The tail behaviour of a random sum of subexponential random variables and vectors

Let $\left\{ X,X_{i},i=1,2,...\right\} $ denote independent positive random variables having common distribution function (d.f.) F(x) and, independent of X, let ν denote an integer valued random variable. Using X ₀=0, the random sum Z=∑ _i=0 ^ν X _i has d.f. $G(x)=\sum_{n=0}^{\infty }\Pr\{\nu =n\}F^{n\ast }(x)$ where F ^n?(x) denotes the n-fold convolution of F with itself. If F is subexponential, Kesten’s bound states that for each ε>0 we can find a constant K such that the inequality $$ 1-F^{n\ast }(x)\leq K(1+\varepsilon )^{n}(1-F(x))\, , \qquad n\geq 1,x\geq 0 \, , $$ holds. When F is subexponential and E(1 +ε) ^ν <∞, it is a standard result in risk theory that G(x) satisfies $$ 1 - G{\left( x \right)} \sim E{\left( \nu \right)}{\left( {1 - F{\left( x \right)}} \right)},\,\,x \to \infty \,\,{\left( * \right)} $$ In this paper, we show that (*) holds under weaker assumptions on ν and under stronger conditions on F. Stam (Adv. Appl. Prob. 5:308–327, 1973) considered the case where $ \overline{F}(x)=1-F(x)$ is regularly varying with index –α. He proved that if α>1 and $E{\left( {\nu ^{{\alpha + \varepsilon }} } \right)} < \infty $ , then relation (*) holds. For 0<α<1, it is sufficient that Eν<∞. In this paper we consider the case where $\overline{F}(x)$ is an O-regularly varying subexponential function. If the lower Matuszewska index $\beta (\overline{F})<-1$ , then the condition ${\text{E}}{\left( {\nu ^{{{\left| {\beta {\left( {\overline{F} } \right)}} \right|} + 1 + \varepsilon }} } \right)} < \infty$ is sufficient for (*). If $\beta (\overline{F} )>-1$ , then again Eν<∞ is sufficient. The proofs of the results rely on deriving bounds for the ratio $\overline{F^{n\ast }}(x)/\overline{F} (x)$ . In the paper, we also consider (*) in the special case where X is a positive stable random variable or has a compound Poisson distribution derived from such a random variable and, in this case, we show that for n≥2, the ratio $\overline{F^{n\ast }}(x)/\overline{F}(x)\uparrow n$ as x↑∞. In Section 3 of the paper, we briefly discuss an extension of Kesten’s inequality. In the final section of the paper, we discuss a multivariate analogue of (*).     

Heterodyne frequency measurements of COF2, CD3F,13CD3F, and C2H5I lasers

Heterodyne frequency measurements are reported for twenty optically pumped laser lines from various molecules including COF₂, CD₃F,¹³CD₃F, and C₂H₅I.     

Light on dark laser addressed smectic liquid crystal projection displays

Abstract

Two techniques for producing dense uniform bulk scattering in cells containing smectic A liquid crystals have been examined for use in a large area liquid crystal projection display. Textures obtained using both thermal pulses and by dynamic scattering in the smectic A phase were evaluated for their uniformity and scattering density. The conditions required for optimum scattering are described in terms of the electrical characteristics of the scattering pulses used, the effects of different surface alignment treatments and of the nematic bandwidth of the materials employed. The two techniques are compared with respect to their suitability for large area, high information content, white on black, laser addressed liquid crystal light valves.     

A single trapped atom in front of an oscillating mirror

We investigate the Wigner-Weisskopf decay of a two-level atom in front of an oscillating mirror. This work builds on and extends previous theoretical and experimental studies of the effects of a static mirror on spontaneous decay and resonance fluorescence. The spontaneously emitted field is inherently non-stationary due to the time-dependent boundary conditions and in order to study its spectral distribution we employ the operational definition of the spectrum of non-stationary light due to the seminal work by Eberly and Wódkiewicz. We find a rich dependence of this spectrum as well as of the effective decay rates and level shifts on the mirror-atom distance and on the amplitude and frequency of the mirror’s oscillations. The results presented here provide the basis for future studies of more complex setups, where the motion of the atom and/or the mirror are included as quantum degrees of freedom.     

