A universal strong law of large numbers for conditional expectations via nearest neighbors

For k_n-nearest neighbor estimates of a regression Y on X (d-dimensional random vector X, integrable real random variable Y) based on observed independent copies of (X,Y), strong universal pointwise consistency is shown, i.e., strong consistency P_X-almost everywhere for general distribution of (X,Y). With tie-breaking by indices, this means validity of a universal strong law of large numbers for conditional expectations E(Y|X=x).     

On the stability of sparse convolutions

We give a stability result for sparse convolutions on ${?}^2(G)\times {?}^1(G)$ for torsion-free discrete Abelian groups G such as $\mathbb{Z}$. It turns out, that the torsion-free property prevents full cancellation in the convolution of sparse sequences and hence allows to establish stability, that is, injectivity with an universal lower norm bound, which only depends on the support cardinalities of the sequences. This can be seen as a reverse statement of the Young inequality for sparse convolutions. Our result hinges on a compression argument in additive set theory.     

Nonparametric quantile estimation using importance sampling

Nonparametric estimation of a quantile of a random variable m(X) is considered, where \(m: \mathbb {R}^d\rightarrow \mathbb {R}\) is a function which is costly to compute and X is a \(\mathbb {R}^d\)-valued random variable with a given density. An importance sampling quantile estimate of m(X), which is based on a suitable estimate \(m_n\) of m, is defined, and it is shown that this estimate achieves a rate of convergence of order \(\log ^{1.5}(n)/n\). The finite sample size behavior of the estimate is illustrated by simulated data.     

On the asymptotic behaviour of linear learning processes with partial forgetting

For linear learning processes with partial forgetting a recursion of the form $$X_{n + 1} = \frac{1}{n}\sum\limits_{k = j_n }^n {(B_k X_k + b_k ) with \frac{{j_n }}{n}} \to \alpha \in [0,1)$$ in a real separable Banach spaceB is investigated. Results on a.s. convergence and rate of convergence of (X _n ) (loglog and distributional invariance principle) are obtained under corresponding assumptions on the sequences of arithmetic means of the random elementsb _n andB _n inB andL(B), resp., together with suitable spectral conditions on the (deterministic) Cesàro limit of (B _n ). By the special casej _n ≡1 results on stochastic approximation are comprehended. The limit process in the invariance principles satisfies a stochastic differential equation with delay.     

Spectral,thermal and magnetic studies of 2-chloro-5-nitrobenzoates of rare earth elements(iii)

The conditions of the formation of yttrium(III) and lanthanide(III) 2-chloro-5-nitrobenzoates were studied and their quantitative composition and solubilities in water at 298 K were determined. The compounds are dihydrates and their solubilities are of the order of 10^-3 mol dm^-3. The FTIR, FIR and X-ray spectra for the complexes were recorded. All complexes are crystalline compounds. Their thermal decomposition was studied only in the range 293-523 K since on heating above 523 K the complexes decompose explosively. Hydrated 2-chloro-5-nitrobenzoates of rare earth elements lose two crystallization water molecules in one step. The enthalpy of dehydration process was established and the magnetic moments of the complexes were determined in the range 77-300 K. The compounds obey the Curie-Weiss law. The 4f electrons do not participate in the formation of the Ln-O bonds. This revised version was published online in August 2006 with corrections to the Cover Date.     

Inverspositive Operatoren und Taubersätze in der Erneuerungstheorie

A generalization of ordinary renewal processes has been treated byHinderer and the author: for the stochastic process with values in ₊ the aim is (a,a+) with 0a<, 0< instead of (a, ), and a possible passing over the aim is replaced by a phase of stagnation. In the present paper for the open case of an infinite mean length of the undisturbed steps the relationV(a)/a0(a), with fixed, for the mean waiting timeV(a) until reaching the aim is obtained. The proof uses the theory of inversepositive operators ofCollatz andSchröder. This concept also yields an elementary proof of the so-called elementary renewal theorem. Finally the Tauberian theorem ofIkehara inAgmon's version is generalized; it yields the so-called general renewal theorem in the case of finite mean length of the steps, which is also a corollary of a general Tauberian theorem ofBene.

