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Let $\{X_i\}$ be a collection of i.i.d. Exp$(1)$ random variables. For $k \in \mathbb{Z}_{>0}$, define $$S_k = \sum_{i=1}^k X_i$$ and note that $\mathbb{E}[S_k] = k$. I was wondering if there is any w …

In this paper by Diaconis and Zabell from 1982, Theorem 2.1 and the remark after essentially stated that Given two probability measures $P$ and $Q$ on the same probability space $\Omega$. If $Q\ll P$ …

Suppose we have a random walk $S_n$ with i.i.d. steps $X_i$. We assume that $$\mathbb{E}[X_i] = -\mu, \text{Var}[X_i] = 1,$$ where $\mu$ is close (or going) to zero. We also assume that the moment gen …

Define the planar percolation where each unit edge is open with probability $p$ very close to $1$. Looking at the event where there exists a directed open path between $(0,0)$ and $(n,n)$. This event …

Look at Bernoulli percolation on $\mathbb{Z}^2$ with $p> p_c$ ($p$ can be arbitrarily close to 1). I am interested in the probability that there exists an infinite cluster starting at $(0,0)$ and it …

Given $X_i = A_i - B_i$ where $A_i\sim \text{ Exp}(\alpha)$ and $B_i \sim \text{ Exp}(\lambda)$. Define $S_k = \sum_{i=1}^k X_i$ with $S_0 = 0$, and $$M_n = \max_{1\leq k \leq n} S_k.$$ Is it possibl …

Let $Z_i$ be i.i.d. random variables with $\mathbb{E}[Z_i] = 0$ and $\mathbb{E}|Z_i|^p< \infty$ for $p=1,2,3,\cdots$. I am looking for the following type of estimate if possible, and it is not like th …