Multiple Hermite polynomials are an extension of the classical Hermite polynomials for which orthogonality conditions are imposed with respect to r > 1 normal (Gaussian) weights w j (x) = e −x 2 +c j x with different means c j /2, 1 ≤ j ≤ r. These polynomials have a number of properties, such as a Rodrigues formula, recurrence relations (connecting polynomials with nearest neighbor multi-indices), a differential equation, etc. The asymptotic distribution of the (scaled) zeros is investigated and an interesting new feature happens: depending on the distance between the c j , 1 ≤ j ≤ r, the zeros may accumulate on s disjoint intervals, where 1 ≤ s ≤ r. We will use the zeros of these multiple Hermite polynomials to approximate integrals of the form ∞ −∞ f (x) exp(−x 2 + c j x) dx simultaneously for 1 ≤ j ≤ r for the case r = 3 and the situation when the zeros accumulate on three disjoint intervals. We also give some properties of the corresponding quadrature weights. * Work supported EOS project 30889451 and FWO project G.0864.16N
We prove a Central Limit Theorem (CLT) for Martin-Löf Random (MLR) sequences. Martin-Löf randomness attempts to capture what it means for a sequence of bits to be "truly random". By contrast, CLTs do not make assertions about the behavior of a single random sequence, but only on the distributional behavior of a sequence of random variables. Semantically, we usually interpret CLTs as assertions about the collective behavior of infinitely many sequences. Yet, our intuition is that if a sequence of bits is "truly random", then it should provide a "source of randomness" for which CLT-type results should hold. We tackle this difficulty by using a sampling scheme that generates an infinite number of samples from a single binary sequence. We show that when we apply this scheme to a Martin-Löf random sequence, the empirical moments and cumulative density functions (CDF) of these samples tend to their corresponding counterparts for the normal distribution.
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