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Register a new userModerate deviations for Ewens-Pitman exchangeable random partitions
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Consider a population of individuals belonging to an infinity number of types, and assume that type proportions follow the two-parameter Poisson-Dirichlet distribution. A sample of size n is selected from the population. The total number of different types and the number of types appearing in the sample with a fixed frequency are important statistics. In this paper we establish the moderate deviation principles for these quantities. The corresponding rate functions are explicitly identified, which help revealing a critical scale and understanding the exact role of the parameters. Conditional, or posterior, counterparts of moderate deviation principles are also established.
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Gibbs-type random probability measures and the exchangeable random partitions they induce represent an important framework both from a theoretical and applied point of view. In the present paper, motivated by species sampling problems, we investigate some properties concerning the conditional distribution of the number of blocks with a certain frequency generated by Gibbs-type random partitions. The general results are then specialized to three noteworthy examples yielding completely explicit expressions of their distributions, moments and asymptotic behaviors. Such expressions can be interpreted as Bayesian nonparametric estimators of the rare species variety and their performance is tested on some real genomic data.
Let $M_{l,n}$ be the number of blocks with frequency $l$ in the exchangeable random partition induced by a sample of size $n$ from the Ewens-Pitman sampling model. We show that, as $n$ tends to infinity, $n^{-1}M_{l,n}$ satisfies a large deviation principle and we characterize the corresponding rate function. A conditional counterpart of this large deviation principle is also presented. Specifically, given an initial sample of size $n$ from the Ewens-Pitman sampling model, we consider an additional sample of size $m$. For any fixed $n$ and as $m$ tends to infinity, we establish a large deviation principle for the conditional number of blocks with frequency $l$ in the enlarged sample, given the initial sample. Interestingly, the conditional and unconditional large deviation principles coincide, namely there is no long lasting impact of the given initial sample. Potential applications of our results are discussed in the context of Bayesian nonparametric inference for discovery probabilities.
Suppose $Pi$ is an exchangeable random partition of the positive integers and $Pi_n$ is its restriction to ${1, ..., n}$. Let $K_n$ denote the number of blocks of $Pi_n$, and let $K_{n,r}$ denote the number of blocks of $Pi_n$ containing $r$ integers. We show that if $0 < alpha < 1$ and $K_n/(n^{alpha} ell(n))$ converges in probability to $Gamma(1-alpha)$, where $ell$ is a slowly varying function, then $K_{n,r}/(n^{alpha} ell(n))$ converges in probability to $alpha Gamma(r - alpha)/r!$. This result was previously known when the convergence of $K_n/(n^{alpha} ell(n))$ holds almost surely, but the result under the hypothesis of convergence in probability has significant implications for coalescent theory. We also show that a related conjecture for the case when $K_n$ grows only slightly slower than $n$ fails to be true.
Consider the state space model (X_t,Y_t), where (X_t) is a Markov chain, and (Y_t) are the observations. In order to solve the so-called filtering problem, one has to compute L(X_t|Y_1,...,Y_t), the law of X_t given the observations (Y_1,...,Y_t). The particle filtering method gives an approximation of the law L(X_t|Y_1,...,Y_t) by an empirical measure frac{1}{n}sum_1^ndelta_{x_{i,t}}. In this paper we establish the moderate deviation principle for the empirical mean frac{1}{n}sum_1^npsi(x_{i,t}) (centered and properly rescaled) when the number of particles grows to infinity, enhancing the central limit theorem. Several extensions and examples are also studied.
We obtain sharp upper and lower bounds for the moderate deviations of the volume of the range of a random walk in dimension five and larger. Our results encompass two regimes: a Gaussian regime for small deviations, and a stretched exponential regime for larger deviations. In the latter regime, we show that conditioned on the moderate deviations event, the walk folds a small part of its range in a ball-like subset. Also, we provide new path properties, in dimension three as well. Besides the key role Newtonian capacity plays in this study, we introduce two original ideas, of general interest, which strengthen the approach developed in cite{AS}.
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