Abstract:
West African Ilisha was the third most abundant species of Clupeidae off Benin coastal waters after Sardinella spp and Ethmalosa frimbriata. The fishing effort of these fisheries increased with a dominance of small-sized specimens in the catches. This paper allowed updating some demographic parameters and the exploitation rate of Ilisha africana collected between July 2013 and June 2014 from the coastal waters of Benin for management and conservation of these fisheries. The growth pattern showed a negative allometric growth with an abundance of small-sized specimens. The von Bertalanffy growth function (VBGF) estimations were: L_{∞}= 21.31 cm standard length; K = 1.20 year^{-1}; and t_{0} = -0.138 year. The total mortality rate (Z), natural mortality rate (M) and fishing mortality rate (F) were 4.040 year^{-1}, 2.27 year^{-1} and 1.77 year^{-1} respectively. The Z/K ratio was 3.667 and the exploitation rate (E = F/Z) was 0.44 showing an under exploitation of this species. The estimated potential longevity (t_{max}) was 2.5 years. In addition, the fisheries management should be devolved from the state to the local level to compel fishermen to take greater responsibility for the sustainability and conservation of the fisheries such as size-limit regulation by gradually increasing fishing gears mesh size.

Abstract:
This paper answers a question of \'{E}mery [In S\'{e}minaire de Probabilit\'{e}s XLII (2009) 383-396 Springer] by constructing an explicit coupling of two copies of the Bene\v{s} et al. [In Applied Stochastic Analysis (1991) 121-156 Gordon & Breach] diffusion (BKR diffusion), neither of which starts at the origin, and whose natural filtrations agree. The paper commences by surveying probabilistic coupling, introducing the formal definition of an immersed coupling (the natural filtration of each component is immersed in a common underlying filtration; such couplings have been described as co-adapted or Markovian in older terminologies) and of an equi-filtration coupling (the natural filtration of each component is immersed in the filtration of the other; consequently the underlying filtration is simultaneously the natural filtration for each of the two coupled processes). This survey is followed by a detailed case-study of the simpler but potentially thematic problem of coupling Brownian motion together with its local time at $0$. This problem possesses its own intrinsic interest as well as being closely related to the BKR coupling construction. Attention focusses on a simple immersed (co-adapted) coupling, namely the reflection/synchronized coupling. It is shown that this coupling is optimal amongst all immersed couplings of Brownian motion together with its local time at $0$, in the sense of maximizing the coupling probability at all possible times, at least when not started at pairs of initial points lying in a certain singular set. However numerical evidence indicates that the coupling is not a maximal coupling, and is a simple but non-trivial instance for which this distinction occurs. It is shown how the reflection/synchronized coupling can be converted into a successful equi-filtration coupling, by modifying the coupling using a deterministic time-delay and then by concatenating an infinite sequence of such modified couplings. The construction of an explicit equi-filtration coupling of two copies of the BKR diffusion follows by a direct generalization, although the proof of success for the BKR coupling requires somewhat more analysis than in the local time case.

Abstract:
Benjamini, Burdzy and Chen (2007) introduced the notion of a shy coupling: a coupling of a Markov process such that, for suitable starting points, there is a positive chance of the two component processes of the coupling staying a positive distance away from each other for all time. Among other results, they showed no shy couplings could exist for reflected Brownian motions in C^2 bounded convex planar domains whose boundaries contain no line segments. Here we use potential-theoretic methods to extend this Benjamini et al. result (a) to all bounded convex domains (whether planar and smooth or not) whose boundaries contain no line segments, (b) to all bounded convex planar domains regardless of further conditions on the boundary.

