Random sieves and generalized leader-election procedures

A random sieve of the set of positive integers N is an infinite sequence of nested subsets N = S 0 ⊃ S 1 ⊃ S 2 ⊃ · · · such that S k is obtained from S k − 1 by removing elements of S k − 1 with the indices outside R k and enumerating the remaining elements in the increasing order. Here R 1 , R 2 , . . . is a sequence of independent copies of an infinite random set R ⊂ N . We prove general limit theorems for S n and related functionals, as n → ∞ .


Introduction and motivation
Let R be an arbitrary random infinite subset of the set of positive integers N := {1, 2, 3, . . .} and (R k ) k∈N independent copies of R. A random sieving of the set N by the set R is an infinite chain of countable sets N =: S 0 ⊃ S 1 ⊃ · · · ⊃ S n ⊃ · · · such that the set S k is obtained from S k−1 , k ∈ N, by removing elements of S k−1 with the indices outside R k and enumerating the remaining elements in the increasing order. Formally, for k ∈ N, if There are several examples of random sieving that were previously analyzed in the literature. where (S k ) k∈N is a standard random walk on N with generic step ξ. The analysis of such a sieving is the subject of the classical renewal theory on positive integers.
Example 1.2. For R being the range of an increasing integer-valued random walk, that is are independent copies of a positive integer-valued random variable, say η, the corresponding random sieving and its connection to classical Galton-Watson processes were analyzed in Alsmeyer, Kabluchko, and Marynych (2016). An important particular case is obtained by choosing η to be geometrically distributed on N with a parameter p ∈ (0, 1). This leads to a well-known classical model called 'leader-election procedure', which has been a subject of active research during last decades; see, for example, Bruss and Grübel (2003); Fill, Mahmoud, and Szpankowski (1996); Grübel and Hagemann (2016); Janson and Szpankowski (1997); Prodinger (1993). The integers in the leader-election procedure are typically viewed as players in a game who independently toss a coin so as to determine whether they stay in the game for the next round or not. Restricted from N to {1, 2, . . . , n} the sieving can be thought of as a procedure of selecting a leader (leaders) from the group of n players, whence the name.
Example 1.3. If R is the set of record's positions in an infinite sample from a continuous distribution the corresponding random sieving was studied in Alsmeyer, Kabluchko, and Marynych (2017).
The above introduced random sieving is intimately connected with several classical models in probability. First of all, let us mention a connection, already observed in Alsmeyer et al. (2016), with the notion of stability of point processes; see Davydov, Molchanov, and Zuyev (2011) and Zanella and Zuyev (2015). Associated with every sieving procedure is the corresponding operator on the space of point processes on [0, ∞). Given a point process X := ∞ k=1 δ X k in [0, ∞) with 0 ≤ X 1 ≤ X 2 ≤ . . ., and an a.s. infinite random set R = {r 1 , r 2 , r 3 , . . .} ⊂ N, we define the thinning of X by R as This random operation transforms X into a 'sparser' point process X • R by removing points of X with indices outside the range of R. In order to compensate such thinning, a second deterministic operation is used for rescaling. Namely, let f be a deterministic function which is 'contractive' in an appropriate sense, and set f (X ) := ∞ k=1 δ f (X k ) . For example, one can take f (x) = ax for some a ∈ (0, 1) or f (x) = log(1 + x). A point process X is called f -stable with respect to thinning by a random set R; see Alsmeyer et al. (2016), if where d = denotes equality in distribution. From this viewpoint a natural problem is to describe the set of point processes which are f -stable with respect to thinning by a random set R. In the setting of Example 1.2, that is, for sieving by a random walk, this problem has been solved in Alsmeyer et al. (2016). To the best of our knowledge no other cases have been addressed so far.
