# Tagged Questions

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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### Can we model this set of experiments as an stochastic process and estimate the sample size?

I have an image with the size 5575x9440 and I'm implementing a modified version of the algorithm used in this paper on it, but because the code performance is low ...
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### Compound poisson process invariant measure

Let $\rho$ be a probability measure in $\mathbb{R}$, $(N_t)$ a standar Poisson process and $(X_i) \stackrel{\text{i.i.d.}}{\sim} \rho$. Then $$Z_t = \sum_{n=1}^{N_t} X_n$$ is call a compound poisson ...
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### conditions in which the repair shop process is recurrent (null\positive) or transient

here's the Story: Let $\epsilon_1.\epsilon_2,...$ be i.i.d numbers of machines for repair to the repair shop on mornings of days $1, 2,...$ . Assume that the shop is capable of repairing exactly K ...
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### Conditions for limiting distribution to equal stationary distribution of SDE

I have SDE of the form $$dX_t=a\mathopen{}\left(X_t\right)dt+b\mathopen{}\left(X_t\right)dW_t,$$ where $W$ is Brownian motion. If the stationary distribution of $X$ exist is it equal to the limiting ...
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### Show that $\hat{Y}$ is an optimal linear estimator of Y

Relevant Information. Let $X(t)$, $t \in T$ be a second order process. Let $M_0$ be the set of random variables of the form $a + b_1X(s_1)+ \cdots + b_nX(s_n)$ for a positive integer $n$ and constants ...
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### How to compute the integral of a renewal process?

Let $\{S_n:n=1,2,\ldots\}$ be a renewal process (with the convention $S_0\equiv 0$) with $\mathbb E[S_1]<\infty$ and $S_1$ absolutely continuous with density $f$. Let $\{N(t):t\geqslant0\}$ be the ...
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### Finding marginal distributions of a process

Suppose we have a process given by $S_t = S_0 \exp(\sigma W_t + (r - \frac{1}{2} \sigma^2 )t)$, and we wish to find the marginal distribution for $S_T$. (Note: $W_t$ is a $\mathbb{Q}$-Brownian Motion) ...
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### I can not deduce this formula from an article…

I do not understand the calculation of a certain article published in J. Chem. Phys. 137 (2012): I have a continous time discrete state model with a total number of for example $N_0$ ...
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### Time scaling birth process in Poisson process

Given a birth process $\{B_t:t\geqslant0\}$ with $\lambda >0$, define $$K_t=\int_{0}^{t}B_s ds=\sum_{i=1}^{n}B_{t_{i}}(t_{i+1}-t_i)$$ if there were $n$ births in $[0,t]$ and let $t_{i}$ be the ...
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### Find (a,b) such that aX+bY is a Brownian motion

Let $$\begin{cases} dX_t = \mathrm{sin}(X_t+Y_t) dW_t \\ dY_t = \mathrm{cos}(X_t+Y_t) dV_t \\ X_0=Y_0=0 \end{cases}$$ Where $(W,V)$ is a two-dimensional Brownian motion and $(X,Y)$ be a strong ...
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### condition for recurrence of semi-stable process

Let $(X_t)$ be a nontrivial $\alpha$-semi-stable process on $\mathbb R$. I want to prove that if $1\le\alpha\le2$, $(X_t)$ is recurrent if and only if it is strictly $\alpha$-semi-stable. I want to ...
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### Levy process measurable past

For a Levy-process $(X_t)_{t\geq 0}$ with stationary indepedent increments which is a markov process, we know that its law is defined by its one dimensional distribution. This is so because for its ...
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