Abstract:
Consider the following random spatial network: in a large disk, construct a network using a stationary and isotropic Poisson line process of unit intensity. Connect pairs of points using the network, with initial / final segments of the connecting path formed by travelling off the network in the opposite direction to that of the destination / source. Suppose further that connections are established using "near-geodesics", constructed between pairs of points using the perimeter of the cell containing these two points and formed using only the Poisson lines not separating them. If each pair of points generates an infinitesimal amount of traffic divided equally between the two connecting near-geodesics, and if the Poisson line pattern is conditioned to contain a line through the centre, then what can be said about the total flow through the centre? In earlier work ("Geodesics and flows in a Poissonian city", Annals of Applied Probability, 21(3), 801--842, 2011) it was shown that a scaled version of this flow had asymptotic distribution given by the 4-volume of a region in 4-space, constructed using an improper anisotropic Poisson line process in an infinite planar strip. Here we construct a more amenable representation in terms of two "seminal curves" defined by the improper Poisson line process, and establish results which produce a framework for effective simulation from this distribution up to an L1 error which tends to zero with increasing computational effort.

Abstract:
This note extends the work of Foss and Tweedie (1997), who showed that availability of the classic Coupling from The Past algorithm of Propp and Wilson (1996) is essentially equivalent to uniform ergodicity for a Markov chain (see also HobertRobert, 2004). In this note we show that all geometrically ergodic chains possess dominated Coupling from The Past algorithms (not necessarily practical!) which are rather closely connected to Foster-Lyapunov criteria.

Abstract:
We exhibit some explicit co-adapted couplings for n-dimensional Brownian motion and all its Levy stochastic areas. In the two-dimensional case we show how to derive exact asymptotics for the coupling time under various mixed coupling strategies, using Dufresne's formula for the distribution of exponential functionals of Brownian motion. This yields quantitative asymptotics for the distributions of random times required for certain simultaneous couplings of stochastic area and Brownian motion. The approach also applies to higher dimensions, but will then lead to upper and lower bounds rather than exact asymptotics.

Abstract:
It is shown how to construct a successful co-adapted coupling of two copies of an $n$-dimensional Brownian motion $(B_1,...,B_n)$ while simultaneously coupling all corresponding copies of L\'{e}vy stochastic areas $\int B_i dB_j-\int B_j dB_i$. It is conjectured that successful co-adapted couplings still exist when the L\'{e}vy stochastic areas are replaced by a finite set of multiply iterated path- and time-integrals, subject to algebraic compatibility of the initial conditions.

Abstract:
The stationary isotropic Poisson line process was used to derive upper bounds on mean excess network geodesic length in Aldous and Kendall [Adv. in Appl. Probab. 40 (2008) 1-21]. The current paper presents a study of the geometry and fluctuations of near-geodesics in the network generated by the line process. The notion of a "Poissonian city" is introduced, in which connections between pairs of nodes are made using simple "no-overshoot" paths based on the Poisson line process. Asymptotics for geometric features and random variation in length are computed for such near-geodesic paths; it is shown that they traverse the network with an order of efficiency comparable to that of true network geodesics. Mean characteristics and limiting behavior at the center are computed for a natural network flow. Comparisons are drawn with similar network flows in a city based on a comparable rectilinear grid. A concluding section discusses several open problems.

Abstract:
Consider an improper Poisson line process, marked by positive speeds so as to satisfy a scale-invariance property (actually, scale-equivariance). The line process can be characterized by its intensity measure, which belongs to a one-parameter family if scale and Euclidean invariance are required. This paper investigates a proposal by Aldous, namely that the line process could be used to produce a scale-invariant random spatial network (SIRSN) by means of connecting up points using paths which follow segments from the line process at the stipulated speeds. It is shown that this does indeed produce a scale-invariant network, under suitable conditions on the parameter; indeed that this produces a parameter-dependent random geodesic metric for d-dimensional space ($d\geq2$), where geodesics are given by minimum-time paths. Moreover in the planar case it is shown that the resulting geodesic metric space has an almost-everywhere-unique-geodesic property, that geodesics are locally of finite mean length, and that if an independent Poisson point process is connected up by such geodesics then the resulting network places finite length in each compact region. It is an open question whether the result is a SIRSN (in Aldous' sense; so placing finite mean length in each compact region), but it may be called a pre-SIRSN.