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We study the one-dimensional branching random walk in the case when the step size distribution has a stretched exponential tail, and, in particular, no finite exponential moments. The tail of the step size $X$ decays as $mathbb{P}[X geq t] sim a exp(-lambda t^r)$ for some constants $a, lambda > 0$ where $r in (0,1)$. We give a detailed description of the asymptotic behaviour of the position of the rightmost particle, proving almost-sure limit theorems, convergence in law and some integral tests. The limit theorems reveal interesting differences betweens the two regimes $ r in (0, 2/3)$ and $ r in (2/3, 1)$, with yet different limits in the boundary case $r = 2/3$.
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We prove large deviation results for the position of the rightmost particle, denoted by $M_n$, in a one-dimensional branching random walk in a case when Cramers condition is not satisfied. More precisely we consider step size distributions with stretched exponential upper and lower tails, i.e.~both tails decay as $e^{-|t|^r}$ for some $rin( 0,1)$. It is known that in this case, $M_n$ grows as $n^{1/r}$ and in particular faster than linearly in $n$. Our main result is a large deviation principle for the laws of $n^{-1/r}M_n$ . In the proof we use a comparison with the maximum of (a random number of) independent random walks, denoted by $tilde M_n$, and we show a large deviation principle for the laws of $n^{-1/r}tilde M_n$ as well.
In this paper we consider a d-dimensional scenery seen along a simple symmetric branching random walk, where at each time each particle gives the color record it is seeing. We show that we can a.s. reconstruct the scenery up to equivalence from the color record of all the particles. For this we assume that the scenery has at least 2d + 1 colors which are i.i.d. with uniform probability. This is an improvement in comparison to [22] where the particles needed to see at each time a window around their current position. In [11] the reconstruction is done for d = 2 with only one particle instead of a branching random walk, but millions of colors are necessary.
We work under the A{i}d{e}kon-Chen conditions which ensure that the derivative martingale in a supercritical branching random walk on the line converges almost surely to a nondegenerate nonnegative random variable that we denote by $Z$. It is shown that $mathbb{E} Zmathbf{1}_{{Zle x}}=log x+o(log x)$ as $xtoinfty$. Also, we provide necessary and sufficient conditions under which $mathbb{E} Zmathbf{1}_{{Zle x}}=log x+{rm const}+o(1)$ as $xtoinfty$. This more precise asymptotics is a key tool for proving distributional limit theorems which quantify the rate of convergence of the derivative martingale to its limit $Z$. The methodological novelty of the present paper is a three terms representation of a subharmonic function of at most linear growth for a killed centered random walk of finite variance. This yields the aforementioned asymptotics and should also be applicable to other models.
We derive a lower bound for the probability that a random walk with i.i.d. increments and small negative drift $mu$ exceeds the value $x>0$ by time $N$. When the moment generating functions are bounded in an interval around the origin, this probability can be bounded below by $1-O(x|mu| log N)$. The approach is elementary and does not use strong approximation theorems.
The stochastic solutions to the Wigner equation, which explain the nonlocal oscillatory integral operator $Theta_V$ with an anti-symmetric kernel as {the generator of two branches of jump processes}, are analyzed. All existing branching random walk solutions are formulated based on the Hahn-Jordan decomposition $Theta_V=Theta^+_V-Theta^-_V$, i.e., treating $Theta_V$ as the difference of two positive operators $Theta^pm_V$, each of which characterizes the transition of states for one branch of particles. Despite the fact that the first moments of such models solve the Wigner equation, we prove that the bounds of corresponding variances grow exponentially in time with the rate depending on the upper bound of $Theta^pm_V$, instead of $Theta_V$. In other words, the decay of high-frequency components is totally ignored, resulting in a severe {numerical sign problem}. {To fully utilize such decay property}, we have recourse to the stationary phase approximation for $Theta_V$, which captures essential contributions from the stationary phase points as well as the near-cancelation of positive and negative weights. The resulting branching random walk solutions are then proved to asymptotically solve the Wigner equation, but {gain} a substantial reduction in variances, thereby ameliorating the sign problem. Numerical experiments in 4-D phase space validate our theoretical findings.
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