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Questions tagged [stochastic-processes]

For questions about stochastic processes, for example random walks and Brownian motion.

A stochastic process is a collection of random variables representing the evolution of a system of random variables over time. A typical example is a .

A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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What does limiting fraction of time mean in Poisson processes and renewal processes?

I'm reading on Poisson and Renewal processes and I encountered the term limiting fraction of time. Although it was somewhat defined I couldn't really grasp the meaning of it. Edit Here is an example of what I mean: The weather in a certain locale…
A.K
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Covariance function meaning

I have this sentence in a report but I don't quiet know what it means. I am familier with covariance and covariance matrices but not with covariance functions. $f(t)$ is a continuous-time white-noise process with zero mean and the covariance…
WG-
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Error committed by linear interpolation of Levy process trajectory

Consider an independent increment process $X_t$, such that $X_t$ follows a continuous distribution. Examples would be a Brownian motion, Gamma process or a stable Levy process. When sampling from these processes on a regular time grid, linear…
Sasha
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Considering the backwards motion of a particle described by $X(t+\Delta t)=X(t)+f(X(t),t)\Delta t+g(X(t),t)\sqrt{\Delta t}\zeta$

If we have a particle whose position $X(t)$ is stochastically described by: $$X(t+\Delta t)=X(t)+f(X(t),t)\Delta t+g(X(t),t)\sqrt{\Delta t}\zeta$$ Where $\zeta \sim N(0,1)$. What is we want to consider the particle's movement in terms of $X(s)$ and…
Freeman
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How to calculate the average waiting time from an in-homogeneous Poisson process.

From the in-homogeneous Poisson process $\lambda(t)=10-5cos(2\pi t)dt $, we know $\lambda(t)$ is the arrival rate at time $t$ and $\int_{0}^{t}10-5cos(2\pi t)$ is the average arrivals during time $0$ to $t$. I'm wondering how can we calculate the…
user137988
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$X$ is a random variable which is measurable with respect to $\mathcal{F}_\infty$

Let $X_1, X_2, \dots $ be a sequence of independent random variables $\mathcal{F}_\infty$ denote the tail $\sigma$-algebra. If $X$ is a random variable which is measurable with respect to $\mathcal{F}_\infty$, then $X$ is almost surely constant,…
user53970
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A problem in Poisson Processes

Let $X_n$ be the interarrival times for a Poisson process $\{N_t; t \geq 0\}$ with rate $\lambda$. Is it possible to calculate the probability $P\{ X_k \leq T \text{ for } k \le n, \sum_{k=1}^{n}{X_k} = t, X_{n+1}>T\}$ for given $t$ and $T$ (suppose…
epsilon
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Homogeneous Poisson Process Questions

Assume that when the German plays Hungary in soccer, each team scores independently as a homogeneous Poisson process with rates $\lambda (\text{Germany}) = 1$ and $\lambda(\text{Hungary}) = 3$ goals per game. a) Expected number of total goals in a…
user129119
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Time average of a function of a process

A probably trivial question, but I don't understand how to solve it. Given a stochastic process $(X_t)_{t \geq 0}$ on a certain probability space $(\Omega, \mathcal{E}, P)$ with values in $[0,1]$, and a function $\varphi \in C^2([0,1], \mathbb{R})$,…
tomino
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How to estimate with random system matrix?

If the system is given by $y=Ax+z$ where $z$ is white Gaussian noise and $A$ is a random matrix with i.i.d. distribution with zero mean how can we estimate $x$ from received vector $y$? I tried linear minimum mean squared error (LMMSE) estimation…
triomphe
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Optimization problem solved via dynamic programming

Consider a situation where decisions are made in stages. The outcome of each decision is not fully predictable but can be anticipated to some extent before the next decision is made. The objective is to minimize a certain cost - a mathematical…
aaaaaa
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Some questions on Doob-Meyer decomposition

I'm confused with different version os the Doob-Meyer decomposition. For example, in the book by Protter, p.116, Theorem 16 it is given for every cadlag supermartingale $Z=Z_0+M-A$ where $M$ is local martingale and $A$ is increasing predictable…
paul
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Point process question

Suppose there is a homogeneous Poisson point process (PPP) $ \phi $ in the plane $\mathbb R^2$. Can we prove that the expected area in the plan that is closer to a given point $x_i \in \phi$ than to any other point in $\phi$, is finite? i.e.,…
triomphe
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The sum of two random processes with independent increment also has independent increment?

$X_n$ and $Y_n$ are random processes. Both of them have independent increment. Does $X_n + Y_n$ has that property?
zZzhuoer
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Probability of $i$ elements in a Branching process

I have a branching process that's family size each generation is Binomially distributed. How do I calculated the probability of the family size $Z$ in stage $n$ is $i$: $P(Z_n=i)$. At the begining of analyzing the problem, it seemed pretty…
Arturo
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