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Mathematics 2018, 6(4), 49; doi:10.3390/math6040049
Large Deviation Results and Applications to the Generalized Cramér Model †
Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, I-56127 Pisa, Italy
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Rome, Italy
Correspondence: [email protected]
The support of INdAM (Fondi GNAMPA) and Università di Pisa (Fondi di Ateneo) is acknowledged. The first version of this paper was written during the staying of the first author at the University Jean Monnet (St. Etienne).
Received: 2 March 2018 / Accepted: 27 March 2018 / Published: 2 April 2018
In this paper, we prove large deviation results for some sequences of weighted sums of random variables. These sequences have applications to the probabilistic generalized Cramér model for products of primes in arithmetic progressions; they could lead to new conjectures concerning the (non-random) set of products of primes in arithmetic progressions, a relevant topic in number theory.
Keywords:arithmetic progressions; first Chebyshev function; products of primes; regularly varying functions; slowly varying functions
The aim of this paper is to prove asymptotic results for a class of sequences of random variables, i.e.,for suitable sequences of real numbers and (see Condition 1 in Section 3) and suitable random independent variables defined on the same probability space . We also present analogue results for the slightly different sequence
More precisely we refer to the theory of large deviations, which gives an asymptotic computation of small probabilities on an exponential scale (see, e.g.,  as a reference on this topic). We recall  as a recent reference on large deviations for models of interest in number theory.
The origin and the motivation of our research rely on the study of some random models similar in nature to the celebrated Cramér model for prime numbers: i.e., what we have called the generalized model (for products of prime numbers in arithmetic progressions). We are not aware of any work where these probabilistic models are studied. Details on these structures will be given in Section 2. Here we only point out that, as the classical probabilistic model invented by Cramér has been used to formulate conjectures on the (non-random) set of primes (see  for details), in a similar way we can draw out conjectures also for the non-random sets of products of primes or products of primes in arithmetic progressions. The large deviation results for the sequences concerning these structures will be given in Corollary 1.
We also remark that the particular form of the sequence (1) is motivated by analogy with the first Chebyshev function, as will be explained in Section 2.
It is worth noting that also some moderate deviation properties can be proved (in terms of suitable bounds on cumulants and central moments) for the centered sequences
Such propositions will not be dealt with in the sequel since, though some specific assumptions must be made in the present setting, these results are in the same direction as those of the paper , where moderate deviations from the point of view of cumulants and central moments are fully investigated.
It should be noted that our results are a contribution to the recent literature on limit theorems of interest in probability and number theory; here, we recall , where the results are formulated in terms of the mod- convergence (see also  where the simpler mod-Gaussian convergence is studied).
We here introduce some terminology and notation. We always set , for , and for all . Moreover, we write
- to mean that ;
- , for , to mean that ;
- , for , to mean that for all integers .
On large deviations.
We refer to  (pages 4–5). Let be a topological space equipped with its completed Borel -field. A sequence of -valued random variables satisfies the large deviation principle (LDP) with speed function and rate function I if the following is true: , and the function is lower semi-continuous.
A rate function I is said to be good if its level sets are compact.
Throughout this paper, we prove LDPs with . We recall the following known result for future use.
Theorem 1 (Gärtner–Ellis Theorem).
Let be a sequence of real valued random variables. Assume that the function defined byexists; assume, moreover, that Λ is essentially smooth (see e.g., Definition 2.3.5 in ) and lower semi-continuous. Then satisfies the LDP with speed function and good rate function defined by
See, e.g., Theorem 2.3.6 in . ☐
The main application of Theorem 1 in this paper concerns Theorem 2, where we have
The LDP in Theorem 3 will instead be proved by combining Theorem 4.2.13 in  with Theorem 2, i.e., by checking the exponential equivalence (see, e.g., Definition 4.2.10 in ) of the involved sequences.
On the generalized Cramér model (for products of primes in arithmetic progressions).
The Cramér model for prime numbers consists in a sequence of independent random variables such that, for every ,
This model can be justified by the prime numbers theorem (PNT), which roughly asserts that the expected density of primes around x is : the cardinality of prime numbers isand, with the words of  (see footnote on p. 6), “the quantity appears here naturally as the derivative of evaluated at ”. Since , another way of stating the PNT is
A first extension of this formula concerns the case of integers n which are products of exactly r prime factors (). More precisely, we consider the setswhere is the number of distinct prime factors of n, and counts the number of prime factors of n (with multiplicity); this means that, letting (by the canonical prime factorization of n) , where are the distinct prime factors of n, we have
A result proved by Landau in 1909 (see, e.g., ) states that the cardinalities and of and respectively verifysee also, e.g., Theorem 437 in  (Section 22.18, page 368) or  (II.6, Theorems 4 and 5). Note that this formula for reduces to Equation (6) when .