Another well-known probabilistic concept related to random sieves is theory of iterated random function systems; see Diaconis and Freedman (1999) and also Marynych and Molchanov (2021) for sieving procedures of such systems. Classic theory of iterated random function systems is mainly concerned with a contractive random mapping Φ defined on some complete separable metric space. For contractive mappings under mild additional assumptions the sequence of forward iterations Φ k • Φ k−1 • · · · • Φ 1 (x 0 ) converges in distribution, as k → ∞, to a random limit for every starting point x 0 , whereas backward iterations Φ 1 • Φ 2 • · · · • Φ k (x 0 ) converge a.s. The most prominent example of this kind is called perpetuity in which Φ is an affine mapping on R, that is, Φ(x) = Ax + B, x ∈ R with (A, B) being an arbitrary random vector. A criterion for convergence of such iterations can be found in Goldie and Maller (2000). To see the connection with our model one may look at the countable sets R k , k ∈ N, and S k , k ∈ N 0 , as random mappings from N to N or, equivalently, as elements of N ∞ . Taking this point of view, we write and recast equation (1) as follows where • denotes superposition of two mappings defined in the usual way (f • g)(x) = f (g(x)). Upon iterating we write Despite seeming backward form of the process S k in (3), S k , regarded as an element of N ∞ , is actually a k-fold forward iteration of i.i.d. random mappings ϕ k : and applied to the starting point Id, the identity mapping. Thus, In contrast to contractive random mappings, every random sieving of N is, in a sense, expanding. More precisely, for every x ≥ x 0 := x 0 (R), where (nonrandom) x 0 is defined by 1 To see this just note that the left-hand side of (5) is a.s. nondecreasing in n because P{R(x) ≥ x} = 1 and diverges to infinity in probability because P{R(x) > x} > 0 for x ≥ x 0 . In view of (5) it is natural to ask what is a correct normalization of S n (x) = R 1 •R 2 •· · ·•R n (x) ensuring convergence to a nondegenerate limit and how to characterize the limit as a function of x, provided it exists. To get a better feeling of what a possible answer can be, let us look at the aforementioned Examples 1.1, 1.2 and 1.3. In what follows we denote by =⇒ weak convergence of probability measures on R ∞ endowed with the product topology.
In the setting of Example 1.1 R(x) = x + ξ and S n (x) = x + S n , x ∈ N, where S n is a random walk. The question of convergence in distribution of S n (x), as n → ∞, is the most classical topic in probability theory and is fully understood. If existent, the limit of properly centered and/or normalized S n (x) is a stable distribution and does not depend on x. However, as we shall see, this type of behavior is not representative for random sieves of N.
In the setting of Example 1.2 R is the range of a random walk on N. Limit theorems for this type of random sieving were established in Alsmeyer et al. (2016). If Eη log η < ∞, then with µ := Eη we have where (Z f 1 (x)) is a certain random walk with a.s. positive i.i.d. steps; see Theorem 2.1 in Alsmeyer et al. (2016). Moreover, (Z f 1 (x)) satisfies a stochastic fixed-point equation where R on the right-hand side is independent of (Z f 1 (x)) x∈N . On the other hand, assume that the distribution of η has infinite mean and satisfies Davies' assumption; see Davies (1978): for some 0 < α < 1, x 0 ≥ 0, and a nonincreasing, non-negative function γ(x) such that x γ(x) is nondecreasing and ∞ x 0 γ(exp(e x )) dx < ∞. Then Theorem 2.8 in Alsmeyer et al. (2016) says (after exponentiation) that where (Z f 2 (x)) x∈N is the running maximum process of i.i.d. random variables with values in (1, ∞). Here the limit satisfies a stochastic fixed-point equation where again R on the right-hand side is independent of (Z f 2 (x)) x∈N .
In the setting of Example 1.3 R(n) is the position of nth record, n ∈ N, in an infinite sample from a continuous distribution. In this case Theorem 2.8 in Alsmeyer et al. (2017) says that where L n (x) is the n-fold iteration of the function x → log(1 + x) with itself and the limit satisfies a stochastic fixed-point equation where, as before, R on the right-hand side is independent of (Z f 3 (x)) x∈N .
A common feature of limit relations (6), (9) and (11) is that they can be written in a unified way as follows: where is a strictly increasing unbounded concave function given by (9) holds, log(1 + t), if i = 3 and (11) holds.
(13) Furthermore, the fixed-point equations satisfied by the limits take the form The main goal of this paper is to establish general conditions ensuring, for a given random set R, existence of a normalizing function f such that for some limit (Z f (x)) x∈N . The fact that whenever convergence (14) holds the limit should satisfy a stochastic fixed-point equation is obvious. Note that (15) is a particular case of (2) when restricted to point processes on N.
As a byproduct we also derive limit theorems for two related characteristics which describe the speed of sieving: • N Note that the above quantities are connected via duality relations: Let us introduce the following shorthand notation: given a sequence of either deterministic or random functions f n : X → X, where X is an arbitrary set, we put f (k↑n) := f k • · · · • f n and f (n↓k) := f n • · · · • f k for k ≤ n. For n < k, we stipulate that f (k↑n) and f (n↓k) denote the identity map on X. Further, we shall use the notation for the n-fold iteration of f with itself.

