A stochastic process is a collection of random variables usually indexed by a totally ordered set.

**63**

votes

**8**answers

11k views

### Is there a natural random process that is rigorously known to produce Zipf's law?

Zipf's law is the empirical observation that in many real-life populations of n objects, the $k^{th}$ largest object has size proportional to $1/k$, at least for $k$ significantly smaller than $n$ ...

**44**

votes

**5**answers

3k views

### Escape the zombie apocalypse

Consider zombies placed uniformly at random over $\mathbb{R}^2$ with asymptotic density $\mu$ zombies/area. You are placed at a random point and can move with speed $1$. Zombies move with speed $v\leq ...

**38**

votes

**4**answers

2k views

### Polynomials on the Unit Circle

I asked this question in math.stackexchange but I didn't have much luck. It might be more appropiate for this forum. Let $z_1,z_2,…,z_n$ be i.i.d random points on the unit circle ($|z_i|=1$) with ...

**36**

votes

**6**answers

1k views

### Tetris-like falling sticky disks

Suppose unit-radius disks fall vertically from $y=+\infty$,
one by one, and create a random jumble of disks above the $x$-axis.
When a falling disk hits another, it stops and sticks there.
Otherwise, ...

**33**

votes

**1**answer

1k views

### Modeling question: how often does “the world's oldest person” die?

This story yesterday (no need to follow the link to understand the question!)
http://www.cnn.com/2011/US/02/01/texas.oldest.person.dies/index.html?hpt=T2
reminds me that I've often wondered about ...

**30**

votes

**6**answers

2k views

### Deep Learning / Deep neural nets for mathematician

I am interested in finding out the math ideas behind the technologies that are under the umbrella of "Deep Learning" or "Deep neural nets".
Most of the papers/books that are often quoted in ...

**26**

votes

**3**answers

2k views

### Expectation of a random sum

Let $X_1, X_2, X_3,\dots$ be an i.i.d. sequence of random variables with finite mean. Write $S_n=X_1+X_2+\dots+X_n$.
Let $N$ be a non-negative integer-valued random variable with finite mean. $N$ may ...

**21**

votes

**2**answers

1k views

### Drawing natural numbers without replacement.

Suppose we start with an initial probability distribution on $\mathbb{N}$ that gives positive probability to each $n$. Let's call this random variable $X_1$ so we have $P(X_1=n)=p_{1,n}>0$ for all ...

**18**

votes

**3**answers

921 views

### Growing random trees on a lattice $\rightarrow$ Voronoi diagrams

Imagine growing trees from $k$ seeds on a square $n \times n$ region
of $\mathbb{Z}^2$.
At each step, a unit-length edge $e$ between two points of
$\mathbb{Z}^2$ is added.
The edge $e$ is chosen ...

**18**

votes

**1**answer

899 views

### Existance of certain almost invariant functions related to amenability and piece-wise transformations

We would like very much to know the answer to the following question:
Let $\|\cdot\|$ be any norm on $\mathbb{Z}^d$ and let $W(\mathbb{Z}^d)$ be the group of all bijections of $\mathbb{Z}^d$ such ...

**17**

votes

**2**answers

903 views

### Age of Stochasticity?

One user on MSE made an interesting question, which was unanswered so I suggested him to post it here but he refused for personal reasons and said I could ask it here.
The question is this:
Today ...

**17**

votes

**2**answers

831 views

### Random Walk on $\mathbb{R}$ with Uniformly Distributed Steps and “Reflective” Boundary at Origin

A particle lies on the real number line at the origin. For each step taken, the particle moves from its current position a distance (and direction) chosen equi-probably from range $[-1,r]$. However, ...

**16**

votes

**1**answer

394 views

### How much universality is there for contact processes?

A couple of weeks ago I had to pick my daughter up from her nursery because of suspected chicken pox. It turned out to be a false alarm, but while I was waiting at the doctor's surgery to establish ...

**14**

votes

**4**answers

2k views

### A Markov process which is not a strong markov process?

Can anyone give an example of a Markov process which is not a strong Markov process? The Markov property and strong Markov property are typically introduced as distinct concepts (for example in ...

**13**

votes

**4**answers

2k views

### Correlated Brownian motion and Poisson process

Is there an (easy) way to construct, on the same filtered probability space,a Brownian motion $W$ and a Poisson process $N$, such that $W$ and $N$ are not independent ?
I first asked this question ...

**13**

votes

**6**answers

943 views

### Optimal pebble-packing shape

Suppose you throw many ($n$) congruent convex bodies (in $\mathbb{R}^3$) of unit volume (or of unit area in $\mathbb{R}^2$) into a large container, and shake it until little else changes.
Q. ...

**13**

votes

**8**answers

2k views

### Brownian bridge interpreted as Brownian motion on the circle

Is it reasonable to view the Brownian bridge as a kind of Brownian motion indexed by points on the circle?
The Brownian bridge has some strange connections with the Riemann zeta function (see ...

