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3 votes
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Local probabilities for lattice random walk

For the one dimensional case, a quite nice bound is in Theorem 4.2 of [1]. See also [2]. The dependence on $\epsilon$ that you seek was first shown by Kesten[3]. The combinatorial approach was ...
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1 vote
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Bernoulli trials with small dependencies: asymptotics (central limit theorem, law of the iterated logarithm)

For $0<\beta \le 1/2$, any limiting distribution of the rescaled process $S_n/n^\beta$ will be fully supported in $[-1,1]$, so it will not be normal. The downward drift will imply (using Hoeffding'...
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2 votes
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Invariance principle: Brownian bridge and random walk conditioned on end point

A more general theorem is proved in [1] for the limits of random walks in the domain of attraction of a stable law. In the case described in the problem, one can also use the strong approximation ...
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9 votes
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Average and max. hitting time to a specific vertex

Notation: Let $G=(V,E)$ be an undirected simple graph of $n$ nodes. If $\tau_x$ is the (random) time it takes the walk to reach the node $x$, then write $H(v,x)=E_v(\tau_x)$. Denote $H_{\max}(x):=\...
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1 vote
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Simple random walk return time

This can be done by the reflection principle. Also, one can use Theorem 0.6, which implies $$P(\tau_0^+>k)=\tfrac1k\,E|S_k|.$$ By the central limit theorem and uniform integrability, for $k\to\...
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