arXiv subjects
Publications and source records attributed to Youzhou Zhou.
In this paper, we derive an integral representation for the distribution of the number of types $K_n$ in the Ewens-Pitman model. Based on this representation, we also establish precise large deviations and precise moderate deviations for $K_n$. After careful examination, we find that the rate function exhibits a second-order phase transition and the critical point is $\alpha=\frac{1}{2}$.
The two-parameter Poisson-Dirichlet diffusion takes values in the infinite ordered simplex and extends the celebrated infinitely-many-neutral-alleles model, having a two-parameter Poisson-Dirichlet stationary distribution. Here we identify a dual process for this diffusion and obtain its transition probabilities. The dual is shown to be given by Kingman's coalescent with mutation, conditional on a given configuration of leaves. Interestingly, the dual depends on the additional parameter of the stationary distribution only through the test functions and not through the transition rates. After discussing the sampling probabilities of a two-parameter Poisson-Dirichlet partition drawn conditionally on another partition, we use these notions together with the dual process to derive the transition density of the diffusion. Our derivation provides a new probabilistic proof of this result, leveraging on an extension of Pitman's Polya urn scheme, whereby the urn is split after a finite number of steps and two urns are run independently onwards. The proof strategy exemplifies the power of duality and could be exported to other models where a dual is available.
The goal of cryptocurrencies is decentralization. In principle, all currencies have equal status. Unlike traditional stock markets, there is no default currency of denomination (fiat), thus the trading pairs can be set freely. However, it is impractical to set up a trading market between every two currencies. In order to control management costs and ensure sufficient liquidity, we must give priority to covering those large-volume trading pairs and ensure that all coins are reachable. We note that this is an optimization problem. Its particularity lies in: 1) the trading volume between most (>99.5%) possible trading pairs cannot be directly observed. 2) It satisfies the connectivity constraint, that is, all currencies are guaranteed to be tradable. To solve this problem, we use a two-stage process: 1) Fill in missing values based on a regularized, truncated eigenvalue decomposition, where the regularization term is used to control what extent missing values should be limited to zero. 2) Search for the optimal trading pairs, based on a branch and bound process, with heuristic search and pruning strategies. The experimental results show that: 1) If the number of denominated coins is not limited, we will get a more decentralized trading pair settings, which advocates the establishment of trading pairs directly between large currency pairs. 2) There is a certain room for optimization in all exchanges. The setting of inappropriate trading pairs is mainly caused by subjectively setting small coins to quote, or failing to track emerging big coins in time. 3) Too few trading pairs will lead to low coverage; too many trading pairs will need to be adjusted with markets frequently. Exchanges should consider striking an appropriate balance between them.
Volatility clustering is a common phenomenon in financial time series. Typically, linear models can be used to describe the temporal autocorrelation of the (logarithmic) variance of returns. Considering the difficulty in estimating this model, we construct a Dynamic Bayesian Network, which utilizes the conjugate prior relation of normal-gamma and gamma-gamma, so that its posterior form locally remains unchanged at each node. This makes it possible to find approximate solutions using variational methods quickly. Furthermore, we ensure that the volatility expressed by the model is an independent incremental process after inserting dummy gamma nodes between adjacent time steps. We have found that this model has two advantages: 1) It can be proved that it can express heavier tails than Gaussians, i.e., have positive excess kurtosis, compared to popular linear models. 2) If the variational inference(VI) is used for state estimation, it runs much faster than Monte Carlo(MC) methods since the calculation of the posterior uses only basic arithmetic operations. And its convergence process is deterministic. We tested the model, named Gam-Chain, using recent Crypto, Nasdaq, and Forex records of varying resolutions. The results show that: 1) In the same case of using MC, this model can achieve comparable state estimation results with the regular lognormal chain. 2) In the case of only using VI, this model can obtain accuracy that are slightly worse than MC, but still acceptable in practice; 3) Only using VI, the running time of Gam-Chain, in general case, can be reduced to below 5% of that based on the lognormal chain via MC.
In this note, we extend the Vandermonde with Arnoldi method recently advocated by P. D. Brubeck, Y. Nakatsukasa and L. N. Trefethen to dealing with the confluent Vandermonde matrix. To apply the Arnoldi process, it is critical to find a Krylov subspace which generates the column space of the confluent Vandermonde matrix. A theorem is established for such Krylov subspaces for any order derivatives. This enables us to compute the derivatives of high degree polynomials to high precision. It also makes many applications involving derivatives possible, as illustrated by numerical examples. We note that one of the approaches orthogonalizes only the function values and is equivalent to the formula given by P. D. Brubeck and L. N. Trefethen. The other approach orthogonalizes the Hermite data. About which approach is preferable to another, we made the comparison, and the result is problem dependent.