Limit theorems for random sieves
Let F be a family of nondecreasing unbounded concave functions f : The key role of the family F R is revealed by Proposition 2.2 below. However, we shall first need to ensure that F R is non-empty.

Lemma 2.1. For an arbitrary infinite random set
A proof of Lemma 2.1 will be given in Section 4.
Proposition 2.2. Let G n be a sigma-algebra generated by the mappings R 1 , R 2 , . . . , R n , n ∈ N, G 0 the trivial sigma-algebra. For every f ∈ F R and x ∈ N, the sequence is a positive supermartingale with respect to the filtration (G n ) n∈N 0 and, thus, converges a.s.
We are ready to formulate our first main result.
Proof. The result follows from Proposition 2.2 upon noticing that for every fixed n ∈ N.
Theorem 2.3 is not completely satisfactory, since the limit (Z f (x)) x∈N might be trivial (a.s. equal to a fixed point of f ) if f ∈ F R is chosen incorrectly. To avoid such trivialities we naturally want to take f ∈ F R 'as large as possible'. To formalize the latter notion, we endow the set F R with a pointwise partial order ⪯: Note that, for every n ∈ N and f ∈ F R , n ≤ R(n) =⇒ f (n) ≤ Ef (R(n)) ≤ n, and, therefore, sup which shows that all the functions in F R are uniformly locally bounded.

It is clear that
Recall that a function f * is called a maximal element of (F R , ⪯) if f * ⪯ g, for some g ∈ F R , implies g ⪯ f * . Let M R be the set of maximal elements in (F R , ⪯). The next proposition, whose proof is postponed to Section 4, demonstrates that the set M R is non-empty.

Proposition 2.4. Every chain in the partially ordered set (F R , ⪯) possesses an upper bound.
Thus, (F R , ⪯) contains at least one maximal element.
The elements of M R seem to be the best candidates for deriving (14) with a non-trivial limit. However, Proposition 2.4 is a result on existence and does not provide any way to find at least one element of M R explicitly. Therefore, we shall formulate the second theorem, which provides us with sufficient conditions for (14) with a non-trivial limit.
Note that every function f in F R is strictly increasing on [0, ∞) and continuous on (0, ∞) . Put x f := f (0+). There exists a unique strictly increasing convex function f ← defined on Extend f ← to a convex function on [0, ∞) by putting f ← (x) = 0, for x ∈ [0, x f ). For n ∈ N and f ∈ F R , introduce a stochastic process X n := (X n (x)) x≥1 defined by where we stipulate R(0) = 0. Note that The processes X n , n ∈ N, are independent but not identically distributed. By Proposition 2.2 The next theorem says, in essence, that if (X n ) n∈N converges uniformly to the identity function, as n → ∞, then the solution Z f (x) satisfies a strong law of large numbers, as x → ∞. The proof will be given in Section 4.
Further, assume that the series which is comprised of independent random variables, converges almost surely. Then In particular, the nondecreasing process Z f is a.s. unbounded.
Remark 2.6. Necessary and sufficient conditions for the a.s. convergence of the series defined in (21) are given by the celebrated Kolmogorov three series theorem. Put The series (21)  Using Theorem 2.5 and duality relations (18) we immediately obtain the following limit theorems for the functionals N  (16) and (17), respectively. For the limit process (Z f (x)) x∈N in Theorem 2.5 define the counting process Furthermore, if x 0 = 1, then for every fixed y > y 0 and x ∈ Z, Remark 2.9. The distribution of X (n↓1) , n ∈ N, can be regarded as a convolution of probability measures on the semigroup of self-maps on [0, ∞) endowed with the composition operation. General results on convergence of convolutions of probability measures on semigroups can be found in (Högnäs and Mukherjea 2013, Section 2.4). However, they do not seem to be applicable in our setting.
As was pointed out to us by the referee, another closely related concept is known in theory of dynamical systems. One can think of the sequence X (n↓1) , n ∈ N, as a random discrete dynamical system which is non-autonomous in a sense that X n 's are not identically distributed. This type of dynamical systems has received some attention in the literature but from a different perspective than considered here; see, for example, Cui and Langa (2017) and (Kloeden, Pötzsche, and Rasmussen 2013, Section 8).

