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Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
193
votes
Accepted
Why do probabilists take random variables to be Borel (and not Lebesgue) measurable?
One should be careful with the definitions here. Notation: Given measurable spaces $(X, \mathcal{B}_X), (Y, \mathcal{B}_Y)$, a measurable map $f : X \to Y$ is one such that $f^{-1}(A) \in \mathcal{B} …
25
votes
"Are you taller than the average of those who are taller than the average?"
We have $\mu_n \uparrow \infty$. Proof: let
$$G(y) = \frac{\int_y^\infty x f(x) dx}{\int_y^\infty f(x) dx}$$
so that $\mu_{n+1} = G(\mu_n)$. Clearly $G$ is a continuous function and $G(y) > y$ for …
20
votes
Accepted
Measure induced on [0, 1] by infinite tosses of biased coin
For $\omega \in [0,1]$, let $X_i(\omega)$ be the $i$th binary digit of $\omega$. (If $\omega$ is a dyadic rational and thus has two binary expansions, let's say we choose the expansion that ends with …
19
votes
Sum of independent random variables
Yes, they are normally distributed. This is the Lévy-Cramér theorem.
18
votes
Mixtures of Gaussian distributions dense in distributions?
One way to say this is: Given any random variable $X$, there is a sequence of random variables $X_n$ whose distributions are finite mixtures of Gaussians, such that $X_n \Rightarrow X$ (i.e. $X_n$ con …
16
votes
Accepted
Does there exist an event independent of a given sigma-algebra?
No. For a very simple example, take $\Omega = \{a,b,c\}$ consisting of three points, with $\mathcal{F} = 2^\Omega$ and $P(A) = |A|/3$ the uniform measure. Let $\mathcal{G} = \{\{a\}, \{b,c\}, \Omega …
14
votes
Limit of distance between two random points in a unit $n$-cube
I think this easy probability argument, using the natural coupling, gives us the limit and the growth rate.
Let $\{U_i, V_i, i=1,2,\dots\}$ be iid uniform on $[0,1]$. Let $$D_n := \sqrt{(U_1 - V_1)^ …
12
votes
Accepted
Constructing an independent uniform random variable from two independent ones
I think this works for a continuous $h$.
Let $f : \mathbb{R} \to [0,1]$ be the "triangle wave" function given on $[0,1]$ by
$$f(u) = \begin{cases}1-2u, & 0 \le u \le \frac{1}{2} \\ 2u-1, & \frac{1}{2 …
11
votes
Accepted
What does convergence in distribution "in the Gromov–Hausdorff" sense mean?
Following the notation of the paper, let $\mathbb{K}$ be the metric space of all compact metric spaces, equipped with the Gromov-Hausdorff metric $\mathrm{d_{GH}}$. Then we can express convergence in …
10
votes
How to sample pairwise independent gaussians
I'm not sure what you mean by "is there a way to sample". But the following fact may be of interest:
Proposition. Let $F_1, F_2, \dots$ be any sequence of distributions (possibly infinite), and …
10
votes
What is convolution intuitively?
Just to add yet another answer:
A heuristic that I like is that convolution is like measuring temperature with a large thermometer.
In one dimension, imagine a trough filled with some sort of warm g …
9
votes
Does the strong law of Large Number hold for an infinite dimensional Brownian motion?
Yes.
One way to prove it is using Fernique's theorem:
Let $\mu$ be a centered Gaussian measure on a separable Banach space $(X, \|\cdot\|)$. Then there exists $\alpha > 0$ such that $$\int_X e^{ …
9
votes
Accepted
Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)
Your intutive reasoning is leading you astray because you are thinking of Brownian motion as behaving like a smooth curve, for which there is a well-defined "direction" in which it is heading. Browni …
8
votes
Accepted
Stochastic integral with respect to discontinuous martingale
Yes, you do need the integrand $f(X_t)$ to be predictable. If it is merely adapted you may not get a martingale.
Intuitively, the stochastic integral $\int Y_t\,dM_t$ tells you the profit from a sto …
8
votes
Accepted
Does a Central Limit Theorem imply a series is $O(\sqrt{N})$?
The sharp general result in this direction is the classical law of the iterated logarithm (LIL). Suppose, after renormalizing if necessary, that the $x_n$ are iid with zero mean and unit variance. T …