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Ulam has defined a history-dependent random sequence of integers by the recursion $X_{n+1}$ $= X_{U(n)}+X_{V(n)}, n geqslant r$ where $U(n)$ and $V(n)$ are independently and uniformly distributed on ${1,dots,n}$, and the initial sequence, $X_1=x_1,dots,X_r=x_r$, is fixed. We consider the asymptotic properties of this sequence as $n to infty$, showing, for example, that $n^{-2} sum_{k=1}^n X_k$ converges to a non-degenerate random variable. We also consider the moments and auto-covariance of the process, showing, for example, that when the initial condition is $x_1 =1$ with $r =1$, then $lim_{nto infty} n^{-2} E X^2_n = (2 pi)^{-1} sinh(pi)$; and that for large $m < n$, we have $(m n)^{-1} E X_m X_n doteq (3 pi)^{-1} sinh(pi).$ We further consider new random adding processes where changes occur independently at discrete times with probability $p$, or where changes occur continuously at jump times of an independent Poisson process. The processes are shown to have properties similar to those of the discrete time process with $p=1$, and to be readily generalised to a wider range of related sequences.
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In the last decade, Hawkes processes have received a lot of attention as good models for functional connectivity in neural spiking networks. In this paper we consider a variant of this process, the Age Dependent Hawkes process, which incorporates individual post-jump behaviour into the framework of the usual Hawkes model. This allows to model recovery properties such as refractory periods, where the effects of the network are momentarily being suppressed or altered. We show how classical stability results for Hawkes processes can be improved by introducing age into the system. In particular, we neither need to a priori bound the intensities nor to impose any conditions on the Lipschitz constants. When the interactions between neurons are of mean field type, we study large network limits and establish the propagation of chaos property of the system.
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In this paper we deal with the classical problem of random cover times. We investigate the distribution of the time it takes for a Poisson process of cylinders to cover a set $A subset mathbb{R}^d.$ This Poisson process of cylinders is invariant under rotations, reflections and translations, and in addition we add a time component so that cylinders are raining from the sky at unit rate. Our main results concerns the asymptotic of this cover time as the set $A$ grows. If the set $A$ is discrete and well separated, we show convergence of the cover time to a Gumbel distribution. If instead $A$ has positive box dimension (and satisfies a weak additional assumption), we find the correct rate of convergence.
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