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.
8,960
questions
9
votes
1
answer
254
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One flip coin game
Nate has $n \geq 2$ coins $\{C_i\}_{0 \leq i \leq n-1}$ that each turn up heads with probability $\frac{i}{n-1}$ each, but he is not sure which ones are which.
He has \$1 with which to bet with. On ...
- 6,036
9
votes
0
answers
162
views
Does there exist such a probability distribution?
Does there exist a probability distribution over the set $\{(x,y,z)\in[0,1]^3\colon x+y+z=3/2\}$ whose projection on each of the three coordinate axes is the uniform distribution over the interval $[0,...
- 125k
4
votes
2
answers
303
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Which coupling of uniform random variables maximises the essential infimum of the sum?
Recall that a coupling of probability measures $\mu_i$ is a set of random variables $X_i$ defined on the same probability space $\Omega$ such that $X_i \sim \mu_i$.
Question: Let $\mu_1, \dots, \mu_n$ ...
- 6,036
0
votes
0
answers
30
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Can one parameterize transition rate matrices such that the stable distribution becomes independent of the transition rates?
I am trying to model a problem in which I need to describe a set of continuous time markov chains that depend on some parameter $v$. Thus, for each $v$, let $K(v)$ be $n\times n$ transition matrix ...
- 101
0
votes
1
answer
46
views
Lower bounding an alternating series with signs from a martingale difference sequence
Let $\epsilon_n \in \{-1, 1\}$ be a martingale difference sequence, in the sense that
$$M_n := \sum_{i = 0}^n \epsilon_i$$
is a martingale.
We assume $\epsilon_0 = \pm 1$ with probability $\frac{1}{2}$...
- 6,036
7
votes
2
answers
662
views
Why is $\mathbb R^{\mathbb N}$ not high-dimensional enough?
In this paper [1], the authors consider the limiting distribution of $$S_{n,p}:=\frac{1}{\sqrt n}\sum_{k=1}^nX_k$$ for $p\rightarrow\infty$ as $n\rightarrow\infty$, where $X_1, X_2,\dots, X_n$ are ...
- 181
2
votes
0
answers
102
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dimensionality reduction of Markov chains
Suppose that $M$ is a time-homogeneous (and, for simplicity, stationary) Markov chain on $d$ states, which induces the probability measure $P$ on paths of length $n$. I seek a Markov chain $M'$ on $d'&...
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4
votes
0
answers
66
views
Absolute Continuity of the Karhunen-Loeve expansion coefficients
The Karhunen-Loeve theorem (see these notes or the wikipedia page, for example) states the following:
Theorem: For a continuous, square-integrable, centered stochastic process $(X_t)_{t \in T}$ (with ...
- 41
0
votes
1
answer
80
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Exchanging the integral and infimum on the space of couplings
Let $\mu,\nu$ be probability measures on $\mathbb{R}^d$ with finite $p$-th moment ($p\in [1,\infty)$) and define the set of couplings by $\mathcal{C}(\mu,\nu)$ i.e. the set of probability measures on ...
- 305
2
votes
1
answer
124
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Measurability of $X$ with respect to $Y$ in conditional probability distributions
Let $\pi$ be a probability measure on $\mathbb{R}^2$ with respective marginals $\mu$ and $\nu$ such that $(X,Y) \sim \pi$.
Notation:
$\pi_{X=x}$ be the conditional distribution of $Y$ given $X=x$,
$\...
2
votes
0
answers
65
views
A Talagrand inequality for the supremum of partial sums over function classes under dependence. (Reference request)
As a consequence to the Talagrand concentration inequality, it is well known that for a measurable space $(S,\mathcal{S})$ and an i.i.d. sample $X_1,...,X_n$ of $S$-valued random variables, if $\...
- 131
3
votes
0
answers
110
views
Distribution of Brownian motion conditional on linear growth
Let $W$ be a standard $d$-dimensional Brownian motion with $W_0 = 0$ almost surely.
Fix a constant $\lambda > 0$ and timeframe $T > 0$, and consider the event
$$ E_T := \{|B_s| \geq \lambda s\ \...
- 6,036
12
votes
1
answer
297
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+50
Show there is no positive r.v. $U$ such that $\frac{1}{2} = \frac{\mathbb{E}[U^k 1_{U \ge (k+1)/2 }]}{\mathbb{E}[U^k]}, \, \forall k \in \mathbb{N}_0$
Let $U$ be a non-negative random variable such that for all $k \in \mathbb{N}_0$
\begin{align}
\frac{1}{2} = \frac{\mathbb{E}[U^k 1_{U \ge \frac{k+1}{2} }]}{\mathbb{E}[U^k]}.
\end{align}
In ...
- 611
1
vote
0
answers
74
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Supremum of sums of functions in $L^1$ taking random signs
Consider the Banach space $X=L^1([0,1])$, and let $n\gg1$ and $x_1, ..., x_n$ be any points in the unit sphere of $X$.
Is there any reasonable lower bound for $$\sup_{(\epsilon_i)_{i=1}^n \in \{-1,+1\}...
- 393
8
votes
1
answer
224
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Higher or lower? (#2)
$N \geq 2$ players play a game - at the start of the game, they are each given independently and uniformly a number from $[0, 1]$. On each round, they are to guess whether their number is higher or ...
- 6,036