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The brownian cactus I. Scaling limits of discrete cactuses

Nicolas Curien, Jean-François Le Gall, Grégory Miermont (2013)

Annales de l'I.H.P. Probabilités et statistiques

The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space E , one can associate an ℝ -tree called the continuous cactus of E . We prove under general assumptions that the cactus of random planar maps distributed according to Boltzmann weights and conditioned to have a fixed large number of vertices converges in distribution to a limiting space called the Brownian cactus, in the Gromov–Hausdorff sense. Moreover, the Brownian cactus...

The law of large numbers and a functional equation

Maciej Sablik (1998)

Annales Polonici Mathematici

We deal with the linear functional equation (E) g ( x ) = ∑ i = 1 r p i g ( c i x ) , where g:(0,∞) → (0,∞) is unknown, ( p ₁ , . . . , p r ) is a probability distribution, and c i ’s are positive numbers. The equation (or some equivalent forms) was considered earlier under different assumptions (cf. [1], [2], [4], [5] and [6]). Using Bernoulli’s Law of Large Numbers we prove that g has to be constant provided it has a limit at one end of the domain and is bounded at the other end.

The quenched invariance principle for random walks in random environments admitting a bounded cycle representation

Jean-Dominique Deuschel, Holger Kösters (2008)

Annales de l'I.H.P. Probabilités et statistiques

We derive a quenched invariance principle for random walks in random environments whose transition probabilities are defined in terms of weighted cycles of bounded length. To this end, we adapt the proof for random walks among random conductances by Sidoravicius and Sznitman (Probab. Theory Related Fields129 (2004) 219–244) to the non-reversible setting.

The rate of convergence of option prices when general martingale discrete-time scheme approximates the Black-Scholes model

Yuliya Mishura (2015)

Banach Center Publications

We take the martingale central limit theorem that was established, together with the rate of convergence, by Liptser and Shiryaev, and adapt it to the multiplicative scheme of financial markets with discrete time that converge to the standard Black-Scholes model. The rate of convergence of put and call option prices is shown to be bounded by n - 1 / 8 . To improve the rate of convergence, we suppose that the increments are independent and identically distributed (but without binomial or similar restrictions...

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