The first complete identification of a diastereomeric catalyst-substrate (alkoxide) species in an enantioselective ketone hydrogenation. Mechanistic investigations

The enantioselective hydrogenations of the dialkyl 3,3-dimethyloxaloacetate ketone substrates (2, 3, and 4; alkyl = Me, (i)Pr, and (t)Bu, respectively) were catalyzed by [Ru((R)-BINAP)(H)(MeCN)(n)(sol)(3-n)](BF(4)) (1, n = 0-3, sol = THF or MeOH, (R)-BINAP = (R)-2,2'-bis(diphenylphosphino)-1,1'-binaphthyl) in up to 82% ee (R). Reaction of the active catalyst 1 with 1 equiv of substrate (2, 3, or 4) in THF or MeOH solution formed the diastereomeric catalyst-alkoxide complexes [Ru((R)-BINAP)(MeCN)(OCH(CO(2)R)-(C(CH(3))(2)CO(2)R))](BF(4)) (5/6 R = Me, 8/9 R = (i)Pr, and 10 R = (t)Bu, respectively) via hydride addition to the ketone carbonyl carbon and ruthenium addition to oxygen. The absolute configurations at the alkoxide groups ((R)- for the major diastereomers 5, 8, and 10) were determined via cleavage of the ruthenium-alkoxide bond with 1 equiv of HBF(4).OEt(2). The solution structures of the major diastereomer catalyst-alkoxide complexes (5, 8, and 10) were unambiguously determined by variable-temperature NMR spectroscopy. The major diastereomers (5, 8, and 10) had the same absolute configuration as the major product enantiomers from the catalytic hydrogenation of 2, 3, and 4 with 1 as catalyst. The ratio of major to minor alkoxide diastereomers was similar to the ee of the catalytic hydrogenation. The catalyst-alkoxide complexes are formed at temperatures as low as -30 degrees C with no other precursors or intermediates observed by NMR showing that ketone-hydride insertion is likely not the turnover limiting step of the catalytic hydrogenation. Results from the stoichiometric hydrogenolysis of 5/6, 8/9, or 10 indicate that their formation is rapid and only partially reversible prior to the irreversible hydrogenolysis of the ruthenium-oxygen bond. The stereoselectivities of the formation and hydrogenolysis of 5/6, 8/9, and 10 sum up to equal the stereoselectivities of the respective catalytic hydrogenations of 2, 3, and 4. The rates of the hydrogenolysis were consistent with these diastereomers being true catalytic intermediates.     

Reaction of benzilmonohydrazone with S4N4, A correction

Reaction of benzilmonohydrazone(1) with S₄N₄ gives the azine (6) and not the 2-$\text{H}$-imidazole (2); this invalidates the claim that (2) is not the intermediate in the formation of $\text{N}$-benzoyltriphenylimidazole from benzil and ammonia. S₄N₄ converts diphenylketene into thiobenzophenone.     

Low-power nonlinear Fabry-Perot reflection in CdHgTe at 10 µm

Measurements of the intensity dependence of CO2laser radiation reflected from a polished Cd0.23Hg0.77Te etalon at 77 K results in nonlinear features at power levels on the order of 1 mW, in good agreement with a theory which includes bandgap resonant nonlinearities and Auger carrier recombination. A high-reflectivity rear coating enhances these features, giving a characteristic which demonstrates differential optical gain. The effects of heating are discussed in this unusual case in which the electronic and thermal contributions to the intensity dependence have the same sign.     

Stochastically monotone Markov Chains

Summary A real-valued discrete time Markov Chain {X _n} is defined to be stochastically monotone when its one-step transition probability function pr {X _n+1y¦ X _n=x} is non-increasing in x for every fixed y. This class of Markov Chains arises in a natural way when it is sought to bound (stochastically speaking) the process {X _n} by means of a smaller or larger process with the same transition probabilities; the class includes many simple models of applied probability theory. Further, a given stochastically monotone Markov Chain can readily be bounded by another chain {Y _n}, with possibly different transition probabilities and not necessarily stochastically monotone, and this is of particular value when the latter process leads to simpler algebraic manipulations. A stationary stochastically monotone Markov Chain {X _n} has cov(f(X ₀), f(X _n)) cov(f(X ₀), f(X _n+1))0 (n =1, 2,...) for any monotonic function f(·). The paper also investigates the definition of stochastic monotonicity on a more general state space, and the properties of integer-valued stochastically monotone Markov Chains.