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The answer is yes. To see this, take a transient graph $G$ (say $\mathbb{Z}^3$) and glue a copy of $\mathbb{N}$ to some vertex (say $v_0$). The vertices of $\mathbb{N}$ now all have $g(v)=1$. If you …

The crucial requirement is that $\sum_{i=0}^\infty t_i^2 < \infty$. See Kolmogorov's two-series theorem and also the more general Kolmogorov's three-series theorem.

The special case when the $X_i$'s are +1 or -1 with equal probabilities is called Bernoulli Convolution, see the nice survey by Peres, Schlag and Solomyak: SIXTY YEARS OF BERNOULLI CONVOLUTIONS.

The current from a vertex to infinity in any graph is never in $\ell^1(E)$. Any cutset (that is, a set of edges separating the origin vertex from infinity) will contribute at least a constant to the $ …

$\newcommand{\Z}{\mathbb Z}$ $\newcommand{\T}{\mathbb T}$ It is well know that the Poisson boundary of simple random walk on graph is not invariant under quasi isometries. Here's a construction: Tak …

Check out the Law of iterated Logarithm. Is this enough for your purpose?

The fundamental result that completely characterizes recurrent/transient graphs is that a graph is recurrent if and only if the effective resistance of the graph, when considered as an electric networ …

For the first question: yes, suitably scaled, the sequence $S^p_n$ tend to Brownian motion, which is positive infinitely often with probability 1. More precisely, for any fixed $p$, $e(n,p)$ will be a …

I'm not sure, but I think what you meant to ask was a practical question: how many random walk samples do I need in order to get a certain precision in estimating $\mathbb{E}(f)$? This question has b …

To answer your first question (regarding existence of the limit): I never remember references, but all you have to do here is show that for any two far enough starting points, there is a coupling of t …