Moderate deviations for densities in Rk

Let {Xⁱ=(X ₁ ⁱ , ..., X _k ⁱ ), i=1, 2, ...} be a sequence of independent identically distributed random vectors with values in the k-dimensional Euclidean space R^k We assume that X¹ has density p(x) and we denote by the density of the normal law in R^k with mean 0=(0, ..., 0) and covariance matrix E and by P_n(x) the density of the random vector . In the paper one finds conditions which are necessary and sufficient in order to have the relation uniformly with respect to x, x , where.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 119, pp. 7–13, 1982.     

Oxidation of 1-vinyl-2-thiabicyclo[3.2.0]-hept-3-ene

Conclusions Oxidation of 1-vinyl-2-thiabicyclo[3.2. 0]hept-3-ene by hydrogen peroxide leads to 1-vinyl-2-thiabicyclo-[3.2.0]hept-3-ene 2-oxide (cis- and trans-isomers) and 1-vinyl-2-thiabicyclo[3.2. 0]hept-3-ene 2,2-dioxide.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 8, pp. 1883–1885, August, 1984.     

Trifluoroacetylation of divinyl sulfide

Conclusions The electrophilic attack of trifluoroacetic anhydride on divinyl sulfide results in replacing the-olefinic proton of only one double bond to give vinyl (2-trifluoroacetylvinyl) sulfide.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 10, pp. 2402–2403, October, 1981.     

Probabilities of moderate deviations

Let X₁, X₂, ... be a sequence of independent, identically distributed random variables, and let. The rate of convergence of probabilities of the form andis studied for any > ₀ and some r and ₀. Moreover, necessary and sufficient conditions are given that the relations be satisfied uniformly with respect to x in the region 0 x clog n, where and c are some positive constants, and. Local limit theorems are also presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituts im. V. A. Steklova AN SSSR, Vol. 85, pp. 6–16, 1979.     

The mechanism of η(η5:η5-fulvalene)-di-μ-chloro-bis (η5-cyclopentadienyltitanium) formation in the system Cp2TiCl2 : LiAlH4 : Olefinic solvent

Summary The reduction of Cp₂TiCl₂ with LiAlH₄ in olefinic solvents at temperatures above 100° leads to the formation of µ-( ⁵ : ⁵-fulvalene)-di-µ-chloro-bis( ⁵-cyclopentadienyl-titanium) (1) and finally to µ-( ⁵ : ¹⁵-fulvalene)-di-µ-hydrido-bis ( ⁵-cyclopentadienyltitanium). The first step of the reduction includes mainly the formation of the dimer of bis( ⁵-cyclopentadienyl)titanium(III) chloride, (Cp₂TiCl)₂. The mechanism of its conversion into (1) involves the formation of an intermediate bis( ⁵-cyclopentadienyl)titanium(III) hydride species which reacts immediately with the olefinic solvent to give bis( ⁵-cyclopentadienyl)alkyltitanium(III) compounds. The formation of the fulvalene ligand in (1) is explained by the interaction of the alkyltitanium compound with the cyclopentadienyl ligands of (Cp₂TiCl)₂ resulting in hydrogen transfer to the olefinic solvent.