Variations of Central Limit Theorems and Stirling numbers of the First Kind

We construct a new parametrization of double sequences $\{A_{n,k}(s)\}_{n,k}$ between $A_{n,k}(0)= \binom{n-1}{k-1}$ and $A_{n,k}(1)= \frac{1}{n!}\stirl{n}{k}$, where $\stirl{n}{k}$ are the unsigned Stirling numbers of the first kind. For each $s$ we prove a central limit theorem and a local limit theorem. This extends the de\,Moivre--Laplace central limit theorem and Goncharov's result, that unsigned Stirling numbers of the first kind are asymptotically normal. Herewith, we provide several applications.


Introduction
The central limit theorem is one of the most remarkable theorems in science [Fe45,Fe71,Fi11].From the de Moivre-Laplace theorem in combinatorics and probability, involving binomial distributions, the central limit theorem culminated in a universal law of nature ([KM20], section 3).In this paper, we construct a parametrization of double sequences {A n,k (s)} n,k between A n,k (0) = n−1 k−1 and A n,k (1) = 1 n! n k , where n k are the unsigned Stirling numbers of the first kind.Finally, we apply Harper's method [Ha67] and prove a central limit theorem and a local limit theorem for each s.
Let Z n ∈ {0, 1, . . ., n} denote a random variable with binomial distribution where 0 < p < 1 and q = 1 − p.The central limit theorem by de Moivre-Laplace states, that the normalized random variables Z n , converge in distribution against the standard normal distribution N (0, 1): In asymptotic analysis [Be73,Ca15], one is interested in the asymptotic normality of sequences.Goncharov [Go44,Go62] proved in 1944, that the unsigned Stirling numbers of the first kind n k have this property.More than 20 years later, Harper [Ha67] discovered a more conceptional method, proving also that the Stirling numbers of the second kind are asymptotically normal.When describing Goncharov's proof, Harper writes "Goncharov . . .by brute force torturously manipulates the characteristic functions of the distributions until they approach exp (−x 2 /c), c a positive constant."Recently, Harper's method has been applied by Gawronski and Neuschel [GN13], investing in Euler-Frobenius numbers and by Kahle and Stump [KS19].Note, that in several cases, also other non-gaussian distributions need to be considered (e. g. limiting Betti distributions of Hilbert schemes of n points, where the Gumbel distribution was the right limit distribution, Griffin et al. [GORT22]).
Moreover, related to the topic, since we deal with unimodal sequences, we suggest the analysis of properties of the modes.We utilize a result of Darroch [Da64,Be96] and study the modes of {A n,k (s)}.This is connected to Erdős' proof [Er53] of the Hammersley conjecture [Ha51], related to the peaks of { n k }.For other sequences, we refer to Bringmann et al. [BJSMR19].

Let
n−1 k=0 P hs k (x) with initial value P hs 0 (x) := 1.We are interested in the coefficients of these polynomials: Let n ≥ 1.Then A n,n (s) = (n!) −s and A n,0 (s) = 0. We deduce from [HN21], Example 1 and Example 3 for the values s = 0 and s = 1: (2.1) The unsigned Stirling numbers of the first kind n k , denote the number of all permutations of a set of n elements with exactly k distinct cycles.We refer to Bóna [Bo12].
We mainly focus on s ∈ [0, 1] due to (2.1).Then, we obtain central and local limit theorems for the double sequence {A n,k (s)} n,k .The case s ∈ [−1, 0] can be reduced to the case [0, 1], since 2.1.Central limit theorem.The classical central limit theorem of de Moivre (1738) and Laplace (1812), was developed from the results in probability theory [Fi11,KM20] to a general theorem, without direct reference to concepts as random variable, expected value, and variance.We refer to Feller [Fe45] and Canfield [Ca15] for excellent surveys.
The modern version of the central limit theorem can also be considered as a theorem on the asymptotic normality of a sequence of non-negative numbers in singularity analysis [Be73].In this spirit, we state our first result in the most general form. Theorem Theorem 2.1 is proven with Harper's method [Ha67].The sequences {a n (s)} n and {b n (s)} n are provided by the expected values and variances of a suitable sequence of random variables.Let h s (0) := 0. It is essential, that (2.3) Further, we utilize the Berry-Esseen theorem [Ca15] to control the convergence rate.The corresponding expected values and variances are 2 and µ n (1) = H n , where H n is the nth harmonic number.Moreover, let generally H (r) Therefore, we obtain: .
The standard deviation σ n (s) approaches infinity.
Remarks.a) Theorem 2.2 implies Theorem 2.1.In singularity analysis [Be73,Ca15], one is most interested in the asymptotic behaviors of {a n } n and {b n } n , rather than in the concrete realization, as given in Theorem 2.2.b) The constant be chosen as C = 0.7975 (we refer to [vB72], and the survey article Here we denote by ζ(s) the Riemann zeta function.2.2.Local limit theorem.We refer to Section 4 for an introduction.We prove: Theorem 2.3.Let s ∈ R. Then there exists a universal constant K > 0, so that , 2.3.Peaks and plateaux.The polynomials P hs n (x) are real-rooted.Therefore, a theorem by Newton implies, that the sequence {A n,k (s)} k is unimodal and has two modes at most.Either we have one peak, or a plateau.
In the case s = 0, we have a peak for n odd at k = n+1 2 and a plateau for n even at k = n 2 and n+2 2 .This is obvious, since the A n,k (0) are binomial coefficients.The case s = 1 is more delicate.Let n ≥ 3. Hammersley [Ha51] conjectured in the context of Stirling numbers of the first kind that there is always a peak.This was proved by Erdős [Er53].The proof depends on the fact, that n k k are natural numbers.This allows Erdős to apply special results related to the prime number theorem and certain divisibility properties of the Stirling numbers of the first kind.Our goal is to contribute to the case s ∈ (0, 1) and obtain information for s = 1.But this seems to be very difficult, since in general, the numbers A n,k (s) are not integers.Nevertheless, by utilizing a theorem by Darroch [Da64] we obtain: Theorem 2.4.Let n ≥ 6.Let k 0 ∈ N be any integer associated with s ∈ (0, 1) so that Then the sequence {A n,k (s)} has a peak at k 0 .The number of possible k 0 is given by the number of integers between H n and n+1 2 .

