# Tag Info

### Limit distribution of the self-normalized sum of Cauchy random variables

It converges to 0 in probability. $\frac{\Sigma_1^nX_i}n$ always has the same cauchy distribution while $\frac{\Sigma_1^n|X_i|}n \rightarrow \infty$ by the law of large numbers
• 1,129

### Upper-bound of the tail of a weighted sum of iid random variables

This is a partial answer, in the regime that the probability in question is close to $1$. I normalize so that $EY_i=1$. The example $a_i=1/\sqrt{n}$ shows that you need to take $t\geq \sqrt{n}/2+O(1)$...
• 7,364

• 288

### Expectation of top-K selection of squared Gaussian random variables

Let $Z \sim \mathcal{N}(0,\sigma^2 I_n)$ be an $n$-dimensional Gaussian vector with independent $\mathcal{N}(0,\sigma^2)$ components. Consider the family of binary vectors in $\mathbb{R}^n$ with ...
• 156
1 vote

• 115k

### How to optimally bet on a biased coin?

It is optimal to always bet the full amount iff heads has come up most of the time, and zero otherwise (breaking ties arbitrarily). To see this, first note that any strategy can be scaled linearly, so ...
• 1,678

### Does some published texbook take a particular approach (described here) to the transition from discrete to continuous probability distributions?

Such a transition from the discrete to the continuous is precisely the point of Nelson's Radically Elementary Probability Theory (REPT). It recently turned out that when viewed as a subsystem of BST,...
• 15.2k

### Deriving the distribution of standardized variables with empirical mean and standard deviation

Anthony's comment has the correct answer: $Z=(z_1,\ldots ,z_N)$ is uniformly distributed on $\|Z\|^2=N$, $\langle e, Z\rangle=0$, $e=(1,1,\ldots, 1)$ (that is, the joint distribution is the $(N-2)$-...
• 22.4k
1 vote
Accepted

### Weak convergence of random measures generated by non-negative martingales?

Partial answer For every $a \in [0,1]$, $(\mu_n([0,a]))_{n \ge 0}$ is still a non-negative martingale, hence it converges almost surely to some random variable $L_a$ with values in $[0,+\infty]$. One ...
• 3,619
Accepted

### Show convergence result

$\newcommand\de\delta\newcommand\ep\varepsilon$You wrote Could you help me to show that under Ass1 and Ass2 $$d_H(A, A_n)\rightarrow_{a.s.} 0$$ Of course, this is not true in such generality. For ...
• 115k

### Show convergence result

EDIT: That whole answer assumes the $p_n$'s are uniformly and independently distributed on $[0,1]$. I have no idea on how to extend it to the general case of other distributions. It gets elementary ...
1 vote

• 115k
Accepted

• 22.4k
1 vote

Let $X=[0,1]$ the unit interval. For every $n\in\mathbb N$ let $a_{n,1},\dots a_{n,n}$ distinct points in $[0,\frac1n]$, chosen such that all $a_{n,k}$ are pairwise different. Outside the set $A=\{a_{... • 247 2 votes ### Functional dependence not preserved in the weak limit Let$M=[0,1]$,$f$is the indicator function of$[0,1/2)$, and$\mu_n=0.5 \delta_{1/2-1/2n}+0.5 \delta_{1/2+1/2n}$. Then$\mu=\delta_{1/2}$but$\nu$is uniform on$\{0,1\}$. • 12.4k 1 vote ### Evolution of the empirical mean of a list as we remove elements proportional to their value I doubt this is an easy analysis, as the result is going to be affected by the distribution used for$P(k)$. I doubt there is a general form, but empirically there may be a well-behaved example. In ... • 830 3 votes ### Constructing a family of$3$-wise independence functions from$\mathbb{Z}_p^n \rightarrow \mathbb{Z}_p$Let$\mathrm{tr}:GF(p^n)\rightarrow GF(p)$be the trace function which is equidistributed. Choose a basis so that you consider$GF(p^n)$as the vector space$GF(p^n)$. For any$c \in GF(p^n)^\ast,\$ ...
• 10k

Top 50 recent answers are included