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Herrn Professor Collatz zum 65. Geburtstag     

Deamidation as a widespread phenomenon in two-dimensional polyacrylamide gel electrophoresis of human blood plasma proteins

The human plasma protein patterns obtained by two-dimensional polyacrylamide gel electrophoresis (2-D PAGE) is a good model system for post-translational modifications because of the existence of several "ladders" of protein spots [Anderson, N. L., Anderson, N. G., Electrophoresis 1991, 12, 883-906], so-called "trains" of spots. Our investigation of several proteins, amongst others beta2-microglobulin and the haptoglobin chains, found the differences in isoelectric points (p/) to be due to deamidation of asparagines. After enzymatic cleavage with endopeptidases in the 2-D polyacrylamide gel, the asparagine and deamidated asparagine containing peptides were separated and quantified by reversed-phase HPLC. In order to separate these peptides, a neutral pH system was established and, as a result, the differences in hydrophobicity of asparagine-containing and deamidated asparagine-containing peptides increased. But how do deamidated asparagines contribute to the observed spot pattern? One spot in the 2-D gel consists of a mixture of protein species with the same number of deamidated asparagines but on different sequence position sites. The difference between the spots in the "ladder" is a growing number of negative charges introduced in the protein by an increasing number of deamidated asparagines. As a consequence, the mass difference between two spots is exactly 1 Da, which is shown in this paper for intact protein masses and the corresponding deamidated peptides.     

Thermal,spectral and magnetic properties of 3,5-dimethoxybenzoates of Co(II), Ni(II) and Cu(II)

The complexes of 3,5-dimethoxybenzoates of Co(II), Ni(II) and Cu(II) have been synthesized as hydrated polycrystalline solids and characterized by elemental analysis, IR, FIR and electronic spectroscopy, magnetic studies and X-ray diffraction measurements. They possess colours typical of the M(II) ions: Cu-blue, Ni-green, Co-pink. The carboxylate groups bind as monodentate or a symmetrical, bidentate chelating or bridging ligands. The thermal stabilities were determined in air. When heated they dehydrate to form anhydrous salts which are decomposed to the oxides of respective metals. The magnetic susceptibilities of the complexes were measured over the range 77-300 K and the magnetic moments were calculated. The results reveal the complexes of Ni(II) and Co(II) to be high-spin complexes and that of Cu(II) to form dimer.     

Thermal, spectral and magnetic characterization of Co(II), Ni(II) AND Cu(II) 4-chloro-2-nitrobenzoates

Physico-chemical properties of 4-chloro-2-nitrobenzoates of Co(II), Ni(II), and Cu(II) were studied. The complexes were obtained as mono- and trihydrates with a metal ion to ligand ratio of 1:2. All analysed 4-chloro-2-nitrobenzoates are polycrystalline compounds with colours depending on the central ions: pink for Co(II), green for Ni(II), and blue for Cu(II) complexes. Their thermal decomposition was studied only in the range of 293–523 K, because it was found that on heating in air above 523 K 4-chloro-2-nitrobenzoates decompose explosively. Hydrated complexes lose crystallization water molecules in one step and anhydrous compounds are formed. The final products of their decomposition are the oxides of the respective transition metals. From the results it appears that during dehydration process no transformation of nitro group to nitrite takes place. The solubilities of analysed complexes in water at 293 K are of the order of 10^–4–10^–2 mol dm^–3. The magnetic moment values of Co²⁺, Ni²⁺ and Cu²⁺ ions in 4-chloro-2-nitrobenzoates experimentally determined at 76–303 K change from 3.89 to 4.82 μ_B for Co(II) complex, from 2.25 to 2.98 μ_B for Ni(II) 4-chloro-2-nitrobenzoate, and from 0.27 to 1.44 μ_B for Cu(II) complex. 4-chloro-2-nitrobenzoates of Co(II), and Ni(II) follow the Curie–Weiss law. Complex of Cu(II) forms dimer.