From the Poisson-Dirichlet diffusions to the $Z$-measure diffusions, they all have explicit transition densities. In this paper, we will show that the transition densities of the $Z$-measure diffusions can also be expressed as a mixture of a sequence of probability measures on the Thoma simplex. The coefficients are still the transition probabilities of the Kingman coalescent stopped at state $1$. This fact will be uncovered by a dual process method in a special case where the $Z$-measure diffusions is established through up-down chain in the Young graph.
Schrödinger-Föllmer sampler (SFS) is a novel and efficient approach for sampling from possibly unnormalized distributions without ergodicity. SFS is based on the Euler-Maruyama discretization of Schrödinger-Föllmer diffusion process $$\mathrm{d} X_{t}=-\nabla U\left(X_t, t\right) \mathrm{d} t+\mathrm{d} B_{t}, \quad t \in[0,1],\quad X_0=0$$ on the unit interval, which transports the degenerate distribution at time zero to the target distribution at time one. In \cite{sfs21}, the consistency of SFS is established under a restricted assumption that %the drift term $b(x,t)$ the potential $U(x,t)$ is uniformly (on $t$) strongly %concave convex (on $x$). In this paper we provide a nonasymptotic error bound of SFS in Wasserstein distance under some smooth and bounded conditions on the density ratio of the target distribution over the standard normal distribution, but without requiring the strongly convexity of the potential.
Kingman Coalescent was first proposed by Kingman [7] in population genetics to describe population's genealogical structure. Now it becomes a bench-mark model for coalescent process. Extensive studies have been conducted on Kingman coalescent. In particular, its explicit finite time distribution was obtained by Tavaré [12]. However, very few people use this explicit distribution to do analysis for it is an intractable infinite series. In this article, we are going to establish a complex integral representation for the finite time distribution, then we use steepest descent method to analyze this integral representation to obtain local central limit theorem at small time regime.
Ewens-Pitman model has been successfully applied to various fields including Bayesian statistics. There are four important estimators $K_{n},M_{l,n}$,$K_{m}^{(n)},M_{l,m}^{(n)}$. In particular, $M_{1,n}, M_{1,m}^{(n)}$ are related to discovery probability. Their asymptotic behavior, such as large deviation principle, has already been discussed in [4],[1] and [2]. Moderate deviation principle is also discussed in [3] with some speed restriction. In this article, we will apply complex asymptotic analysis to show that this speed restriction is unnecessary.
The Fleming-Viot process with parent-independent mutation process is one particular neutral population genetic model. As time goes by, some initial species are replaced by mutated ones gradually. Once the population mutation rate is high, mutated species will elbow out all the initial species very quickly. Small time behaviour in this case seems to be the key to understand this fast transition. The small-time asymptotic results related to time scale $\frac{t}θ$ and $a(θ)t$, where $\lim_{θ\to\infty}θa(θ)=0$, are obtained in \cite{MR1815182},\cite{MR1649005}, \cite{MR1887170} and \cite{MR2184086}, respectively. Only the behaviour under the scale $t(θ)$, where $\lim_{θ\to\infty}θa(θ)=\infty$, was left untouched. In this paper, the weak limits under various small time scales are obtained. Of particular interest is the large deviations for the small-time transient sampling distributions, which reveal interesting phase transition. Interestingly, such a phase transition is uniquely determined by some species diversity indices.
The Pitman-Yor process is a random discrete measure. The random weights or masses follow the two-parameter Poisson-Dirichlet distribution with parameters $0<α<1, θ>-α$. The parameters $α$ and $θ$ correspond to the stable and gamma components, respectively. The distribution of atoms is given by a probability $ν$. In this article we consider the limit theorems for the Pitman-Yor process and the two-parameter Poisson-Dirichlet distribution. These include law of large numbers, fluctuations, and moderate or large deviation principles. The limiting procedures involve either $α$ tends to zero or one. They arise naturally in genetics and physics such as the asymptotic coalescence time for explosive branching process and the approximation to generalized random energy model for disordered system.
This paper considers the limiting distribution of $π_{λ,θ}$, the stationary distribution of the infinitely-many-alleles diffusion with symmetric overdominance \cite{MR1626158}. In \cite{MR2519357} the large deviation principle for $π_{λ,θ}$ indicates that there are countably many phase transitions for the limiting distribution of $π_{λ,θ}$, and the critical points are $λ=k(k+1), k\geq1$. The asymptotic behaviours at those critical points, however, are unclear. This article provides a definite description of the critical cases.
In this article, three models are considered, they are the infinitely-many-neutral-alleles model \cite{MR615945}, infinite dimensional diffusion associated with two-parameter Poisson-Dirichlet distribution \cite{MR2596654} and the infinitely-many-alleles model with symmetric dominance \cite{MR1626158}. The new representations of the transition transition densities are obtained for the first two models. Lastly, the ergodic inequalities of these three models are provided.