Going a little further, for fixed integers a and q, we can consider the sets of products of primes in arithmetic progressions
One can prove (by similar methods as in [10,11]) that, for any a and q with , the cardinalities and of and respectively verifywhere is Euler’s totient function. Notice that, for , we recover the sets of primes in arithmetic progressions, considered for instance in [8,10] II.8, or ; the case is studied in ; the general case is considered in the recent preprint ; for , we recover the sets and the formulas for the model described above.
Therefore, following Cramér’s heuristic, Equation (5), we can define the generalized Cramér model for products of r prime numbers (or products of r prime numbers in arithmetic progression) as a sequence of independent random variables such that
Obviously in Equation (7) we take , where is an integer, such that for ; the definition of for is arbitrary.
Large deviation results for this model will be presented in Corollary 1 as a consequence of Theorem 3 and Remark 2, withthus, the sequences in Equations (1) and (2) becomerespectively. Moreover, by taking into account Remark 3 presented below, the sequences in Equation (9) converge almost surely to 1 (as ).
On the first Chebyshev function.
The first Chebyshev function is defined bywhere the sum is extended over all prime numbers .
Therefore, when considering the classical Cramér model, this function is naturally modeled with (and we obtain the numerator of the first fraction in Equation (9)).
It must be noted that T. Tao, in his blog (see ), considers the same random variable and proves that almost surely one hasfor all (where the implied constant in the notation is allowed to be random). In particular, almost surely one has
It appears clearly that in this setting we have a sequence of the form of Equation (1), with the particular choices and . What we are going to investigate in the sequel is how the sequence of random variables and the two sequences of numbers and must be connected in order to obtain large deviations and convergence results (see also Equations (8) and (9) above).
On slowly and regularly varying functions (at infinity).
Here we recall the following basic definitions. A positive measurable function H defined on some neighborhood of of infinity is said to be slowly varying at infinity (see, e.g., , page 6) if
Similarly, a positive measurable function M defined on some neighborhood of of infinity is said to be regularly varying at infinity of index (see, e.g., , page 18) if
Obviously, we recover the slowly varying case if . Recall the following well-known result for slowly varying functions.
Lemma 1 (Karamata’s representation of slowly varying functions).
A function H is slowly varying at infinity if and only ifwhere and for some (as ).
See, e.g., Theorem 1.3.1 in . ☐
In view of what follows we also present the following results. They are more or less known; but we prefer to give detailed proofs in order to ensure that the paper is self-contained.
Let M be a regularly varying function (at infinity) of index . Then,
It is well-known (see, e.g., Theorem 1.4.1 in ) that we have for a suitable slowly varying function H. Thus, it is easy to check that it suffices to prove the result for the case (namely for a slowly varying function H), i.e.,
By Lemma 1, for all , we havefor . Obviously, (as ). Moreover, for all , we havefor , and (as ); thus,by the arbitrariness of . Thus, Equation (10) holds, and the proof is complete. ☐
Let H be a slowly varying function (at infinity). Then,
By the representation of H in Lemma 1, for all there is an integer such that, for all , we have and . Then, we take , and
The first summand in the right hand side can be ignored since, if we take , for a sufficient high x, we havewhich yields for a suitable constant (and as ). Therefore, we concentrate our attention on the second summand and, by taking into account again the representation of H in Lemma 1, for a sufficiently high x, we have
Moreover,andand the proof is complete by the arbitrariness of . ☐
In this section we present large deviation results for Equations (1) and (2). We start with the case of Poisson distributed random variables (see Theorem 2 and Remark 1), and later we consider the case of Bernoulli distributed random variables (see Theorem 3 and Remark 2). Our large deviation results yield the almost sure convergence to 1 (as ) of the involved random variables (see Remark 3 for details). In particular, the results for Bernoulli distributed random variables can be applied to the sequences of the generalized Cramér model in Equation (9) (see Corollary 1).
In all our results, we assume the following condition.
The sequence is eventually positive; is eventually positive and non-decreasing.
In general, we can ignore the definition of and for a finite number of indices; therefore, in order to simplify the proofs, we assume that and are positive sequences and that is non-decreasing.
We start with the case where are (independent) Poisson distributed random variables.
Theorem 2 (the Poisson case; the sequence in Equation (1)).
Let and be two sequences as in Condition 1. Assume that
Moreover, assume that are independent and for all , where are positive numbers such that
We point out that Equation (12) is satisfied if the sequence is nondecreasing and is the restriction (on ) of a regularly varying function with positive index (at infinity); this is a consequence of Lemma 2.
