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The probability of a random walk returning to its origin is 1 in two dimensions (2D) but only 34% in three dimensions: This is Pólya's theorem. I have learned that in 2D the condition of returning to ...

I have a random walk on $\mathbb{Z}^2$ that takes a step with equal probability in the three directions that avoid retracing the previous step. The walk proceeds until it returns to a lattice point ...

Let $S = (S_t, t \geq 0)$ be a simple one-dimensional continuous-time random walk with total jump rate one, $S_0 = 0$. Denote by $T_k$ the time when $S$ exits the interval $I_k = [-k,k] \cap \...

Earlier I asked a question: Distribution of the $k$-th largest eigenvalue of in the sample covariance matrix?, but I forgot to mention that I'd like results for asymtotic regime. So, I'm posting here ...

Let us assume we've a rectangular data matrix $X=[x_1 \dots x_n] \in \mathbb{R}^{p \times n}$, where the $x_i \in \mathbb{R}^{p \times 1}$ are iid column vectors. I'm not assuming here that the ...