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One reason is that probabilists often consider more than one measure on the same space, and then a negligible set for one measure (added in a completion) might be not negligible for the other. The sit …

Given $n^2$ i.i.d. uniform points in $[0,1]^2$, the goal is to draw a configuration of $cn$ vertical lines and $cn$ horizontal lines such that in each small rectangle there is at most one marked point …

The asymptotic frequency of square-free integers is known to be $6/\pi^2$, see [1]. Denote by $P_n$ the uniform distribution on $[1,n]$ and by $E_n$ the corresponding expectation. Then $$E_n(r)=\sum_{ …

If the random vector $(X,Y)$ in the plane has independent coordinates and a rotation-invariant distribution, then it is Gaussian.

The expected value is asymptotic to $(\log n)/n$ as $n$ tends to infinity (By "asymptotic" I mean that the ratio tends to 1). One way to see this is to use the representation of order statistics of un …

This question has been studied extensively in the computer science literature under the name "balls in bins"; see [1] which gives quite tight bounds in Theorem 1, page 161 and also describes prior wo …

Of course the $X_k$ are not independent as random variables. So I assume you are referring to some notion of asymptotic independence, and it would help if you state your conjecture more precisely. One …

Observe that $X_n=X_{n-1}(1+Z_n)$ where $\{Z_k\}_{k \ge 1}$ are i.i.d. standard normal. Hence to analyze the asymptotic distribution of $|X_n|$, pass to logarithms, to get $$\log(|X_n|)= \log(|X_1|) + …

This is known as the support theorem for Brownian motion. Besides the proof in the answer of Iosif Pinelis and the proof in Exercise 1.8 of [1], there is also a proof on page 59 of [2]. Generalization …

The answer is negative: It is possible that there is no good choice of $i,j$. Let $T$ be a uniform spanning tree in the infinite ladder ${\bf Z} \times \{0,1\}$. To be precise, this is a weak limit o …

The Borel distribution with parameter $x \in [0,1]$ is given by $$p_n= \frac{e^{-x n}(x n)^{n-1}}{n!}\,.$$ for $n \ge 1$. This distribution represents the probability that in a Galton Watson branchin …

There is a simple reduction of the Birkhoff ergodic Theorem for $L^1$ functions to the bounded case using Kakutani-Rokhlin towers, that I learned from H. Furstenberg and B. Weiss decades ago. We use …

The canonical argument proving the upper bound is in the comment by Mathworker21, which has now been posted as an answer. I just want to add that the constant $e$ obtained there is sharp. Indeed, if …

Yes, the local time (at zero) maps the zero set of Brownian motion to an interval. See e.g. Lemma 6.9 page 159 in [1] for continuity. [1] Brownian motion, by Peter Mörters and Yuval Peres. Cambridge U …

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):=\m …