# 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.

8,578
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### The probability that iid draws from a mean zero random variable sum to zero

Suppose we have a probability distribution $p(\cdot)$ supported on the integers between $-m$ and $m$ for some positive integer $m$, with $\sum_k kp(k) = 0$. Suppose furthermore that all $p(k)$ are ...

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### Do we need to assume that $y$ is bounded or subgaussian?

Suppose that $X_1,\dots, X_n$ are iid $P$ on $\mathcal{X}$. The empirical measure $\mathbb{P}_n$ is defined by $$\mathbb{P}_n:=\frac{1}{n}\sum_{i=1}^n\delta_{X_i}$$
For a real-valued function $f$ on $\...

7
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147
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### Optimally betting a beta-biased coin

This question is inspired by How to optimally bet on a biased coin? by Nate River but generalized slightly. I decided the generalization might be interesting enough to be its own question.
A number $p$...

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45
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### Upper-bound of the tail of a weighted sum of iid random variables

I have a question related to this one. $X_i$ are n iid random variables with CDF $1_{[0,+\infty[}(x) \Phi(x)$, i.e. it is a mixture between a folded Gaussian and a delta in $0$, both with weight $1/2$....

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### Random walks on groups

I recently started reading Wolfgang Woess' book titled "Random Walks on Infinite Groups". In the section where he introduces Markov chains and random walks on a set $X$, he has defined a ...

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1
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88
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### Stochastic order on weighted sum of iid random variables

$X_i$ are n iid random variables with CDF $1_{[0,+\infty[}(x) \Phi(x)$, i.e. it is a mixture between a half Gaussian and a delta in $0$, both with weight $1/2$.
I would like to show that, $\forall a \...

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1
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94
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### Does a random matrix over $\mathbb{Z}_q$ map linearly independent vectors to statistically independent vectors?

Suppose $A \in \mathbb{Z}_q^{n \times m}$ is a random $n \times m$ matrix whose entries are i.i.d. uniform over $\mathbb{Z}_q=\mathbb{Z}/q\mathbb{Z}$, where $q\geq2$. Let $\mathbf{x}_1, \ldots, \...

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1
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154
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### Can the Wiener Chaos expansion converge in $L^p$?

For a function $F$ satisfying $|F(x)|\le C(1+|x|)^M$ for some $C,M>0$, and a mollifier $\rho$, we define $F_{\epsilon}=F\ast\rho_{\epsilon}$, where $\rho_{\epsilon}(x)=\epsilon^{-1}\rho(\frac{x}{\...

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78
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### A Kolmogorov inequality for sums of contiguous subsequences

If $X_1, \ldots X_n$ are independent real-valued random variables such that $E[X_k] = 0$ and $E[X_k^2]$ is finite for each $k$, Kolmorogov's inequality gives an upper bound on $P[\max_{1\le k \le n}|...

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137
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### Hunting an invisible target

An invisible target on the integer line starts at $0$. On each round it either stays put, moves to the left or moves to the right by $1$ with probability $\frac{1}{3}$ each. You are then asked to ...

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1
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117
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### Expectation of top-K selection of squared Gaussian random variables

Let us have
$$
Z = [z_1, z_2, \dots, z_n],
$$ where $z_i \sim N(0, \sigma^2)$ and are iid. Additionally, consider
$$
X_k := \{ x \in \{0, 1\}^n : e^T x = k \}
$$ If $Y = \max_{X \in X_k} |Z^T X|^2,$ ...

19
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2
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### How to optimally bet on a biased coin?

A number $p$ is drawn uniformly at random from $[0, 1]$. You are then given a biased coin that turns up heads with probability $p$, but the number $p$ is not known to you.
You start with a total ...

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78
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### The uniform odd and even subgraph of $\mathbb{Z}^2$

Given a (first finite and later infinite) graph $G =(V,E)$ the uniform even graph is the uniform probability measure on the set of spanning even subgraphs. That is subgraphs (V, E') with $E' \subset E$...

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473
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### Does some published texbook take a particular approach (described here) to the transition from discrete to continuous probability distributions? [closed]

(I posted this question at matheducators.stackexchange.com and it seems to be considered an inappropriate question for that site. I don't understand why.)
Imagine an introductory probability course ...

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66
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### Some stability and estimate of the optimal transport map (Brenier map)

Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...

2
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1
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69
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### Deriving the distribution of standardized variables with empirical mean and standard deviation

I'm working with a set of independent and identically distributed random variables $\{ x_i \}_{i=1}^N$, where each $x_i$ follows a Gaussian distribution $P_X(x) = \mathcal{N}(x; \mu, \sigma^2)$. This ...

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1
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151
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### Minimum of exponential distribution

Consider $n$ independent random variables $𝑋_𝑖\sim\exp(𝜆_𝑖)$ for $I=1\ldots,n$. Let $\lambda = \sum_{i=1}^n\lambda_i$. Of course, the minimum of these exponential distributions has distribution:
$...

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2
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388
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### Show that the distance converges to zero

Consider the following sets:
$$
A = \Big\{ x\in X: \Pr\bigg(\lim_{n \to \infty}d\big(p_n, [\ell(x), u(x) ] \big)= 0\bigg)=1 \Big\},
$$
and
$$
A_n = \Big\{ x\in X: d\big(p_n, [\ell(x), u(x) ] \big)...

