 Main
COUPLINGS, COMPONENT COUNTING PROCESSES, AND PROBABILISTIC NUMBER THEORY
 Squillace, Joseph Paul
 Advisor(s): Cranston, Michael;
 Kaplan, Nathan
Abstract
Our results are concerned with couplings, component counts of combinatorial objects, and
probabilistic number theory. In the theory of couplings, we are concerned with the general
problem of proving the existence of joint distributions p(i,j) of two discrete random variables M and
N subject to infinitely many constraints of the form p(M=i, N=j)=0. The constraints placed on the joint distributions will require, for many
elements j in the range of N, p(M=i, N=j)=0 for infinitely many values of i in the
range of M, where the corresponding values of i depend on j. To prove the existence of
such joint distributions, we apply a theorem proved by Volker Strassen on the existence of joint
distributions with prespecified marginal distributions. In the case in which N is uniformly
distributed in a combinatorial structure with C_i components of size i, we seek to measure
the amount of dependence in the process (C_i )_{i\le n} by coupling N with a variable M such that
M has Z_i components of size i and the Z_i's are independent, with
sum_{i\le n}(C_i  Z_i)^+ <2.
In the combinatorial example of noncrossing partitions, we provide
two derivations of the probability distribution of the component counts of a uniformly distributed noncrossing partition. Upon applying a bijection between the set of noncrossing partitions
and Dyck paths consisting of upsteps and downsteps, our results specify the joint and marginal distributions of the block counts of the number of consecutive upsteps in a uniformly random chosen Dyck path.
In number theory, we give an analogue of the Erd\"{o}sKac
Theorem by providing a family of integervalued random variables on
{1,\ldots,n} whose number of distinct prime factors
is roughly log\log n+X\cdot\sqrt{\log\log n} for large values
of n, where X is a standard normal variable. Our final result
involves couplings of a Zetadistributed variable. Given s>1 and
n\in\mathbb{N}, consider a Zeta(s)distributed integervalued
random variable with prime factorization Z(s)=\prod_{p}p^{\alpha_{p}(s)}
and the truncation Z_{n}(s)\coloneqq\prod_{p\le n}p^{\alpha_{p}(s)
The prime powers \alpha_{p}(s) are independent with
\alpha_{p}(s)\sim\text{Geometric}(1/p^s),
and we also consider a random variable M(n)=\prod_{p\le n}p^{Z_{p}},
where the Z_{p}'s are independent with Z_{p}\sim\text{Geometric}(1/p).
We apply the concept of pivot mass and a theorem proved by Strassen
in order to prove the existence of couplings of a Zetadistributed random variable Z(s) and M(n)
in which we can make probabilistic divisibility statements of the form "Z_{n}(s) divides M(n)P(n)"
for some random prime P(n)\le n. In particular, we will
prove that for each n\in\mathbb{N} and an integer k\ge 4, there exists an \varepsilon(k)>0
such that when s\in\left(1,1+\varepsilon(k) we can
couple Z(s) and M\left(n\right) such that if Z(s)\le k,
then Z_{n}(s) always divides M(n)P(n)
for some random prime P(n)\le n.
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