We derive explicitly the coupling property for the transition semigroup of a Lévy process and gradient estimates for the associated semigroup of transition operators. This is based on the asymptotic behaviour of the symbol or the characteristic exponent near zero and infinity, respectively. Our results can be applied to a large class of Lévy processes, including stable Lévy processes, layered stable processes, tempered stable processes and relativistic stable processes.

We present a practical high-speed quantum random number generator, where the timing of single-photon detection relative to an external time reference is measured as the raw data. The bias of the raw data can be substantially reduced compared with the previous realizations. The raw random bit rate of our generator can reach 109 Mbps. We develop a model for the generator and evaluate the min-entropy of the raw data. Toeplitz matrix hashing is applied for randomness extraction, after which the final random bits are able to pass the standard randomness tests.

BACKGROUND: Recent genome-wide association studies of lung cancer have shown that the CHRNA5-A3 region on chromosome 15q24-25.1 is strongly associated with an increased risk of lung cancer and nicotine dependence, and is thought to be associated with chronic obstructive pulmonary disease as well. However, it has not been established whether the association between genetic variants and lung cancer risk is a direct one or one mediated by nicotine dependence. METHODS: The authors applied a rigorous statistical approach, mediation analysis, to examine the mediating effect of smoking behavior and self-reported, physician-diagnosed emphysema (chronic obstructive pulmonary disease [COPD]) on the relation between the CHRNA5-A3 region genetic variant rs1051730 and the risk of lung cancer. RESULTS: Our results showed that rs1051730 is directly associated with lung cancer risk, but it is also associated with lung cancer risk through its effect on both smoking behavior and COPD. Furthermore, we showed that COPD is a mediating phenotype that explains part of the effect of smoking behavior on lung cancer. Our results also suggested that smoking behavior is a mediator of the relation between rs1051730 and COPD risk. CONCLUSIONS: Smoking behavior and COPD are mediators of the association between the single nucleotide polymorphism (SNP) rs1051730 and the risk of lung cancer. Also, COPD is a mediator of the association between smoking behavior and lung cancer. Finally, smoking behavior also has mediating effects on the association between the SNP and COPD. Cancer 2010;116:3458-62.

We establish the exponential convergence with respect to the L 1 -Wasserstein distance and the total variation for the semigroup corresponding to the stochastic differential equationwhere (Z t ) t≥0 is a pure jump Lévy process whose Lévy measure ν fulfillsfor some constant κ 0 > 0, and the drift term b satisfies that for anywith some positive constants K 2 , l 0 and positive measurable function Φ 1 . The method is based on the refined basic coupling for Lévy jump processes. As a byproduct, we obtain sufficient conditions for the strong ergodicity of the process (X t ) t≥0 .where Φ 1 and Φ 2 are two nonnegative measurable functions, and l 0 ≥ 0 is a constant. For example, when Φ 2 (r) = K 2 r for some positive constantThis holds if the drift term b is dissipative outside some compact set. In particular, when Φ 1 (r) = K 1 r for some constant K 1 ≥ 0, it follows from (1.3) that for anywhich, along with (1.3), yields that the SDE (1.1) has a non-explosive and pathwise unique strong solution, see [1, Chapter 6, Theorem 6.2.3] (in the standard Lipschitz case) or [11, Theorem 2] and [25, Chapter 3, Theorem 115] (in the one-sided Lipschitz case). Note that, sincewe are sometimes concerned with only measurable drift term b, non-Lipschitz condition like B(K 1 , K 2 r, l 0 ) will also be adopted in our results below. The reader can refer to [8,20,22,26,32] and references therein for recent studies on the existence and uniqueness of strong solution to (1.1) with non-regular drift term. In particular, assuming that Z is the truncated symmetric α-stable process on R d with α ∈ (0, 2), and b is bounded and β-Hölder continuous with β > 1 − α/2, it was proved in [8, Corollary 1.4(i)] that the SDE (1.1) has a unique strong solution for each x ∈ R d . Furthermore, in the one-dimensional case, if α > 1, then the SDE (1.1) also enjoys a unique strong solution for each x ∈ R, even if the drift b is only bounded and measurable (see [26, Remark 1, p. 82]). Denote by ν the Lévy measure of the pure jump Lévy process Z. We assume that there is a constant κ 0 > 0 such that(1.4)Condition (1.4) was first used in [23] to study the coupling property of Lévy processes. It is satisfied by a large class of Lévy measures. For instance, iffor some z 0 ∈ R d and some ε > 0 such that ρ 0 (z) is positive and continuous on B(z 0 , ε), then such Lévy measure ν fulfills (1.4), see [24, Proposition 1.5] for details. Actually, as shown in Proposition 6.5, the condition (1.4) implies that there is a nonnegative measurable function ρ on R d such that ν(dz) ≥ ρ(z) dz andLet (P t ) t≥0 be the transition semigroup associated with the process (X t ) t≥0 . In this paper we are interested in the asymptotics of the Wasserstein-type distances (including the L 1 -Wasserstein distance and the total variation) between probability distributions δ x P t = P t (x, ·) and δ y P t = P t (y, ·) for any x, y ∈ R d , when the drift term b is dissipative outside some compact set, i.e. b satisfies B(Φ 1 (r), K 2 r, l 0 ) for some positive measurable function Φ 1 , and some const...