144 research outputs found
On the nonuniform Berry--Esseen bound
Due to the effort of a number of authors, the value c_u of the absolute
constant factor in the uniform Berry--Esseen (BE) bound for sums of independent
random variables has been gradually reduced to 0.4748 in the iid case and
0.5600 in the general case; both these values were recently obtained by
Shevtsova. On the other hand, Esseen had shown that c_u cannot be less than
0.4097. Thus, the gap factor between the best known upper and lower bounds on
(the least possible value of) c_u is now rather close to 1.
The situation is quite different for the absolute constant factor c_{nu} in
the corresponding nonuniform BE bound. Namely, the best correctly established
upper bound on c_{nu} in the iid case is about 25 times the corresponding best
known lower bound, and this gap factor is greater than 30 in the general case.
In the present paper, improvements to the prevailing method (going back to S.
Nagaev) of obtaining nonuniform BE bounds are suggested. Moreover, a new method
is presented, of a rather purely Fourier kind, based on a family of smoothing
inequalities, which work better in the tail zones. As an illustration, a quick
proof of Nagaev's nonuniform BE bound is given. Some further refinements in the
application of the method are shown as well.Comment: Version 2: Another, more flexible and general construction of the
smoothing filter is added. Some portions of the material are rearranged. In
particular, now constructions of the smoothing filter constitute a separate
section. Version 3: a few typos are corrected. Version 4: the historical
sketch is revised. Version 5: two references adde
A Berry-Esseen bound for the uniform multinomial occupancy model
The inductive size bias coupling technique and Stein's method yield a
Berry-Esseen theorem for the number of urns having occupancy when
balls are uniformly distributed over urns. In particular, there exists a
constant depending only on such that \sup_{z \in
\mathbb{R}}|P(W_{n,m} \le z) -P(Z \le z)| \le C \left(
\frac{1+(\frac{n}{m})^3}{\sigma_{n,m}} \right) \quad \mbox{for all $n \ge d$
and $m \ge 2$,} where and are the standardized
count and variance, respectively, of the number of urns with balls, and
is a standard normal random variable. Asymptotically, the bound is optimal up
to constants if and tend to infinity together in a way such that
stays bounded.Comment: Typo corrected in abstrac
Classical and Almost Sure Local Limit Theorems
We present and discuss the many results obtained concerning a famous limit
theorem, the local limit theorem, which has many interfaces, with Number Theory
notably, and for which, in spite of considerable efforts, the question
concerning conditions of validity of the local limit theorem, has up to now no
satisfactory solution. These results mostly concern sufficient conditions for
the validity of the local limit theorem and its interesting variant forms:
strong local limit theorem, strong local limit theorem with convergence in
variation. Quite importantly are necessary conditions, and the results obtained
are sparse, essentially: Rozanov's necessary condition, Gamkrelidze's necessary
condition, and Mukhin's necessary and sufficient condition. Extremely useful
and instructive are the counter-examples due to Azlarov and Gamkrelidze, as
well as necessary and sufficient conditions obtained for a class of random
variables, such as Mitalauskas' characterization of the local limit theorem in
the strong form for random variables having stable limit distributions. The
method of characteristic functions and the Bernoulli part extraction method,
are presented and compared. A second part of the survey is devoted to the more
recent study of the almost sure local limit theorem, instilled by Denker and
Koch. The almost sure local limit theorems established already cover the i.i.d.
case, the stable case, Markov chains, the model of the Dickman function. Our
aim in writing this monograph was notably to bring to knowledge many
interesting results obtained by the Lithuanian and Russian Schools of
Probability during the sixties and after, and which are essentially written in
Russian, and moreover often published in Journals of difficult access
Stein approximation for functionals of independent random sequences
We derive Stein approximation bounds for functionals of uniform random
variables, using chaos expansions and the Clark-Ocone representation formula
combined with derivation and finite difference operators. This approach covers
sums and functionals of both continuous and discrete independent random
variables. For random variables admitting a continuous density, it recovers
classical distance bounds based on absolute third moments, with better and
explicit constants. We also apply this method to multiple stochastic integrals
that can be used to represent U-statistics, and include linear and quadratic
functionals as particular cases
Quantitative de Jong theorems in any dimension
We develop a new quantitative approach to a multidimensional version of the
well-known {\it de Jong's central limit theorem} under optimal conditions,
stating that a sequence of Hoeffding degenerate -statistics whose fourth
cumulants converge to zero satisfies a CLT, as soon as a Lindeberg-Feller type
condition is verified. Our approach allows one to deduce explicit (and
presumably optimal) Berry-Esseen bounds in the case of general -statistics
of arbitrary order . One of our main findings is that, for vectors of
-statistics satisfying de Jong' s conditions and whose covariances admit a
limit, componentwise convergence systematically implies joint convergence to
Gaussian: this is the first instance in which such a phenomenon is described
outside the frameworks of homogeneous chaoses and of diffusive Markov
semigroups.Comment: 40 pages, to appear in: Electronic Journal of Probabilit
Fourth moment theorems on the Poisson space in any dimension
We extend to any dimension the quantitative fourth moment theorem on the
Poisson setting, recently proved by C. D\"obler and G. Peccati (2017). In
particular, by adapting the exchangeable pairs couplings construction
introduced by I. Nourdin and G. Zheng (2017) to the Poisson framework, we prove
our results under the weakest possible assumption of finite fourth moments.
This yields a Peccati-Tudor type theorem, as well as an optimal improvement in
the univariate case. Finally, a transfer principle "from-Poisson-to-Gaussian"
is derived, which is closely related to the universality phenomenon for
homogeneous multilinear sums.Comment: Minor revision. to appear in Electron. J. Proba
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