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

**10**

votes

**2**answers

367 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 ...

**21**

votes

**2**answers

900 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 ...

**10**

votes

**3**answers

6k 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

**3**answers

432 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 ...

**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 ...

**12**

votes

**3**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 ...

**4**

votes

**2**answers

248 views

### Wiener measure and Bochner Minlos

I am reading probability theory and I have a question. The Bochner-Minlos theorem roughly says that if we have $E \subset H \subset E^*$ where $H$ is a Hilbert space, then there is a unique measure ...

**3**

votes

**2**answers

566 views

### Finite time hitting probabilities for Brownian motion in the plane

Consider a Brownian particle in the plane with a circular trap at the origin. If we give the particle enough time it falls into the trap (since Brownian motion is space filling in 2D). However, ...

**5**

votes

**3**answers

671 views

### One can earn nothing on the Brownian motion, true ?

Consider any discrete time stochastic process $p(n)$ (price) with independent increments $\xi_k$ and $E(\xi_k)=0$. E.g. Brownian motion (i.e. $\xi_k = N(0,1)$).
Consider some "trading strategy" ...

**5**

votes

**1**answer

291 views

### How does the distribution of Erdős number evolve over time ? How to build a model to fit the real data ?

Let $E(n,t)$ be the number of mathematicians with finite positive Erdős number $n$ at time $t$. As old mathematicians leave, and new mathematicians come, how does $E(n,t)$ change over time ?
We can ...

**4**

votes

**1**answer

147 views

### Concurrency related problems in $n$ independent, parallel $M/M/1$ queues

Queueing Model:
Consider $n$ independent, parallel $M/M/1$ queues with identical arrival rate $\lambda$ and service rate $\mu$. For each $M/M/1$ queue, we use the FCFS (First Come First Served) ...

**3**

votes

**1**answer

138 views

### Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes

I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to ...

**3**

votes

**2**answers

818 views

### stopping time expectation for gambler's ruin

2 players A and B start with x & y dollars respectively, and they bet against each other 1 dollar each time by tossing a fair coin.
I let $X_n = x + \sum_{i=1}^{n}\xi_i$ where $\xi_i$ are i.i.d. ...

**4**

votes

**1**answer

167 views

### A queuing process where customers must be detected

Imagine a scenario where customers arrive in some queue according to a Poisson process with rate parameter $\lambda_{arr}$, and where the process of responding to the customers has a kind of ...

**3**

votes

**1**answer

265 views

### Casino does not win, while clients do lose ? Prob_loss(T) = 1 - .8/sqrt(T)?

Setup. Let casisno generate a color: black or red with equal probability.
Let client try to guess the color. If guess is correct - he earns 1 coin from casino, if not - he gives one to casino. If he ...

**3**

votes

**1**answer

429 views

### Topological conditions of Kolmogorov Extension Theorem

KET is often used to construct stochastic processes in continuous time when the state space is $\Bbb R^d$. As far as I am familiar with its proof, it uses standard monotonic class-like arguments ...

**2**

votes

**1**answer

153 views

### Empirical estimator fot the total variation distance on a finite space

I have two probability measures $p$ and $p'$ on a finite set $X$ which I do not know precisely, but which I can sample from. I would like to estimate their total variation (omitting multiplier $2$):
...

**2**

votes

**1**answer

1k views

### Right-continuity of natural filtrations

My question: Is the natural filtration of a right-continuous process also right-continuous?
I would say yes, but don't know where to start proving it.
Thanks for your help/ideas!

**1**

vote

**1**answer

492 views

### Hitting time probability in a Random Walk with possibility to die.

A Random Walker can move of one unit to the right with probability $p$, to the left with probability $q$ and it can jump again to the starting point with probability $r$ and die. Naturally $p+q+r=1$. ...

**1**

vote

**2**answers

584 views

### Change of time or change of measure

Consider simple diffusion $dX_t = \sigma dw_t$ and a parameter $a>0$ and $X_0=x$. Let us denote $Y_t = X_{at}$ - thus we made a change of time. Let us denote an original measure as $P$. How to find ...

**0**

votes

**2**answers

313 views

### Representation theorem for continuous uniformly integrable martingales

For some time $u$ and positive continuous process $a_t$ adapted to $\mathcal{F}_t$ I have a (continuous-time) martingale defined as:
$$M_t(u) = \mathbb{E}[a_u | \mathcal{F}_t]$$
for $t\leq u$. I ...