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 ...
6
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2answers
385 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 ...
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1answer
82 views

Is it true that all stationary measurable stochastic processes are “measurably stationary”?

(Philosophically, the following question is of a similar flavour to A stochastic process that is 1st and 2nd order (strictly) stationary, but not 3rd order stationary, but more "advanced".) Let ...
21
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2answers
959 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 ...
11
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3answers
8k 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
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3answers
452 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 ...
13
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4answers
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 ...
8
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2answers
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 ...
5
votes
3answers
529 views

A non-degenerate martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_t$ is a standard Brownian motion. Let $Y_t$ be a martingale given by ...
6
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2answers
757 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
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2answers
299 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 ...
6
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1answer
354 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 ...
5
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3answers
719 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
2answers
249 views

Liverani's CLT (a question)

Let $(\Omega,\mathcal{F},P)$ be a probability space where $\Omega$ is a complete separable metric space, let $T:\Omega\to \Omega$ ` be an ergodic transformation, let $\hat{T}:L^{2}_{_P}(\Omega)\to ...
4
votes
1answer
175 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) ...
4
votes
1answer
837 views

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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2answers
1k 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. ...
2
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0answers
57 views

Do the Birkhoff averages of a measurable stationary homogeneous Markov process in continuous time “converge to the right limit”?

Let $\,(P_x^t)_{x \in \mathbb{R} , t \geq 0}\,$ be a measurable Markovian family of transition probabilities - that is, a family of Borel probability measures $P_x^t$ on $\mathbb{R}$ such that for ...
2
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1answer
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!
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1answer
630 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 $p+q+r=1$. ...
4
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1answer
171 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
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1answer
171 views

Characterizations of the GOE/GUE family of distributions

This question is somewhat related to this one. Loosely speaking, when should I expect a GOE/GUE distribution? The angle of my approach to this is not through statements such as "there is a natural ...
3
votes
1answer
276 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
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1answer
642 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
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1answer
179 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
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1answer
351 views

Probability that a “closable” self-avoiding random walk forms a polygon

Consider a self-avoiding random walk on an infinite graph (for concreteness, the grid of 2-dimensional lattice points $\mathbb{Z}^2$), in which on each step, the next position is chosen uniformly at ...
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2answers
622 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
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2answers
365 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 ...
-1
votes
1answer
119 views

Property of relative entropy [closed]

For $X$ a measurable space and $P,Q$ two probability measures on $X$ s.t. $Q$ is absolutely continuous with respect to $P$, the relative entropy is defined as $$D(Q\|P)=\int_X \log(\frac{dQ}{dP})dQ,$$ ...