We consider the persistence probability of a certain fractional Gaussian process M H that appears in the Mandelbrot-van Ness representation of fractional Brownian motion. This process is self-similar and smooth. We show that the persistence exponent of M H exists and is continuous in the Hurst parameter H.Further, the asymptotic behaviour of the persistence exponent for H ↓ 0 and H ↑ 1, respectively, is studied. Finally, for H → 1/2, the suitably renormalized process converges to a non-trivial limit with non-vanishing persistence exponent, contrary to the fact that M 1/2 vanishes.

We consider the one-sided exit problem for (fractionally) integrated random walks and Lévy processes. We prove that the rate of decrease of the non-exit probability -the so-called survival exponent -is universal in this class of processes. In particular, the survival exponent can be inferred from the (fractionally) integrated Brownian motion. This, in particular, extends Sinai's result on the survival exponent for the integrated simple random walk to general random walks with some finite exponential moment. Further, we prove existence and monotonicity of the survival exponent of fractionally integrated processes. We show that this exponent is related to a constant appearing in the study of random polynomials.

We consider the one-sided exit problem for fractional Brownian motion (FBM), which is equivalent to the question of the distribution of the lower tail of the maximum of FBM on the unit interval. We improve the bounds given by and shed some light on the relation to the quantity I studied there.

Abstract-Current Gigabit-class passive optical networks (PONs) evolve into next-generation PONs, whereby high-speed Gb/s time division multiplexing (TDM) and long-reach wavelength-broadcasting/routing wavelength division multiplexing (WDM) PONs are promising near-term candidates. On the other hand, next-generation wireless local area networks (WLANs) based on frame aggregation techniques will leverage physical-layer enhancements, giving rise to Gigabit-class very high throughput (VHT) WLANs. In this paper, we develop an analytical framework for evaluating the capacity and delay performance of a wide range of routing algorithms in converged fiber-wireless (FiWi) broadband access networks based on different next-generation PONs and a Gigabit-class multiradio multichannel WLAN-mesh front end. Our framework is very flexible and incorporates arbitrary frame size distributions, traffic matrices, optical/wireless propagation delays, data rates, and fiber faults. We verify the accuracy of our probabilistic analysis by means of simulation for the wireless and wireless-optical-wireless operation modes of various FiWi network architectures under peer-to-peer, upstream, uniform, and nonuniform traffic scenarios. The results indicate that our proposed optimized FiWi routing algorithm (OFRA) outperforms minimum (wireless) hop and delay routing in terms of throughput for balanced and unbalanced traffic loads, at the expense of a slightly increased mean delay at small to medium traffic loads.Index Terms-Availability, fiber-wireless (FiWi) access networks, frame aggregation, integrated routing algorithms, next-generation passive optical networks (PONs), very high throughput wireless local area network (VHT WLAN).

We derive general results on the small deviation behavior for some classes of iterated processes. This allows us, in particular, to calculate the rate of the small deviations for n-iterated Brownian motions and, more generally, for the iteration of n fractional Brownian motions. We also give a new and correct proof of some results in [21].• to show how the technique can be modified if Y has jumps. This is illustrated by several examples, among them the α-time Brownian motion, previously studied in [21]. Here, we give a correct proof of (a weaker version of) the results from [21].Small deviation problems (also called small ball problems or lower tail probability problems) were studied intensively during recent years, which is due to many connections to other subjects such as the functional law of the iterated logarithm of Chung type, strong limit laws in statistics, metric entropy properties of linear operators, quantization, and several other approximation quantities for stochastic processes. For a detailed account, we refer to the surveys [15] and [13] and to the literature compilation [16].The interest in iterated processes, in particular iterated Brownian motion, started with the works of Burdzy (cf.[6] and [7]). Iterated processes have interesting connections to higher order PDEs, cf. [1] and [22] for some recent results. Small deviations of iterated processes or the corresponding result for the law of the iterated logarithm are treated in [9] (X and Y Brownian motions), [10] (X Brownian motion, Y = |Y ′ | with Y ′ being Brownian motion), [21] (see Section 5 below), [18] (X fractional Brownian motion, Y a subordinator), and, most recently, [19] (X fractional Brownian motion, Y a subordinator, and the sup-norm is taken over a possibly fractal index set).In Section 2, we give general results under the assumption that the small deviation probabilities of X and Y , respectively, are known to some extent and that Y has a continuous modification. The proofs for these results are given in Section 3 and the results are illustrated with several examples in Section 4. In Section 5, we treat examples where Y has jumps, in particular, the so-called α-time Brownian motion, studied earlier in [21]. Finally, we mention some possible extensions and applications of our results and collect some open questions in Section 6. General resultsBefore we formulate our main results, let us define some notation. We write f g or g f if lim sup f /g < ∞, while the equivalence f ≈ g means that we have both f g and g f . Moreover, f g or g f say that lim sup f /g ≤ 1. Finally, the strong equivalence f ∼ g means that lim f /g = 1.We say that a process X is H-self-similar if (X(ct)) d = (c H X(t)) for all c > 0, where d = means that the finite-dimensional distributions coincide. Recall that, for example, fractional Brownian motion with Hurst parameter H is H-self-similar. However, there are many interesting self-similar processes outside the Gaussian framework, e.g. a strictly α-stable Lévy process is 1/α-self-similar ([24], [8], [23]). Let ...