Tagged Questions
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
-2
votes
0answers
22 views
Problem in understanding means of i.i.d. variable
Let $X_i$ be i.i.d. and $X_i(\omega)= \omega, \omega \in (0,1)$, $P :=$ Lebesgue measure.
Then I have $\bar X_n (\omega) = \frac{\sum^n_{i=1} X_i(\omega)}{n} = \frac{n \omega }{n} = \omega$.
However,...
0
votes
1answer
46 views
Expected sum of chosen coordinates in a random subset of a Hamming hypercube
Let $S$ = $\{v_1, v_2, ..., v_n\}$ denote a random subset of a Hamming hypercube of dimension $d$, where $n = |S|$ and $n \leq 2^d$. If $v_i$ = $\langle x^i_1x^i_2... x^i_d\rangle$ for all $i \in [1,n]...
4
votes
1answer
112 views
Nonlinear boolean functions
Let $\mathbb{F}_2=\{0,1\}$ be the field with two elements. I wonder if there is any known algorithm/construction that, given any $n\geq 1$, returns a boolean function $f:\mathbb{F}^n_2\rightarrow \...
-2
votes
0answers
37 views
What is the theoretical probability? [on hold]
I have created a variation of the dice game PIG. In the game, you roll 1 die. If you roll a 1, your point total for that round is 0. If you roll a 2, your overall point total goes back to zero. ...
1
vote
1answer
59 views
Generalization of inverse transform sampling
If X is a random variable over an arbitrary alphabet, is there a (deterministic) function f() such that X = f(U), where U is a uniform random variable over the unit-interval?
0
votes
0answers
57 views
Expected value of eigenvalue of matrix
Let $A = (X_{ij})_{ij}$ a square matrix of size $n$ where the $X_{ij}$ are (discrete) real random non-negative entries. Denote by $\lambda_1(A) \geq \dots \geq \lambda_n(A)$ the (random) ordered ...
5
votes
2answers
151 views
Probability of at least two of $n$ independent events occurring subject to some conditions
Given a set of independent Bernoulli random variables $\{x_1, \dots, x_n\}$, let $p = \sum_{0<i\leq n}\Pr[x_i = 1]$ and $X=\sum_{0<i\leq n} x_i$. We know that for any $i$, we have $\Pr[x_i = 1]\...
-2
votes
0answers
31 views
Sum of probabilities of events vs. probability of at least one event [on hold]
$P_i$ is the probability of one event. As defined below, $a$ is the sum of all probabilities of events (that may or may not be independent), and $c$ is the overall probability that at least one event ...
3
votes
0answers
81 views
Collecting proofs of the birth of the giant component
I want to collect different proofs of Erd?s-Rényi result on the double jump of the largest connected component on $G(n,p)$ (or in $G(n,M)$.
I know the original proof of Erd?s-Rényi, the proof that ...
1
vote
1answer
48 views
gaussian isoperimetric result for minimal measure under translation
Consider two spherical Gaussian distributions in $\mathbb{R}^n$, $A = \mathcal{N}(x, I)$ and $B=\mathcal{N}(y, I)$ where the difference in means is $\delta = y - x$.
Let $S \subset \mathbb{R}^n$ be a ...
3
votes
0answers
157 views
+50
What happens in the martingale CLT if I norm by the conditional variance instead?
TLDR: I'm a statistician (bear with me!) trying to use the martingale CLT but I only can estimate the conditional variance instead of the unconditional one. Can I do anything to get a CLT with norming ...
-1
votes
0answers
22 views
Independence of stationary process and it's derivative
Let $X(t)$ be a centred stationary gaussian process on the reals, with differentiable sample paths, with covariance function $r(t)$
Are $X(0)$ and $X'(0)$ independent? Why?
Are they independent only ...
-1
votes
0answers
28 views
Sigma algebra generated by two partitions [on hold]
If $(A_1, ..., A_m)$ and $(B_1, ... B_n)$ are two partitions of $\Omega$, show that:
(a) $(Ai\cap Bj)$ is a partition of $\Omega$.
(b) $\sigma \{ \sigma\{A_i\} \cup \sigma\{B_j\} \} = \sigma \{ A_i ...
1
vote
1answer
41 views
References for Hellinger distance/affinity involving mixture distributions
For two continuous probability distributions $F,G$ and their densities, $f,g$, the (squared) Hellinger distance/affinity is given by $d^2_H(F,G)=1-\int_{\mathbb{R}} \sqrt{fg}~dx$. Suppose that $f,g$ ...
2
votes
1answer
73 views
The effect of random projections on matrices
Let $A\in\mathbb{R}^{n\times n}$ be a given normal matrix, i.e. $A^TA=AA^T$. Let $P_s\in\mathbb{R}^n$ be a random projection matrix to an $s$-dimensional subspace in $\mathbb{R}^n$.
Suppose $\frac{A+...
