , .

Lemma 6. Let be the sequence as in Theorem 1. Let also be such that, for all , Then,