# Tagged Questions

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

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### 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 it'...
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### 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, ...
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### 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 ...
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### 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 ...
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### Collecting stones in n buckets

There are $n$ stones distributed in $n$ buckets (initially one stone in each bucket). At each step the content of each bucket is put in a random bucket, chosen independently out of a set of $n$ new ...
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### Calculating the probability of an event defined by a condition on a Gaussian random process

Although the question itself can be expressed succinctly, I couldn't come up with a nice self-explanatory title - suggestions are welcome. Motivation/Background I was investigating whether it would ...
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### 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 ...
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### 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" ...
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### 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) ...
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### Area enclosed by Brownian motion (without winding number)

The question Average Value of Area Closed by Brownian Motion turned out to be about the Lévy area process, which measures "signed area with multiplicity" enclosed by Brownian motion (e.g. each ...
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### A stochastic process that is 1st and 2nd order (strictly) stationary, but not 3rd order stationary

I asked this question on stats.stackexchange.com a little while back but didn't get an answer. It was suggested that I post it here at the time. There appears to be some migratory problems going on ...
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### 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 ...
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### 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!
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### Numerical computation of Skorokhod integral

How can I numerically compute the Skorokhod integral of a non-adapted process? If it is adapted, that is easy since the integral is just an Ito integral. I have found that computing the Malliavin ...
Consider integers $r \geq 1$ and $k \geq 1$ and consider the following game: We start with $r$ tokens and at each round we choose $i \in \{1,...,r\}$ tokens to bet (if we have $N<r$ tokens we can'...