This note strengthens and simpliies Lutz's resource-bounded version of the Borel-Cantelli lemma for density systems and martingales. We observe that the technique can be used to construct martingales that are \additively honest," and also martingales that are \multiplicatively honest." We use this to improve the \Weak Stochasticity Theorem" of Lutz and Mayordomo: their result does not address t...

The converse of the Borel–Cantelli Lemma states that if {Ai}i=1∞ is a sequence independent events such ∑P(Ai)=∞, then almost surely infinitely many these will occur. Erdős and Rényi proved it sufficient to weaken condition independence pairwise independence. Later, several other weakenings appeared in literature. aim this paper provide collection conditions, all which imply occur, determine com...

Using techniques based on the continuous and discrete semimartingale convergence theorems, this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations (SDDEs). The important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of...

In this thesis, we establish extreme value (EV) theory and dynamical BorelCantelli lemmas for a class of deterministic chaotic dynamical systems. We establish the distributional convergence (to the three classical extreme value distributions) of the scaled sequence of partial maxima of some time series arising from an observable on systems such as the planar dispersing billiards, Lozi-like maps...

In this paper, we show that the domination number D of a random graph enjoys as sharp a concentration as does its chromatic number χ. We first prove this fact for the sequence of graphs {G(n, pn}, n → ∞, where a two point concentration is obtained with high probability for pn = p (fixed) or for a sequence pn that approaches zero sufficiently slowly. We then consider the infinite graph G(Z+, p),...

We study the Taylor expansion for the solution of a differential equation driven by a multi-dimensional Hölder path with exponent β > 1/2. We derive a convergence criterion that enables us to write the solution as an infinite sum of iterated integrals on a nonempty interval. We apply our deterministic results to stochastic differential equations driven by fractional Brownian motions with Hurst ...

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Strong forms of the Borel - Cantelli lemma are variants strong law large numbers for sums indicators events such that series from probabilities these diverges. These centered at means and normalized by some function means. In this paper, we derive new under wider restrictions on variations increments than it was done earlier. commonly used investigations properties dynamical systems. We apply o...