# Questions tagged [pr.probability]

Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

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### Uniform iid sequence

The Rademacher functions are an explicit iid sequence with Bernoulli law. Does it exist an explicit construction of an iid sequence with uniform law?
• 8,144
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### Show that the set of strictly stationary, mean zero and finite variance stochastic processes is closed (or not)

Let $\mathcal{P}$ be the set of real-valued and strictly stationary processes with expectation zero and finite variance, i.e.: \mathcal{P}:=\left\{ X = (X_t)_{t \in \mathbb{Z}} \, ...
• 131
1 vote
50 views

### Constructing k-wise independent variables over a general set

We have seen in class a polynomials based construction that builds in $O(n^k)$ time, $n$ random variables, $k$-wise independent, over a field with $n$ elements. More specifically, you generate all the ...
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### Distributions of distance between two random points in Hilbert space

Let $\mu$ be a probability distribution on a separable infinite-dimensional Hilbert space. Let $D$ be the distance between two independent random samples from $\mu$. So $D$ has some probability ...
• 136
1 vote
122 views

### Eigenvalue distribution of random matrices

Given basis $M_1,M_2\dotsc,M_{d^2}$ in $\mathbb C^{d\times d}$, we consider $$\sum_i x_i M_i$$ for random variables $x_i$. What is the distribution of $$\lVert\sum_i x_i M_i\rVert_1=\sum \sigma_k?$$ ...
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### Tail bound on trace norm / nuclear norm / Schatten-1 norm of Rademacher matrix

Let $0 < r \leq d$ integers. Let $X$, $Y$ be $d \times r$ matrices of independent Rademacher variables, that is, $X,Y \in \mathbb{R}^{d \times r}$ with entries $\pm1$ with probability $1/2$. I am ...
• 223
92 views

### An infinite moving average is stationary iff its innovations are stationary

Let $(a_j)_{j \in \mathbb{N}_0}$ be a real-valued sequence such that $\sum_{j = 0}^\infty a_j^2 < \infty$. Further, define an infinite moving average time series $X = \{ X(t), t \in \mathbb{Z}\}$ ...
• 261
467 views

### Relationship between KL, chi-squared, and Hellinger

There are many well-known relationships between the KL divergence, chi-squared ($\chi^2$) divergence, and the Hellinger metric. In the paper "Assouad, Fano, and Le Cam" by Bin Yu, the author ...
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### Likelihood ratio of non-trivial cycles in an inhomogeneous random square lattice graph embedded on a toroidal surface

Consider a square lattice (random) graph $G$ embedded on a toroidal surface. Each edge $(i, j)$ of the graph has an associated likelihood probability $p_{ij}$. The probabilities $p_{ij}$ come from a ...
244 views

### Discrimination between set of binary distributions

Suppose we know two sets of distributions $A=\{p_1,p_2,\cdots,p_k\}$ and $B=\{q_1,q_2,\cdots,q_k\}$. We are given $C=\{r_1,r_2,\cdots,r_k\}$ such that $r_i=p_i$ for all $i$ or $r_i=q_i$ for all $i$. ...
• 1,385
1 vote
106 views

### Vector version of concentration of Lipschitz functions on sphere (Levy's Lemma)

Levy's Lemma asserts Lipschitz functions of vectors chosen uniformly from the unit hypersphere concentrate: Lemma. Suppose $f:\mathbb{S}^{d-1} \to \mathbb{R}$ is $L$-Lipschitz on the unit hypersphere. ...
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