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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.
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 ind
We survey our recent articles dealing with one dimensional attractive zero range processes moving under site disorder. We suppose that the underlying random walks are biased to the right and so hyperbolic scaling is expected. Under the conditions of
We consider the problem of bounding large deviations for non-i.i.d. random variables that are allowed to have arbitrary dependencies. Previous works typically assumed a specific dependence structure, namely the existence of independent components. Bo
We study continuous-time (variable speed) random walks in random environments on $mathbb{Z}^d$, $dge2$, where, at time $t$, the walk at $x$ jumps across edge $(x,y)$ at time-dependent rate $a_t(x,y)$. The rates, which we assume stationary and ergodic
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 unde