Some local limit theorems in probability and number theory

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Some local limit theorems in probability and number theory

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Some local limit theorems in probability and number theory
E. Kowalski
ETH Zu¨rich
September 2012

[Joint work with A. Barbour, F. Delbaen, J. Jacod, A. Nikeghbali]
(J-K-N) Forum Math. 23 (2011), 835–873 (K-N) IMRN 2010 (B-K-N) arXiv 0912.1886 (D-K-N) arXiv 1107.5657 (K-N) J. London Math. Soc. 86 (2012), 291–319
We consider limit theorems in probability theory which have arithmetic incarnations and applications. One basic idea is to find information which lie beyond such universal statements as the Central Limit Theorem.

Example 1
[The Erd¨os-Kac theorem] Consider random variables Nn which are uniformly distributed among integers 1 ≤ k ≤ n. For an integer k ≥ 1, let ω(k) be the number of prime divisors of k. Then
ω(N√n) − log log n l⇒aw N(0, 1). log log n

Example 2

[Selberg’s Normal Limit Theorem]

Consider random variables UT uniformly distributed on [0, T ]. Let

ζ(s) = n−s = s + s +∞ {x }x −s−1dx


s −1


be the Riemann zeta function, meromorphic on C. Then as T → +∞, we have

log |ζ(1/2 + iUT )| l⇒aw N(0, 1). 12 log log T

These two results have the following drawbacks (for certain purposes):
The limit distributions are the same, although the quantities log |ζ(1/2 + it)| and ω(k) are very different; In particular, ω(k) takes discrete values, whereas log |ζ(1/2 + it)| is a continuous quantity, and this distinction is lost; As a consequence, these two theorems do not give much information on the distribution of non-typical values of ω (e.g., of prime powers, such that ω(k) = 1) or of ζ(1/2 + it) (e.g., of zeros of ζ on the critical line).

Refining convergence in law
We attempt to refine convergence in law of normalized sequences Yn − mn
Xn = √ σn
by looking more carefully at the limiting behavior of the characteristic functions ϕn(t) = E(eitYn ) without normalizing. We find that this behavior often contains significant information in addition to a possible Normal Limit Theorem for Xn.

Let n be a random variable counting the number of distinct cycles in a uniformly chosen permutation σ of {1, . . . , n} (e.g., a transposition σ has √ n(σ) = n − 1). One also knows that Xn = ( n − log n)/ log n converges to N(0, 1). But the characteristic function is given exactly by
E(eit n ) = (1 − j −1 + j −1eit )
j =1
from which we can extract information.

The product diverges as n → +∞ for t ∈/ 2πZ. But we can write
E(eit n ) = (1 + (eit − 1)/j )(1 + 1/j)1−eit × exp((eit − 1)Hn)
j =1
where Hn = 1 + 1/2 + · · · + 1/n. The second term is the characteristic function of a Poisson random variable PHn with parameter λ = Hn (recall that in general
P(Pλ = k) = e−λ λkk! for k ≥ 0.)


The first term converges as n → +∞ and in fact

z 1+ j≥1 j

1 −z





Γ(1 + z)

for any z ∈ C (Euler) so that for any t ∈ R, we get

E(eit n ) ∼ exp((eit − 1)Hn) 1 = E(eitPHn ) 1

Γ(eit )

Γ(eit )

as n → +∞.

Remarks on this example

The factorization suggests that there could be a decomposition n = Xn + Yn where Xn l=aw PHn and where Yn is independent of Xn and converges in law to a random variable with characteristic function 1/Γ(eit). But 1/Γ(eit) is not a characteristic function!










We called this type of behavior mod-Poisson convergence with parameters Hn and limiting function 1/Γ(eit). This is a type of Poisson approximation that seems widespread but not much studied. (An exception is an early paper of Hwang with different terminology.)
Limit TheoremsLimit TheoremLawFunctionType