Tail Sum Research Articles

Wittmann's Law of Iterated Logarithm for Tail Sums of B-Valued Random Variables

We present an analogue of Wittmann's law of iterated logarithm (LIL) for tail sums of independent B-valued random variables by using the isoperimetric method and give the precise value of the upper limit for the LIL for tail sums.

SECOND-ORDER REGULAR VARIATION AND THE DOMAIN OF ATTRACTION OF STABLE DISTRIBUTIONS

We characterize second order regular variation of the tail sum of F together with a balance condition on the tails interms of the behaviour of the characteristic function near zero.

Structure Control of Synthetic Bilayer Membranes from Single-Chain Amphiphiles Containing the Schiff Base Segment: I. Conformation Control and Spectral Characterization

Fifteen single-chain ammonium amphiphiles with a rigid Schiff base segment, CnSBCmN+(m= 4, 6, 10;n= 4, 7, 10, 12, 16), were prepared, and their aggregational behaviors in aqueous dispersions were investigated by electron microscopy, differential scanning calorimetry and electronic absorption spectra. The agreement of absorption spectra between aqueous dispersions and cast films, and the temperature-dependent spectral features, have been employed as criteria for bilayer formation. The results showed that except for the five short-chain-lengthed amphiphiles with the sum of tail and spacer not larger than C14, stable bilayers were obtainable for the other amphiphiles. Through the spectral assignments of the planar and nonplanar conformations of the Schiff base segment in the monomer spectra, the electronic absorption spectra of the stable bilayer assemblies were studied. It was observed systematically for the first time that with increasing the alkyl chain length, the spectra change from a single peak at ca. 286 nm characteristic of the loose packing of the nonplanar Schiff base segments in the bilayers, cascading through a doublet band with a 286-nm absorption overlapped by a 330-nm band, which is indicative of the coexistence of the two types of conformations in the bilayer matrix, finally to the fine structured band along with the 286-nm absorption representing the strong aggregation of long-chain-lengthed amphiphiles with the sum of tail and spacer larger than C20. Therefore, the conformation control and spectral characterization of the supramolecular systems containing a nonplanar chromophore have been established.

Structure Control of Synthetic Bilayer Membranes from Single-Chain Amphiphiles Containing the Schiff Base Segment: II. pH and Temperature Dependence of the Aggregational Behaviors

The aggregational behaviors in aqueous solutions of the single-chain ammonium amphiphiles with a rigid Schiff base segment, the diphenylazomethine chromophore, have been investigated by electronic absorption spectroscopy under various chemical (pH) and physical (temperature) environments. pH dependence is not observed for amphiphiles with the sum of tail and spacer not larger than C14. As for bilayer-forming amphiphiles, different pH-dependent spectral features take place at temperatures below and above the phase transition temperature (Tc) of the corresponding bilayer membranes at neutral environment. Basic environment favors the planar conformation of the chromophores in the whole temperature range investigated, and its influence diminishes with increasing alkyl chain length. On the opposite site of pH, i.e., under the strongly acidic condition with a pH of 1.2, a new absorption band attributed to the protonated species appears below room temperature, while it does not show itself at temperature aboveTc. Temperature dependence of the absorption spectra under the strongly acidic condition (pH < 3) reveals that a deprotonation on the amphiphiles takes place at a temperature belowTc. In basic solution (pH > 10.6), planar conformation increases with raising temperature, so leading to the appearance of a strong 330-nm absorption. Under intermediate pHs (3 < pH < 10.6) the bilayer aggregational behaviors are relatively stable. The observed spectral changes with pH were fully reversible.

A complementary note on the supercritical birth, death and catastrophe process

A central limit theorem for the population size of the super-critical linear birth and death process with a linear catastrophe component has previously been obtained under a fourth order moment condition on the increment distribution. In this note we show that this result is valid under a second order moment condition, and that no lesser condition will suffice. This is accomplished by giving a new, self-contained and simple proof of the asymptotic normality of a certain tail sum of independent variates.

Orientation and Spectral Characteristics of the Azobenzene Chromophore in the Ammonium Bilayer Assembly

AbstractAzobenzene‐containing, single‐chain ammonium amphiphiles in which the alkyl chain length in the tail and spacer portions was systematically varied were prepared, and their aggregation behavior in water was studied by electron microscopy, differential scanning calorimetry and absorption spectroscopy. These amphiphiles exist as molecular dispersions, hydrated microcrystals, micelles, or bilayers. The formation of stable bilayers was deduced on the basis of three criteria which include the spectral agreement between aqueous dispersions and the corresponding cast films. Stable bilayers were obtainable when the sum of tail and spacer is not smaller than C14 to C16, and the tail is at least C8. Double‐chain ammonium amphiphiles which possess an octadecyl chain in addition to the azobenzene‐containing chain were prepared and shown to form stable bilayers. The absorption spectrum reflected the aggregate structure. In particular, the bilayer spectra were classified into four types: type A (λmax ca. 300 nm) for the parallel chromophore orientation (H aggregate), type B (330 – 340 nm) for the dimeric chromophore, type C (ca. 355 nm) for the isolated chromophore, and type D (360–390 nm) for the head‐to‐tail chromophore orientation (J‐like aggregate). Liquid‐crystalline bilayers gave type B spectra, while crystalline bilayers gave either of types A, B, and D, depending on the alkyl chain length. Finally, various examples of spectral control were discussed in relation to the present results.

A Log Log Improvement to the Riemann Hypothesis for the Hawkins Random Sieve

This paper is concerned with the Hawkins random sieve which is a probabilistic analogue of the sieve of Eratosthenes. Analogues of the prime number theorem, Mertens' theorem and the Riemann hypothesis have previously been established for the Hawkins sieve. In the present paper we give a more delicate analysis using iterated logarithm results for both martingales and tail sums of martingale differences to deduce a considerably improved $\log\log$ replacement for the Riemann hypothesis result.

Tail sums of convergent series of independent random variables

Let X1, X2, … be a sequence of independent random variables such that, for each n ≥ 1, EXn = 0 and and assume that then converges almost surely as N → ∞. Let and let Fn(x) denote the distribution function of Xn. Loynes (2) observed that the sequence {Sn} is a reversed martingale, and applied his central limit theorem to it: however, stronger results are obtainable, in precise duality with the classical theory of partial sums of independent random variables. These results describe the fluctuations of the sequence {Sn}, and hence the way in which converges to its limit.