The probabilistic viewpoint on asymptotic normality
We begin with a useful tool from probability.
3.1.The Berry-Esseen theorem [Ca15], theorem 3.2.4.Let X be a random variable.We denote by E(X) and V(X) the expected value and variance of X.
Theorem 3.1.Let X n,k for 1 ≤ k ≤ n be independent random variables with values in {0, 1, . . ., n}.Let µ n,k be the expected values, σ 2 n,k the variances, and where ||f (x)|| R denotes the supremum norm of f on R and C > 0 is a universal constant.This constant can be chosen as C = 0.7975 [vB72].
Let P n (x) = n k=0 a n,k x k be a monic polynomial of degree n with a n,k ≥ 0. Suppose, the roots of P n (x) are real and P n (x) = n k=1 (x + r k ).Harper [Ha67] introduced a triangular array of Bernoulli random variables X n,j with distribution P (X n,j = 0) := r j 1 + r j and P (X n,j = 1) := 1 1 + r j .
Pn(1) .Let X n,j be given.Then We apply Harper's method setting P n (x) = (n!) s P hs n (x).The expected value of Z n (s) is given by µ n (s) and the variance by σ n (s) 2 , as recorded in (2.4) and (2.5).
Lemma 3.2.a) Let s ∈ [0, 1).Then . This indicates that for 0 ≤ s < 1 and arbitrarily small ε > 0, there is an Therefore, for 0 ≤ s < 1 and all ε > 0, there is an N ∈ N, such that for all n ≥ N holds 1 − ε ≤ µn(s) , meaning that the same upper bounds also apply here.
Thus, we obtain Corollary 3.3 states, that the variance approaches infinity.This proves Theorem 2.2.

3.3.
Proof of Theorem 2.1.This follows from Theorem 2.2.The crucial part of successfully applying Harper's method and the Berry-Esseen theorem is that the variance approaches infinity.

Local limit theorem
A double indexed sequence {a n,k } n,k satisfies a local limit theorem on a set S of real numbers provided Canfield [Ca15], section 3.7).We recall the following result due to Bender [Be73].
Theorem 4.1 (Bender).Suppose, that the {a (n, k)} k for n ∈ N are asymptotically normal, and σ 2 n → ∞.If for each n the sequence {a (n, k)} k is log-concave in k, then {a (n, k)} k satisfies a local limit theorem on S = R.
4.1.Pólya frequency sequences and limit theorems.We follow the excellent survey by Pitman [Pi97] and apply several results to the sequences {A n,k (s)} k .
Proof of Theorem 2.3.It follows from our previous considerations, that {A n,k (s)} k is a Pólya frequency sequence for all s ∈ R.This implies the theorem.

Peaks of {A n,k (s)} k
We recall Darroch's theorem [Da64].Let (a 0 , a 1 , . . ., a n ) be a Pólya frequency sequence.Let p(x) := n k=0 a k x k with p(1) > 0. Let µ n := p ′ (1) p(1) .Newton's theorem implies, that the sequence {a k } k is unimodal and has two modes at most.Darroch proved, that the modes have distance less than 1 from µ n .Armed with the results of the previous sections we have: Proof of Theorem 2.4.We consider the polynomials P hs n (x).Then µ n (s) = 1 + n−1 k=1 1 1+k s .Suppose, that µ n (s) is an integer.Then {A n,k (s)} k has a peak.Let us restrict µ n to [0, 1].In this case It construes that µ n is differentiable and strictly monotone for n ≥ 3. We have Therefore, µ n is bijective.Let n ≥ 6.Then µ n (0) − µ n (1) > 1.This implies that integers k 0 ∈ µ n (1), µ n (0) exist and are realizable by suitable s ∈ (0, 1): Let such an s be given.Then {A n,k (s)} k has a peak at k 0 .
We add the following approximation of µ n (s).

Table 1 .
[Wi93]able1.Let m n (1) be the unique mode of {A n,k (1)}, as proven by Erdős.Modes for s = 1 and related values.Approximation of Stirling numbers of the first kind.Wilf[Wi93]contributed to the asymptotic behavior of Stirling numbers of the first kind.Several asymptotic formulas are provided.Wilf also compares his results with Jordan's formula in the case We have µ 100 (1) ≈ 5.19 and σ 100 (1) ≈ 1.88477.Wilf considers the value of 100 5 /99! (Table2).

Table 2 .
[Da64,Pi97]roximations of the maximal value for n = 100 (see Wilf[Wi93], page 349 for details).6.2.One mode property of the Stirling numbers of the first kind.Let n ≥ 3. Erdős[Er53]proved that n k k has one mode.We give a new proof for infinitely many n.A variant of Darroch's theorem[Da64,Pi97]states: Let {a k } k be a Pólya frequency sequence.Let µ n = p ′ n (1)