A martingale
Assume that there exists µ > 1 such that ER(x) = µx, for all x ∈ N. In this special case M R contains a function x → µ −1 x. Furthermore, for every fixed x ∈ N, the sequence (µ −n R (n↓1) (x)) is a positive martingale rather than just a positive supermartingale.

Suppose that
and, further, The latter is equivalent to A simple sufficient condition for (23) and (24) is a Marcinkiewicz-Zygmund type strong law for some δ ∈ (0, 1). In particular, if R is the standard random walk with E(R(1)) 1+ε < ∞, for some ε > 0, then (23) and (24) hold true.
Summarizing, under the above assumptions the limit relation (14) holds true with f (x) = µ −1 x.

Records
This model has been treated in details in Alsmeyer et al. (2017). Here we just demonstrate that our Theorem 2.5 is powerful enough to recover the results of Alsmeyer et al. (2017).
Let U 1 , U 2 , . . . be a sequence of independent copies of a random variable with the uniform distribution on [0, 1]. Let R(k) be the index of k-th record in the sample. Thus, According to the next lemma, whose proof will be given in Section 4, the function x → log(1 + x) belongs to F R .
can be checked using the mean value theorem for differentiable functions in conjunction with the fact that see Theorem 2(v) in Gut (1990) for a stronger version of (27). We refer the reader to calculations on pp. 4364-4366 in Alsmeyer et al. (2017) for a derivation of an L 1 -version of (26). Thus, (14) holds true with f (x) = log(1 + x).

Proofs
Proof of Lemma 2.1. We need to find a concave strictly increasing and unbounded function f such that (19) holds. Such a function will be constructed by finding a sequence 0 = t 0 < t 1 < t 2 < · · · such that (t k − t k−1 ) k∈N is nondecreasing and defining f by a linear interpolation of the points (t k , αk), k = 0, 1, 2, . . ., for some α ∈ (0, 1). The resulting function is obviously strictly increasing and unbounded. Furthermore, it is concave, since (t k −t k−1 ) k∈N is assumed nondecreasing.
In order to find (t k ) we argue as follows. Fix α ∈ (0, 1), put Define the sequence (t k ) recursively as follows Then, for every n ∈ N, The proof is complete.
Proof of Proposition 2.4. Let C be a chain (totally ordered subset) in (F R , ⪯). Define a function f * C : [0, ∞) → [0, ∞) as a pointwise supremum: Note that (20) implies that f * C is locally bounded. Obviously, f * C is nondecreasing, unbounded and is an upper bound for the chain C. We need to prove that f * C ∈ F R which amounts to checking that f * C is concave and Ef * C (R(n)) ≤ n, n ∈ N.
We shall first prove concavity. Fix t 1 , t 2 ≥ 0 and ε > 0. Then for some f 1 , f 2 ∈ C. Since C is a chain we have either max(f 1 , f 2 ) = f 1 or max(f 1 , f 2 ) = f 2 . Without loss of generality assume the latter. Then, for every λ ∈ [0, 1], Sending ε → 0+ yields concavity of f * C on [0, ∞). In order to prove (28) note that, for every m ∈ N and every ε > 0, there exists f m ∈ C such that It remains to note that where p n is the index of the ⪯-maximal function in {f 1 , . . . , f n } ⊂ C which exists since C is a chain. Thus, for every fixed n ∈ N, by the monotone convergence theorem, Since ε > 0 is arbitrary, we obtain (28). The second claim of the lemma follows from Zorn's lemma.
We now turn to the proof of Theorem 2.5. We start with an auxiliary lemma.
Proof of Theorem 2.5. The proof consists of several steps.
Step 1. Let us show that, for every fixed n ∈ N, The above formula is equivalent to Proof of Lemma 3.1. We shall rely on Williamson's representation for R, see Eq. (3.4) in Gut (1990), R(1) = 1, R(k + 1) = ⌈R(k)/U k ⌉, k ∈ N, where (U k ) k∈N are i.i.d. random variables with the uniform distribution on [0, 1]. According to this representation, for k ∈ N, Assume that we have proved j≥A+1 log(1 + j) j(j − 1) Then E log(1 + R(k + 1)) ≤ 1 + E log(1 + R(k)), k ∈ N, and the claim of lemma follows by induction. It remains to check (33). To this end, note that The proof of (33) and of the entire lemma is complete.