**13**

votes

**3**answers

1k views

### Expected Degree of a vertex in Delaunay Triangulations

Assume you have a Poisson point process of constant intensity $\lambda$ in the Euclidean plane. From this point process we construct the Delaunay triangulation (or the Voronoi tessellation for that ...

**13**

votes

**1**answer

417 views

### Fictitious density of paths of diffusion processes outside the Cameron--Martin space

Let $X_t$ be an $n$-dimensional diffusion process satisfying the following Itō SDE over $[0,1]$:
$$dX_t = f(X_t)\,dt + dW_t,$$
where $W_t$ is an $n$-dimensional Wiener process and $f$ is of class ...

**13**

votes

**1**answer

984 views

### Bochner integral of stochastic process = path by path Lebesgue integral?

After some helpful comments, I realized that I had to repost this question in a more systematic way.
On a complete probability space, let $\mathcal{H}_0$ denote the Hilbert space of square ...

**13**

votes

**0**answers

339 views

### Why, and how badly, does the proof of “no percolation at the critical point in half-spaces” fail for full spaces?

The proof by Barsky et. al. that there is no percolation in half-spaces proceeds by a dynamic renormalization argument. The proof couples critical percolation in the half-space $\mathbb{H}^d$ with a ...

**12**

votes

**3**answers

9k views

### Maximum of Gaussian Random Variables

Let $x_1,x_2,…,x_n$ be zero mean Gaussian random variables with covariance matrix $\Sigma=(\sigma_{ij})_{1\leq i,j\leq n}$.
Let $m$ be the maximum of the random variables $x_{i}$
$$
...

**12**

votes

**2**answers

660 views

### Is there a percolation threshold in the hard discs model?

Take a random configuration of $n$ non-overlapping discs of radius $r$ in the unit square $[0,1]^2$. (You could think of this as taking $n$ points uniform randomly in $[r,1-r]^2$ and then restricting ...

**12**

votes

**3**answers

475 views

### An “inchworm-like” random walk on an integer interval

Imagine I place $k$ stones on an infinite one-dimensional integer interval $Z$ s.t. no stone is more than some distance $d$ from any other stone. For example, if $d=1$ and $k = 5$, we might place the ...

**12**

votes

**2**answers

796 views

### Non-integrable ergodic theory

Can anyone help me out with proofs/counterexamples? I'm working on an operator-valued multiplicative ergodic theorem and need what may(?) be a well-known fact. This fact (if true) would help me get ...

**12**

votes

**1**answer

473 views

### Random walk origin return monotinicity

Consider a Markov chain on $\mathbb{Z}^d$ with transition kernel $P$ for adjacent vertices (non-diagonal). Essentially this is a $d$ dimensional random walk with the probability of a transition ...

**12**

votes

**0**answers

747 views

### surprisingly difficult filtration problem

I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky:
Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...

**11**

votes

**1**answer

2k views

### Martingales in both discrete and continuous setting

I am wondering, polynomials like
$S_n^4-6n S_n^2+3n^2+2n$ for $$S_n=\sum_{i=1}^n{X_i}$$ where $$\mathbb{P}(X_i=1)=\mathbb{P}(X_i=-1)=\frac{1}{2}$$ is a martingale (under the conventional filtration). ...

**11**

votes

**2**answers

460 views

### Error term for renewal function

Consider a sequence of independent uniform $[0,1]$ random variables, and for nonnegative real $t$, let $m(t)$ be the expected number of terms in the first partial sum that exceeds $t$. For instance ...

**11**

votes

**3**answers

2k views

### Infinitesimal generators of stochastic processes

What's the $L^1$ analogue of Stone's theorem saying that any strongly continuous 1-parameter unitary groups has a unique self-adjoint generator?
More precisely: let $X$ be a measure space ...

**11**

votes

**1**answer

669 views

### How far can a particle travel from its origin if it exhibits self-avoiding Brownian motion in two-dimensions?

Let's say I have a point-like Brownian particle undergoing two-dimensional diffusion on an infinite plane with the caveat is that the particle can never return to a coordinate that it previously ...

**11**

votes

**5**answers

3k views

### Brownian motion, martingales, Markov Chains - Rosetta Stone

What are the most
fundamental/useful/interesting ways in
which the concepts of Brownian motion,
martingales and markov chains are
related?
I'm a graduate student doing a crash course in ...

**11**

votes

**1**answer

524 views

### “continuous” and “discontinuous” phase transitions in branching processes.

Consider a Galton-Watson branching process, with offspring distribution
$\mathbf{p}=(p_0, p_1, \dots, p_n, \dots)$.
Let $O$ be the root of the branching process.
Write $\eta=P(\text{process survives ...

**10**

votes

**2**answers

807 views

### Why do we want maps to be measurable (in countably-additive setting)

When I have to explain things that I am doing to people who did not do (or even did not learn) measure-theoretical probability, I think of getting a question in the title, and I am not sure I have ...

