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I think one cannot find any useful bounds with these assumptions. For example, consider a positive r.v. $\xi$ such that $E\xi^2<\infty$, but $E\xi^3=\infty$. Then, let $\eta_1,\eta_2,\eta_3$ be i.i.d. …

If you are interested in the longest run of 0's in the i.i.d. setting, see this paper: http://gato-docs.its.txstate.edu%2Fmathworks%2FDistributionOfLongestRun.pdf&usg=AFQjCNE8shEgVJmaWNEVSYv5YNRIs08 …

I don't know if there is a nice closed-form expression for this, but you can try to work it out writing $$ P[Z\leq y] = P[X_t\neq 0 \text{ for }x\in (y,1]] = \int_{-\infty}^{+\infty} f_{y}(v) P_v[X_t\ …

This is not true. Consider some random sequence that converges to $0$ a.s., for example, $Y_n=(Z_1+\cdots+ Z_n)^{-1}$, where $Z$'s are i.i.d. (for instance) Exp(1) random variables. Set $$ X_t = \beg …

Assume that $f_1, f_2, f_3,\ldots$ are i.i.d. random functions $[0,1]\mapsto \mathbb{R}$ such that (1) random variables $M_k=\sup_{x\in[0,1]}f_k(x)$ have exponential tails, (2) $f$'s are a.s. Lipsch …

This does not seem to be true. Take $p=2/3$ and $n=k=1000$. Then, you need that $$ \frac{(2/3)^{1000}}{1-(1/3)^{1000}} \geq (2/3)^{999}, $$ which is clearly false.

Let $Z(m)$ be the number of uncovered (= not chosen) sites after $m$ trials. First, we have $$\mathbb{E} Z(N^a(1+\epsilon)\ln N)=N\times(1-N^{-a})^{N^a(1+\epsilon)\ln N}\approx N^{-\epsilon},$$ so you …

Let $(S_n, n\geq 0)$ be the one-dimensional simple random walk started at the origin. Set $X_n=\mathbf{1}\{S_n\geq 0\}$. Then $n^{-1}(X_1+\cdots+ X_n)$ is the proportion of time that the SRW is non-ne …

Is there an explicit formula for the (logarithmic) capacity of a union of two disjoint disks? As far as I understand, one can assume without loss of generality that the disks have the same radii (othe …

See Example 8d of Section 4.8 of "A 1st course in Probability" by Sheldon Ross. Here it is: If independent trials, each resulting in a success with probability $p$, are performed, what is the probab …

Let $S_n$ be the one-dimensional nearest neighbor random walk with $ 1-q=p=P[S_{n+1}=x+1\mid S_n=x]=1-P[S_{n+1}=x-1\mid S_n=x]$, where $p\neq q$. Then, there is a (rather surprising) fact that $Y_n=| …

You may find it useful to look here: http://www.artofproblemsolving.com/community/c7h398470 (in particular, the link in the 1st post, and also fedja's answer).

It seems that formula (5.7) of Chapter XIV of the 1st volume of Feller may be used to obtain the full distribution of $C_n$ (as distribution of sum of two independent r.v.'s with explicit distribution …

There is an example in the book "Essentials of Stochastic processes" of Durrett, which shows that, in general, $m$ need not be equal to $2n-2$ (it is Example 4.5 of the chapter devoted to Markov chain …

Its meaning should be something like "degree distribution seen from a random edge". Indeed, let us first think of an unoriented edge as of two oriented (in opposite directions) ones. Assume that there …