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.
26 questions from the last 30 days
0
votes
0
answers
37
views
Explosion of the moment of double stochastic exponential
We consider the stochastic system
$$\frac{dS_t}{S_t}=-R_t\,dW_t,$$
with
$$dR_t=-R_t\,dt-R_t\,dW_t, \quad R_0>0.$$
We conjecture, and would like to show that
$$\mathbb{E}[S_t^2]
=
S_0^2\,\mathbb{E}\...
- 696
1
vote
0
answers
40
views
Tail log-convexity of moments of an even Hermite polynomial of a Gaussian
Let $G\sim N(0,1)$ and let $\{\mathrm{He}_n\}_{n\ge 0}$ denote the probabilists' Hermite polynomials.
Let $H_n:=\mathrm{He}_n/\sqrt{n!}$ be the orthonormal version, so that
$\mathbb{E}[H_n(G)H_m(G)]=\...
- 73
7
votes
1
answer
261
views
What is the value of two player Liar’s Dice?
Alice and Biboo play a game. Each privately rolls a fair $n$ sided die labelled with $\{0, ..., n-1\}$, visible only to themselves. Players take turns with Alice starting first. Alice starts by making ...
- 11.2k
2
votes
0
answers
91
views
Concentration inequalities for a specific type of random variables
Consider dynamics on the time interval $[0, n]$, $n \in \mathbb{N}$, where we events (a birth) happen after independent and unit-exponentially distributed waiting times. Every time $t$ such a event ...
- 838
6
votes
1
answer
218
views
A too good to be true moment inequality for empirical processes
I have a question regarding an inequality that I obtained which seems to be too good to be true.
Consider a sequence $(X_i)_{i\leq N}$ of independent and identically distributed r.v.s. with law $\mu$ ...
- 273
32
votes
2
answers
713
views
Is the expected distance between two random interior points of a convex body always at most that of two random boundary points?
Let $K \subset \mathbb{R}^2$ be a convex body. Define two quantities:
Interior mean distance. Let $X, Y$ be independent and uniformly distributed in $K$. Set
$$\Delta(K) \;=\; \mathbb{E}\,\|X - Y\|.$$...
- 1,012
13
votes
2
answers
256
views
Behavior of certain random products of power series
Let $\varepsilon_1, \varepsilon_2, \cdots$ be independent random
variables taking the values $\pm 1$ with probability $1/2$ each. What
can be said about the coefficients $a_k$ of the power series ...
- 54.1k
2
votes
2
answers
200
views
Rate of convergence of powers of empirical measures
Consider a product measure $\mu\otimes\mu$ on a "nice" product space (e.g., compact metric space), draw $n$ iid samples $(x_1,y_1),\ldots,(x_n,y_n)\sim\mu\otimes\mu$, and let $\nu_n=\frac{1}{...
- 65
-3
votes
0
answers
50
views
Has the Gram factorisation of the Donsker-Varadhan Hessian's nonequilibrium correction appeared before? [closed]
Consider an irreducible continuous-time Markov chain on $\{1,\dots,n\}$ with generator $Q$ and stationary distribution $\pi$. Pass to the Fisher-conjugated generator $L_\pi = D_\pi^{1/2} Q D_\pi^{-1/2}...
2
votes
0
answers
53
views
On a skew diffusion process
Let $p \in [0,1]$, and let $X=(X_t)_{t \ge 0}$ be a skew Brownian motion with parameter $p$ defined a probability space $(\Omega,\mathcal{F},P)$. It is known that $X$ can be defined as a solution to ...
- 21
7
votes
3
answers
505
views
Nash equilibria in simultaneous blind poker
Three players are playing a game of blind poker. Each is dealt a number uniformly and independently from $[0, 1]$. Each player can see the cards of both the other players but not their own.
Initially ...
- 11.2k
0
votes
0
answers
43
views
Rationality of Joint Subset Coalescence Probabilities for Products of Random $n$-Cycles
Let $\mathcal{C}_n$ denote the set of all $n$-cycles in the symmetric group $\mathfrak{S}_n$, which has cardinality $(n-1)!$. Draw $\sigma$ and $\tau$ independently and uniformly from $\mathcal{C}_n$, ...
- 61
0
votes
0
answers
184
views
+50
Different results for joint measurability of conditional densities
Let $(\Omega, \mathcal{A}, P)$ be a probability space and let $Y:(\Omega, \mathcal{A})\rightarrow (\mathcal{Y}, \mathcal{G})$ and $X:(\Omega, \mathcal{A})\rightarrow (\mathcal{X}, \mathcal{F})$ be ...
- 181
1
vote
1
answer
132
views
Does Bayes theorem imply that if $P_{X|\Theta}$ is a regular cond. distr., that then also $P_{\Theta|X}$ is a regular cond.distr.?
In the book Theory of Statistics, Schervish states in Theorem 1.31 the Bayes theorem very rigorously. To this end let $(\Omega, \mathcal{A}, P)$ be an underlying probability space, let $\Theta:(\Omega,...
- 181
13
votes
4
answers
1k
views
Pi Day: estimating pi using probability
Are there other ways statistics/probability can be used to estimate pi?
The most classic response is Buffon's needle approach ($2/\pi$).
Monte Carlo simulation of uniform-randomly throwing darts at a ...