PageRank as the stationary distribution of a random surfer

Irreducibility. Because $1-d>0$, every entry of $G$ is strictly positive ($G_{ij}\ge(1-d)/N>0$), so from any page there is a positive-probability direct jump to any other page in one step — the underlying graph is trivially strongly connected, hence $G$ is irreducible.

Aperiodicity. A chain where every state can go directly to every state (including itself, since $G_{ii}>0$ too) has return times of every length $1,2,3,\dots$ available, whose greatest common divisor is 1; so $G$ is aperiodic.

Existence, uniqueness, convergence. $G$ is row-stochastic by construction (a convex combination $dP+(1-d)\tfrac1N J$ of two row-stochastic matrices $P$ and $J/N$ is row-stochastic), and it is irreducible and aperiodic as just shown, so the fundamental theorem of finite Markov chains applies directly: a unique stationary $\pi$ exists with $\pi=\pi\Big(dP+(1-d)\tfrac1N J\Big)$, and $G^n$ converges entrywise to a matrix with every row equal to $\pi$.

Power iteration. Since $\pi_k=\pi_0G^k$ for any starting distribution $\pi_0$, and every row of $G^k$ converges to $\pi$ as $k\to\infty$, the weighted average $\pi_0G^k$ converges to $\pi$ as well — this is precisely why repeatedly applying $\pi_{k+1}=\pi_k G$ from an arbitrary starting rank (commonly the uniform distribution) converges to the true PageRank vector.