If a probability density (x) (x ∈ ℝ) is bounded and := ∫e
(x)dx < ∞ for some linear functional u and all ∈ (01), then, for each ∈ (01) and all large enough , the -fold convolution of the -tilted density ˜pt := e
(x)/ is bounded. This is a corollary of a general, “non-i.i.d.” result, which is also shown to enjoy a certain optimality property. Such results and their corollaries stated in terms of the absolute integrability of the corresponding characteristic...
Let X be a compact topological space, and let D be a subset of X. Let Y be a Hausdorff topological space. Let f be a continuous map of the closure of D to Y such that f(D) is open. Let E be any connected subset of the complement (to Y) of the image f(∂D) of the boundary ∂D of D. Then f(D) either contains E or is contained in the complement of E.
Applications of this dichotomy principle are given, in particular for holomorphic maps, including maximum and minimum modulus principles,...
The well-known Bennett–Hoeffding bound for sums of independent random variables is refined, by taking into account positive-part third moments, and at that significantly improved by using, instead of the class of all increasing exponential functions, a much larger class of generalized moment functions. The resulting bounds have certain optimality properties. The results can be extended in a standard manner to (the maximal functions of) (super)martingales. The proof of the main result relies on an...
If a probability density () ( ∈ ℝ) is bounded and
:= ∫e
()d < ∞ for some linear functional and all ∈ (01), then, for each ∈ (01) and all large enough , the -fold convolution of the -tilted density
:= e
()/ is bounded. This is a corollary of a general, “non-i.i.d.” result, which is also shown to enjoy a certain optimality property. Such results and their corollaries stated in terms of the absolute integrability of the corresponding characteristic functions...
It is proved that the best constant factor in the Rademacher-Gaussian tail comparison is between two explicitly defined absolute constants
and
such that
1.01
.
A discussion of relative merits of this result limit theorems is given.
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