We apply Theorem 1, i.e., we check that Equation (3) holds with , , and as in Equation (4) (in fact, Equation (3) holds even without assuming (13); however, Equation (13) must be required in order that be a speed function). We remark that
Equation (3) trivially holds for . The proof is divided in two parts: the proof of the upper bound,and that of the lower bound,
We start with the proof of Equation (15). For , we havesince is nondecreasing, and we obtain Equation (15) by letting n go to infinity and by taking into account Equation (14). For , we take and(possibly infinite). Recalling that is nondecreasing and that (it is a consequence of Lemma 2), we have
Using Equation (12), we conclude by letting .
The proof of Equation (16) is similar with reversed inequalities; hence, we only sketch it here. For , we haveand we obtain Equation (16) by letting n go to infinity and by taking into account (14). For , we take and, for defined as above, after some manipulations, we obtain
We conclude by letting (by Equation (12)). ☐
Remark 1 (The Poisson case; the sequence in Equation (2)).
The LDP in Theorem 2 holds also for the sequence in Equation (2) in place of the sequence in Equation (1). In this case we only need to use Condition 1 and to assume Equations (13) and (14), whereas Equations (11) and (12) (which were required in the proof of Theorem 2) can be ignored. For the proof, we still apply Theorem 1, so we have to check that Equation (3) holds with , , and Λ as in Equation (4). This can be easily checked noting thatwhere the limit relation holds by Equation (14).
The next result is for Bernoulli distributed random variables . Here we shall use the concept of exponential equivalence (see, e.g., Definition 4.2.10 in ). The proof is similar to the one of Proposition 3.5 in  (see also Remark 3.6 in the same reference). We point out that it is not unusual to prove a convergence result for Bernoulli random variables starting from a similar one for Poisson random variables and by setting for all ; see, for instance, Lemmas 1 and 2 in .
Theorem 3 (The Bernoulli case; the sequence in Equation (1)).
Let such that for all (recall that as ), and let be independent random variables such that (for all ), where for (the definition of for is arbitrary). Notice thatbecause (as ) by Equations (13) and (14) and, by the Cesaro theorem,
Hence, the assumption of Equation (14) and Theorem 2 are in force for the sequence (in fact, we have Equation (14) with in place of ) and, if we set (for all ), the sequence is indeed an instance of the sequence appearing in the statement of the present theorem since, by construction, and .
The statement will be proved by combining Theorem 4.2.13 in  and Theorem 2 (for the sequence ). This means that we have to check the exponential equivalence conditionwhere
We remark thatby the monotonicity and the nonnegativeness of ; therefore, if we combine Equation (19) and the Chernoff bound, for each arbitrarily fixed , we obtain
Moreover, if we setwe have
The proof will be complete if we show that, for all ,
In fact, by Equations (14) and (21), we deduce from Equation (20) thatand we obtain Equation (17) by letting go to infinity.
Thus, we prove Equation (21). We remark that because (as ). Hence, for all , there exists such that, for all , we have and
Moreover, (as ) by Equation (13) and
Remark 2 (The Bernoulli case; the sequence in Equation (2)).
Remark 3 (Almost sure convergence to 1 of the sequences in Theorems 2 and 3).
Let be either the sequence in Equation (1) or the sequence in Equation (2), where is as in Theorem 2 or as in Theorem 3 (so we also consider Remarks 1 and 2). Then, by a straightforward consequence of the Borel–Cantelli lemma, the sequence converges to 1 almost surely (as ) if
Obviously this condition holds if because are nonnegative random variables. On the other hand, if is not empty, is finite; moreover, because . Then, by the upper bound of the closed set, for all , there exists such that, for all , we have
Thus, again by the Borel–Cantelli lemma, converges almost surely to 1 (as ) if, for all , we have
We conclude with the results for the generalized Cramér model (the sequences in Equation (9)).
Corollary 1 (Application to the sequences in Equation (9)).
In this proof, the sequences in Equation (9) play the roles of the sequences in Equations (1) and (2) in Theorem 3 and Remark 2, respectively. Therefore, we have to check that the hypotheses of Theorem 3 are satisfied. Condition 1 and Equations (11) and (13) and can be easily checked. Moreover, one can also check Equation (12) with ; note that in this case, we have a regularly varying function with index (as ), and is eventually nondecreasing. Finally, Equation (14), which iscan be obtained as a consequence of Lemma 3; in fact, and are restrictions (on ) of slowly varying functions at infinity. ☐
In conclusion, we can say that, roughly speaking, for any Borel set A such that (where is the closure of A), the probabilities and decay exponentially as (as ). Thus, in the spirit of Tao’s remark, we are able to suggest estimations concerning a sort of “generalized” Chebychev function defined by or by . To our knowledge, such estimations are not available for .
Rita Giuliano and Claudio Macci equally contributed to the proofs of the results. The paper was also written and reviewed cooperatively.
Conflicts of Interest
The authors declare no conflict of interest.
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