Browsing by Author "Huang, Dongzhou"

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    Two Random Walk Problems
    (2022-04-22) Huang, Dongzhou; Ernst, Philip A.
    Among numerous probabilistic objects, random walk is one of the most fundamental but most favourable. This dissertation concerns two problems related to random walk theory. The first problem regards $d$ independent Bernoulli random walks. We investigate the first “rencontre-time” (i.e. the first time all of the $d$ Bernoulli random walks arrive in the same state) and derive its distribution. Further, relying on the probability generating function, we discuss the probability of the first “rencontre-time” equaling infinity, whose positivity depends on the dimension $d$ and the success-parameters of these $d$ Bernoulli random walks. We then investigate the conditional expectations of the first “rencontre-time” by studying their bounds. In the second problem, we investigate Yule's “nonsense correlation” for two independent Gaussian random walks. The original problem, calculating the second moment of Yule's “nonsense correlation” for two independent Bernoulli random walks, has proved elusive. Relevant work in this topic includes two papers by Ernst et al., with the former first calculating explicitly the second moment of its asymptotic distribution and the latter providing the first approximation to the density of the asymptotic distribution by exploiting its moments up to order 16. We replace the Bernoulli random walks with Gaussian random walks. Relying on the property that the distribution of Gaussian random vector is invariant under orthonormal transformation, we successfully derive the distribution of Yule's “nonsense correlation” of Gaussian random walks. We also provide rates of convergence of the empirical correlation of two independent Gaussian random walks to the empirical correlation of two independent Wiener processes. At the level of distributions, in Wasserstein distance, the convergence rate is the inverse $n^{-1}$ of the number of the data points $n$.
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    Yule’s “nonsense correlation” for Gaussian random walks
    (Elsevier, 2023) Ernst, Philip A.; Huang, Dongzhou; Viens, Frederi G.
    This paper provides an exact formula for the second moment of the empirical correlation (also known as Yule’s “nonsense correlation”) for two independent standard Gaussian random walks, as well as implicit formulas for higher moments. We also establish rates of convergence of the empirical correlation of two independent standard Gaussian random walks to the empirical correlation of two independent Wiener processes.