1,721,032 research outputs found
A central limit theorem for some generalized martingale arrays
Full text linkLet and be two arrays of real random variables and a Borel function. Define and where is a standard Brownian motion, a standard normal random variable independent of , and and are distribution functions.
Conditions for , in distribution or stably, are given. Among other things, such conditions apply to certain sequences of stochastic integrals, when the quadratic variations of the integrand processes converge in distribution but not in probability. An upper bound for the Wasserstein distance between the probability distributions of and is obtained as well
An inequality for the total variation distance between high-dimensional centered Gaussian laws
No full textWe propose tight bounds for the total variation measure between two centered Gaussian laws, which improve over the existing inequalities. The suggested inequalities have applications in different contexts when a high-dimensional setting is assumed
Some Skorohod-type results
Full text linkLet be a metric space, a Borel function, and a sequence of tight probability measures on . If on , there are -valued random variables , all defined on the same probability space, such that and for all . Moreover, if and only if for each . This result, proved in \cite{PR2023}, is the starting point of this paper. Three types of contributions are provided. First, is replaced by an arbitrary sub--field . Second, the result is applied to some specific frameworks, including equivalence couplings, total variation distances, and the decomposition of cadlag processes with finite activity. Third, following \cite{HMMS}, the result is extended to models and kernels. This extension has a fairly natural interpretation in terms of decision theory, mass transportation and statistics
On the almost sure convergence of sums
Full text linkTwo counterexamples, addressing questions raised in cite{AD} and cite{PZ}, are provided. Both counterexamples are related to chaoses. Let , where the random variables and belong to different chaoses of uniformly bounded degree. It may be that , and , and yet fails to converge to 0 a.s
On the properties of a Takács distribution
No full textThe Takács law – which includes the generalized Dickman law as a special case – arises as a limiting distribution of normalized sums. We provide some properties of this law and a feasible method for computing the probability density and distribution functions of the Takács random variable
A survey on Skorokhod representation theorem without separability
No full textLet be a metric space, a -field of subsets of and a sequence of probability measures on
. Say that admits a Skorokhod representation if, on some probability space, there are random variables with values in such that
\begin{equation*}
X_n\sim\mu_n\text{ for each }n\ge 0\quad\text{and}\quad X_n\rightarrow X_0\text{ in probability}.
\end{equation*}
We focus on results of the following type: has a Skorokhod representation if and only if , where is a suitable distance (or discrepancy index) between probabilities on . One advantage of such results is that, unlike the usual Skorokhod representation theorem, they apply even if the limit law is not separable. The index is taken to be the bounded Lipschitz metric and the Wasserstein distance
Total variation bounds for Gaussian functionals
No full textLet be a centered Gaussian process with continuous paths, and where the are suitable constants. Fix , and and denote by the centered Gaussian kernel with (random) variance . Under an Holder condition on the covariance function of , there is a constant such that
\begin{gather*}
\norm{P\bigl(\sqrt{c_n}\,I_n\in\cdot\bigr)-E\bigl[N_c(\cdot)\bigr]\,}\le k(\beta)\,\Bigl(\frac{a_n}{n^{1+\alpha}}\Bigr)^\beta+\frac{\abs{c_n-c}}{c}\quad\text{for all }n\ge 1,
\end{gather*}
where \norm{\cdot} is total variation distance and the Holder exponent of the covariance function. Moreover, if and , then converges \norm{\cdot}-stably to , in the sense that
\begin{gather*}
\norm{P_F\bigl(\sqrt{c_n}\,I_n\in\cdot\bigr)-E_F\bigl[N_c(\cdot)\bigr]\,}\rightarrow 0
\end{gather*}
for every measurable with . In particular, such results apply to fractional Brownian motion. In that case, they strictly improve the existing results in \cite{NNP16} and provide an essentially optimal rate of convergence
Bayesian nonparametric inference on a Frechet class
No full textLet and be probability spaces and a sequence of random variables with values in . Let be the collection of all probability measures on such that
In this paper, we build some probability measures on . In addition, for each such , we assume that is exchangeable with de Finetti's measure and we evaluate the conditional distribution . In Bayesian nonparametrics, if are the available data, and can be regarded as the prior and the posterior, respectively. To support this interpretation, it suffices to think of a problem where the unknown probability distribution of some bivariate phenomenon is constrained to have marginals and . Finally, analogous results are obtained for the set of those probability measures on with marginal on (but arbitrary marginal on ). That is, we introduce some priors on and we evaluate the corresponding posteriors
Quantitative bounds in the central limit theorem for m-dependent random variables
Full text linkFor each , let be real random variables and . Let be an integer. Suppose is -dependent, , E(X_{ni}^2)<\infty and \sigma_n^2:=E(S_n^2)>0 for all and . Then,
\begin{gather*}
d_W\Bigl(\frac{S_n}{\sigma_n},\,Z\Bigr)\le 30\,\bigl\{c^{1/3}+12\,U_n(c/2)^{1/2}\bigr\}\quad\quad\text{for all }n\ge 1\text{ and }c>0,
\end{gather*}
where is Wasserstein distance, a standard normal random variable and
U_n(c)=\frac{m_n}{\sigma_n^2}\,\sum_{i=1}^{N_n}E\Bigl[X_{n,i}^2\,1\bigl\{\abs{X_{n,i}}>c\,\sigma_n/m_n\bigr\}\Bigr].
Among other things, this estimate of yields a similar estimate of where is total variation distance
Convergence in total variation of random sums
Full text linkThe rate of convergence of random sums, with respect to total variation distance, is investigated for an exchangeable sequence of real random variables and a sequence of random indices, with independent of
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