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Register a new userBounds on the running maximum of a random walk with small drift
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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.
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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 $r = 2/3$.
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.
Let $xi(n, x)$ be the local time at $x$ for a recurrent one-dimensional random walk in random environment after $n$ steps, and consider the maximum $xi^*(n) = max_x xi(n,x)$. It is known that $limsup xi^*(n)/n$ is a positive constant a.s. We prove that $liminf_n (logloglog n)xi^*(n)/n$ is a positive constant a.s.; this answers a question of P. Revesz (1990). The proof is based on an analysis of the {em valleys /} in the environment, defined as the potential wells of record depth. In particular, we show that almost surely, at any time $n$ large enough, the random walker has spent almost all of its lifetime in the two deepest valleys of the environment it has encountered. We also prove a uniform exponential tail bound for the ratio of the expected total occupation time of a valley and the expected local time at its bottom.
We consider the limit behavior of a one-dimensional random walk with unit jumps whose transition probabilities are modified every time the walk hits zero. The invariance principle is proved in the scheme of series where the size of modifications depends on the number of series. For the natural scaling of time and space arguments the limit process is (i) a Brownian motion if modifications are small, (ii) a linear motion with a random slope if modifications are large, and (iii) the limit process satisfies an SDE with a local time of unknown process in a drift if modifications are moderate.
We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels without the assumption of uniform ellipticity or nearest-neighbour jumps. Specifically, we obtain a strong law of large numbers, a functional central limit theorem and large deviation estimates for the position of the random walker under the annealed law in a high density regime. The main obstacle is the intrinsic lack of monotonicity in higher-dimensional, non-nearest neighbour settings. Here we develop more general renormalization and renewal schemes that allow us to overcome this issue. As a second application of our methods, we provide an alternative proof of the ballistic behaviour of the front of (the discrete-time version of) the infection model introduced in [23].
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