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It can probably be done by looking at the sum of squares of sizes of color clusters and then constructing an appropriate martingale. But here's a somewhat elegant solution: reverse the time! Let's fo …

$\newcommand{\E}{\mathbf{E}}$ $\renewcommand{\P}{\mathbf{P}}$ $\DeclareMathOperator{\var}{Var}$ If we use Taylor expansion (as Anthony suggested) for $\sqrt{x}$ around 1, we get: $$\sqrt{x}\approx 1 + …

No, you cannot get rich with identical copies on the unlabeled tree. This is a special case of the Mass Transport Principle - take a look at the book of Lyons and Peres, chapter 8.

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 …

This happens whenever $n\gg m^5$. To see this, notice that the expected number of balls in each bin is $n/m$ and the variance is also on the order of $n/m$. The distribution "tends" to N(n/m,n/m) (in …

The distribution should be roughly geometric with expectation roughly $\delta^{-2}$. To bound from above, in every $2\delta^{-2}$ steps there is a constant (independent of $\delta$) probability that t …

One very useful construction: if $X_1,\ldots,X_n$ are i.i.d. RVs, uniform in $\{0,\ldots,q-1\}$ ($q$ prime), then two linear combinations $\sum a_i X_i$ and $\sum b_i X_i$ are independent iff the vect …

Here's a counterexample. Let $X$ be equal to $2^k k^{-2}$ with probability $2^{-k}$. The probability that among $n$ i.i.d. copies of $X$ we get at least one with value $2 ^ {2 \log n} (2 \log n)^{-2} …

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.

$\newcommand{\R}{\mathbb R}$ $\newcommand{\P}{\mathbf P}$ $\newcommand{\Z}{\mathbb Z}$ I found this question very interesting and gave it much thought this week. I believe I have a proof now. I think …

Partition the unit square into small squares of area roughly $a$. Your question is equivalent to asking for which $a$ do we typically see about 1 small square with points from each of $X$,$Y$ and $Z$? …

Let's speak momentarily about the space average, rather than the expected degree. That is, consider the (expectation of the) average degree over all vertices in the disc of radius $R$ around the origi …

You're right that something more is needed to conclude that the probability of no streak is small. In this particular case, one can easily get a lower bound by partitioning the sequence of coin flips …

How is the general case different than this example? If $\sigma=0$ then the variance of $S_n/\sqrt{n}$ goes to 0 so $S_n/\sqrt{n} \to 0$ in distribution.

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.