**10**

votes

**1**answer

291 views

### Does Brownian motion immediately visit both sides of a Jordan curve?

Let $C$ be a Jordan curve in $\mathbb{R}^2$. By the Jordan curve theorem, $\mathbb{R}^2 \smallsetminus C$ is uniquely partitioned into two connected regions $A$ and $B$ (the interior and exterior).
...

**10**

votes

**2**answers

322 views

### Minimal expected absolute value of linear combinations of Gaussian random variables

I am interested in the following question. Consider $n$ independent standard normal random variables $g_i$. Cosider a linear combination $w_1g_1+\cdots+w_ng_n$. Can one give a "decent" upper bound for ...

**10**

votes

**2**answers

828 views

### Markov chains: invariant measures and explosion

The following seems like such an elementary question, but I didn't get anywhere with it.
Suppose you are considering a Markov chain in continuous time which is transient and has an invariant measure ...

**10**

votes

**1**answer

826 views

### Hardy spaces: analysis <---> martingales

Let $H^p$ be the Hardy space of analytic functions on the open unit disk $\mathbb{D}$: $f \in H^p$ if $f$ is analytic on $\mathbb{D}$ and $\sup_{r < 1} \int_0^{2\pi} |f(re^{i\theta})|^p d\theta ...

**10**

votes

**1**answer

287 views

### Trapping a particle

A particle starts a brownian walk in the middle of a long tunnel in the plane, at one end of the tunnel is a region Y of given area A.
Does the shape of region Y affect average time for the particle ...

**10**

votes

**1**answer

1k views

### Gluing Markov processes

I am looking for a reference on the gluing together of strong Markov processes to get a new one.
Here is an example of what I have in mind. Let $B^1, B^2, \ldots $ be independent one-dimensional ...

**9**

votes

**2**answers

1k views

### The conditions in the definition of Brownian motion

A (standard, real-valued) Brownian motion $W = \{W(t): t \geq 0\}$ is commonly defined by the following properties: 1) $W(0) = 0$ a.s., 2) the process has independent increments, 3) for all $s,t ...

**9**

votes

**1**answer

335 views

### Extending state space to make a process Feller

Let $X$ be a locally compact Hausdorff space, and let $Y_t$ be a continuous Markov process on $X$ with transition function $P(t, x, \Gamma) := \mathbb{P}_x (Y_t \in \Gamma)$. Let $T_t$ be the ...

**9**

votes

**1**answer

261 views

### a question on 0-1 valued stochastic process

Here's a question on probability theory from a layman (I'm a game theorist). It is very likely that the question will be a straightforward matter for someone who is a probability theorist. I guess I'm ...

**9**

votes

**2**answers

4k views

### Coin Pusher Game

While doing laundry at my local laundromat, I saw a coin pusher game. Below is a picture, and here is a video depicting how it works (disregard non-coins).
Essentially, one has a distribution of ...

**9**

votes

**1**answer

1k views

### Karhunen–Loève approximation of Brownian motion and diffusions

The Karhunen–Loève theorem says that Brownian motion on the interval [0,1] can be represented as follows:
$W_t = \sum_{n=1}^\infty Z_n \frac{\sin((n-1/2)\pi t)}{(n-1/2)\pi},$
where $Z_n \sim ...

**8**

votes

**2**answers

530 views

### Does the strong law of Large Number hold for an infinite dimensional Brownian motion?

For finite-dimensional Brownian motion $W_t$, it is well known that
\begin{equation}
\lim_{t\to \infty}\frac{W_t}{t}=0,\text{ a.s. }\ \ \ \ \hspace{1cm} \langle 1\rangle
\end{equation}
Now suppose we ...

**8**

votes

**2**answers

1k views

### Gaussian processes, sample paths and associated Hilbert space.

Given a Gaussian process on some topological space $T$, with a continuous covariance kernel
$C(\cdot,\cdot)\colon T\times T\to R$, we can associate a Hilbert space, which is the reproducing kernel ...

**8**

votes

**3**answers

571 views

### A question on Cramer's theorem

Almost everybody is familiar with Cramer's theorem: a sum $X+Y$ of of independent random variables is normal if and only if both $X$ and $Y$ are normal. Are there any other classes of distributions ...

**8**

votes

**3**answers

637 views

### Maximum of the expectation of maximum of Gaussian variables

Suppose $X=(X_1,\ldots,X_n)$ is a Gaussian vector with each entry $X_i$ marginally distributed as $\mathcal{N}(0,1)$. Want to find out the possible maximum of
$$\mathbb{E}\max_{1\le i\le n}|X_i|$$
and
...

**8**

votes

**5**answers

908 views

### $E(X_1 | X_1 + X_2)$, where $X_i$ are (integrable) independent infinitely divisible rv's “of the same type”

The following is inspired by this recent question on math.stackexchange.
Two standard exercises in conditional expectation are to find ${\rm E}(X_1|X_1+X_2)$ where:
1) $X_i$, $i=1,2$, are independent ...