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This is tight, at least when $p=\frac12$. You simply need to approximate $\log\big({n \choose \frac{n}{2}-\varepsilon n} \frac1{2^{n}}\big)$ using Stirling's formula and you'll see that the leading co …

Your argument can be extended to any polynomial. Let $q(y_1,y_2)=(p(y_1,y_2))^2$. Then the leading coefficient in $q$, as a polynomial in $y_1$ is a polynomial $r(y_2)$ which is a square. Since $Y_2$ …

One particular (relatively) well studied case is that of Bernoulli convolution, where the RV are supported on just 2 points. Take a look here to get an idea about what is known.

Indeed, as Adam Smith pointed out, you can do this using coupling. Very briefly, since I don't have the time to elaborate, for any two vertices $x,y\in G$ and any two finite configurations of lamps $u …

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 …

There's no reason to believe that the new speed will be $2 \epsilon$ more then the old speed. To give a concrete example, take $\epsilon=0.01$ and let the environment be $p_n=0.01$ when $n$ is a multi …

This particular problem can be solved by dividing it into two parts. First, the probability that the first $\sqrt{n}$ numbers chosen are all distinct. Second, the probability that a random subset of …

let's consider a simpler question: for which values of the parameters does this probability tend to 0 or to 1? Here are some basic estimates for the case where all the parameters tend to infinity and …

$\newcommand{\E}{\mathbf{E}}\renewcommand{\P}{\mathbf{P}}\DeclareMathOperator{\var}{Var}$ Yes. If $s_n=\max_{1\le k \le r_n} \sigma_{nk}$ then we have $$\E\big[|X_{nk}|^2 I_{|X_{nk}|>\eta}\big] \le \s …

http://puhep1.princeton.edu/~mcdonald/examples/EM/atkinson_ajp_67_486_99.pdf I haven't checked its validity, but it has Mathematica code for calculating what you want. As a side notice, the asymptoti …

Suppose for the moment that we're talking about a SRW on $\mathbb{Z}^2$ and the target is the origin. Suppose that you're interested in the time interval $[0,T]$, meaning that you're asking about the …

Poisson Kernel

This isn't true, in general. If you take $p_0=1/n$ and the other $p_i=1$ then you get a constant probability for $X>2\mathbb{E}(X)$.

Some tools from the theory of the classical moments problem are useful here. You can see how they are used and get some bounds on your question in my joint paper with Itai Benjamini and Ron Peled here …

$X$ will "tend" to $N(c,c)$, even though the usual formulation of the CLT does not cover this case, and $f(c)$ will be of order $\sqrt{c}$. In fact, this is true for any $c=\omega(1)$. Notice that eve …