2
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+50

### Asymptotic Independence of random walks from increments?

Suppose we have two random walks $(S_n:n\geq 1)$ and $(T_n:n\geq 1)$ building from independent identically distributed increment vectors $\{(X_k,Y_k):k\geq 1\}$, i.e. $S_n=\sum_{k=1}^n X_k, T_n=\sum_{...

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### Conditional joint distribution decomposition

Let $I$ be a finite index set and $\mathcal X_i$ for $i\in I$ be a standard measurable space (or say a Polish space or $\mathbb R$). Suppose that the probability measure $P(X_I)$ on the product space $...

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### Conditional distributions of random orthogonal projection matrix

I have encountered a rather curious question.
Suppose I have a symmetric idempotent orthogonal projection matrix $A\in\mathbb R^{N\times N}$ that projects onto a uniformly random $n$-dimensional ...

1
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0
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58
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### A small lemma on cache resets (Bloom filters in particular)

Assume a fixed set of message $D$ and an associated distribution for selecting each message $d_i$ such that the total probability $\sum_{i \in D} d_i = 1$. We create a cache with $M$ bits and $k$ ...

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46
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### Moment method / genus expansion for random matrices with i.i.d. entries

Given a (say real) random matrix $M=(M_{i,j})_{1\leq i, j \leq N}$, the moments method consists in computing (the limits in $N$ of) the quantities $$ \mathbb{E} \left(\mathrm{tr} M^k\right)^{1/k}, $$
...

4
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1
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140
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### Probability problem in Sheehan's conjecture

As my first math project, I have been working on Sheehan's Conjecture
and am stuck for weeks. I wonder if I am at a dead end.
Sheehan's Conjecture states that every Hamiltonian 4-regular simple
graph ...

1
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1
answer

107
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### Projection of an element of the $n$-simplex onto subset

Let $\mathbb{S}^{n}$ denote the $n$-dimensional probability simplex and let $\{e_1,...,e_{n+1}\}$ be the canonical basis of $\mathbb{R}^{n+1}$. Consider the subset $\mathbb{S}^{n}(K) \subset \mathbb{S}...

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343
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### What happens when the diffusion term in an SDE becomes zero?

Consider this time-homogeneous SDE, in the Ito sense:
$$dX_t= -(X_t-a)\,dt+\sigma(X_t)\,dW_t,$$
where $W_t$ is standard Brownian motion, $a<b\in\mathbb{R}$, $X_0\leq b$ a.s., and $\sigma(b)=0$. ...

2
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2
answers

283
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### How does pairwise independence restrict dependence to a third variable?

Let $X,Y,Z$ be random variables such that
$X,Y,Z$ have mean $0$ and variance $1$.
$X$ and $Y$ are pairwise independent.
$Z$ can be arbitrarily dependent on $X$ and $Y$.
My question is: What can we ...

2
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0
answers

69
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### Multi-scale 3- and 5-arm exponents for critical planar percolation

Consider critical site percolation on the planar triangular lattice. Denote by $A_j(m,n)$ the event that there are $j$ arms (paths from the inner boundary to the outer boundary) of alternating colour ...

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### Proof that Component-wise MH algorithm is invariant w.r.t. target measure

consider a standard situation in Bayesian modelling,
given real vector parameter $\theta=(\theta_1,\dotsc,\theta_n)$ and observations $x$ we derive a posterior distribution $\pi$ with posterior ...

4
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1
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250
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### Expected number of coin flips before you see a $k$-term arithmetic progression of heads

Let $\{X_i\}_{i \in \mathbb Z_+} $ be independent fair coin flips. Write $S := \{i \in \mathbb Z_+\, | \, X_i \text{ is heads}\}$, and define, for an integer $k \geq 3$,
$$Y := \inf \{n \in \mathbb N \...

6
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1
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### Hopf monads in categorical probability theory

1. Context. According to [1], probability monads are arguably the most important concept in categorical probability theory. In [2] Fritz and Perrone argue that "in order for a monad to really ...

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1
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81
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### Lower bounds for truncated moments of Gaussian measures on Hilbert space

Let $\mu_C$ be a centered Gaussian probability Borel measure on a real separable Hilbert space $\mathcal{H}$ with covariance operator $C$. Denote the ball with radius $r$ in $\mathcal{H}$ centered at ...

2
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97
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### The fluctuations of a random path

Suppose I have a $n \times n$ square grid and for each square, I assign 1 with probability $\frac{1}{2}$ and 0 with probability $\frac{1}{2}$. On the boundary, I put 1s on the lower half and 0s on the ...

4
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### Reference request: Gaussian measures on duals of nuclear spaces

I am interested in constructive quantum field theory where Gaussian measures on duals of nuclear spaces (specifically, the space of tempered distribution $\mathcal{S}'(\mathbb{R}^n)$) play a key role. ...

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0
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53
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### Dimension-free sample complexity for the inverse of Gaussian sample covariance?

Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need the inverse of the sample covariance $\Sigma_m^{-1}$ to be $\varepsilon$-close to true inverse covariance $\Sigma^{-1}$ (in ...

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### Weak limit of pushforward measures with finite second moments is also a pushforward measure with finite second moment

Suppose $\mu \in P(\mathbb{R}^d)$ and for each $n$, $T_n:\mathbb{R^d} \rightarrow \mathbb{R^d}$ is such that the pushforward $T_n \# \mu$ has a finite second moment. If $\{T_n \# \mu\}$ converges ...

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### Non-adjacent permutations

Suppose we have an $N$ by $M$ table. Suppose that $x=(a,b)$ and $y=(c,d)$ are two locations in the table, specified by their row and column indexes. We say that (x,y) is horizontally adjacent if $c=...

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111
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### Property of $p$-norm in the $n$-simplex

Let $\mathbb{S}^{n}$ be the canonical simplex of $\mathbb{R}^{n}$ and let $u = (1/n,\dotsc,1/n)$. Is it true that
$$\lVert x - u \rVert_p \leq \lVert y - u \rVert_p$$
implies that
$$\lVert x\rVert_p \...

3
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1
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151
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### Simple linear asymptotics for leaving time of particle in open-boundary TASEP

EDIT: It appears the hypothesis may not be true - I am not sure. I therefore changed my question.
ORIGINAL QUESTION:
Consider a system $n$ linked discrete cells numbered $1 \ldots n$. Particles are ...

3
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1
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147
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### Concentration for sum of order statistics

Assume we draw $n$ numbers uniform i.i.d. from $[0,1]$, and let the least $k$ of them be $x_1,\dots,x_k$. It is well-known that their expectations are $\frac{1}{n+1},\dots,\frac{k}{n+1}$, so the ...

2
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0
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85
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### Embedding a Markov chain in a Markov process

Let $X_{t\ge 0}$ be a Markov process with values in a metric space $(\mathcal{X},d)$ defined on a probabiltiy space $(\Omega,\mathcal{F},\mathbb{P})$ and let $(\tau_n)_{n=1}^{\infty}$ be a sequence of ...

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### Functional dependence not preserved in the weak limit

Let $M$ be a metric space, $f\colon M \to M$ a measurable function, and $(\mu_n)$ a sequence of probability measures on $M$.
Assume $(\mu_n, f_*\mu_n)$ converges weakly to a measure on $M\times M$ ...

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### Multivariable local CLT for uncorrelated (but dependent) coordinates?

Let $\vec f, \vec g\sim\mathcal{N}(0, \sigma^2I_n)$ be independent Gaussians.
Define $\mathsf{cyc}^i(\vec f) = (\vec f_i, \vec f_{i+1},\dots, \vec f_{n-1}, \vec f_0, \vec f_1,\dots, \vec f_{i-1})$ to ...

5
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3
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514
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### Winning game probability

At each round of a game with two players Alice and Bob, Alice can win with a fixed probability $a$ and Bob can win a fixed probability $b$, such that $a+b < 1$, otherwise there is a draw.
The game ...

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0
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73
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### High probability bound on number of sparse solutions to Gaussian linear system

Suppose we have a random matrix $A \in \mathbb{R}^{m \times n}$ with all entries i.i.d. from the standard Normal distribution $\mathcal{N}(0, 1)$. Suppose $k$ divides $n$, and let $S \subseteq \mathbb{...

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1
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178
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### Constructing a family of $3$-wise independence functions from $\mathbb{Z}_p^n \rightarrow \mathbb{Z}_p$

A family of function hash functions $\mathcal{H}:\{h:N\rightarrow M\}$ is call $k$-wise independent if whenenver $h$ is drawn uniformly from $\mathcal{H}$, let $x_1,\ldots,x_k$ be distinct elements of ...

7
votes

2
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212
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### Evolution of the empirical mean of a list as we remove elements proportional to their value

Consider a list of $N$ integers $k_1,k_2,\dots k_N$, drawn independently from some distribution $P(k)$ with $k_i \geq 1$. We denote its mean with $\langle k\rangle=\sum_{k=1}kP(k)$. The first two ...

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0
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102
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### Statistically stationary properties of expectations conditioned on the value of an Ornstein–Uhlenbeck process

Consider the modified Ornstein–Uhlenbeck process
$$\mathop{dx_t}=\theta(y_t-x_t)\mathop{dt}+\sigma\mathop{dW_t}$$
for a standard Brownian motion $W_t$ and $\theta,\sigma\in\mathbb{R}_{>0}$. Let's ...

1
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1
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74
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### Reference for the 'Brownian Representation Formula'

I am reading a paper ('Hydrodynamics of the N-BBM Process', by De Masi, Ferrari, Presutti, Soprano-Loto) which quotes the 'Brownian representation formula' to represent the solution of a free boundary ...

7
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486
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### A variation on the Borel–Cantelli lemma theme

Let $X,X_0,X_1,\dots$ be nonnegative independent identically distributed (i.i.d.) random variables. Let
\begin{equation*}
E:=\bigcap_{n\ge0}B_n,
\end{equation*}
where
\begin{equation*}
B_n:=\...