# Questions tagged [stochastic-processes]

A stochastic process describes the evolution of random variables/systems over time and/or space and/or any other index set. It has applications in areas such as econometrics, weather, signal processing, etc. Examples: Gaussian process, Markov process, etc.

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### Do measurable maps preserve stationary ergodicity?

In a recent effort to establish stationary ergodicity for a certain stochastic process, I just happened to come across a statement, which I find to be little bit confounding. Given two measurable ...
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### Multiple interval ratio of E[X / (X + Y)]

I have a sequence of interchanging on- and off-intervals, each pair identified by index $i$. The duration of the on-interval $i$ is represented by random variable $X_i$, and the duration of the off-...
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### What is the relation between stationary distribution, limiting distribution, ergodicity and detailed balance equation in a markov chain?

I have studied them from various sources and I am not able to make a strong conclusion with respect to their relation with each other. This is what I understand by these terms Ergodic : If all ...
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The negative binomial distribution with parameters $p\in(0,1)$ and $t>0$ is sometimes defined as the distribution of the number of failures before the $t$th success. This is supported on the set $\{... 0answers 33 views ### Imposing independence in multivariate MA process Consider a 4-dimensional MA(q) process: $$z_t = A(L) \epsilon_t$$ where $$A(L) = \sum_{j=0}^q A_j L^j$$ is a matrix-polynomial in the first$q$powers of$L$. What is the best way to impose ... 0answers 61 views ### Soft Question: What background do I need to understand Feynmann Kac Formulae by Pierre Del Moral? I am attempting to understand Sequential Monte Carlo(SMC) deeply, but with little theoretical background on probability theory and stochastic processes. Usually, the 'statistics' perspective of markov ... 0answers 146 views ### What is the mean and variance of a general stochastic integral? $$X_T = x_t+\int_t^T\mu(s,X_s)ds+\int_t^T\sigma(s,X_s)dW_s$$ where$W$is a Wiener process. What is the variance and mean of this process? It is well known$$E\left[\int_t^T\sigma(s,X_s)dW_s\right]=0.... 0answers 52 views ### Shortest time for$C$distinct objects to arrive Assume there are$N+1$objects denoted by$O_1,O_2,...,O_{N+1}$, and each object has an arrival process. The first$N$of them arrive with rate$\lambda_i,~i=\{1,2,...,N\}$, and$O_{N+1}$arrive with ... 0answers 55 views ### How Stochastic Binary Neurons works with rectified linear (or Linear threshold) case? I'm taking a course in coursera about neural networks. I understand the relation and difference between "Sigmoid Neurons" and "Stochastic Binary Neurons", but I don't how could you adapt "Rectified ... 0answers 218 views ### Integral of a continuous family of i.i.d bernoulli random variables Let$F(t) \sim \operatorname{Ber}(p)$for every$t \in [0,1]$. Let$X = \int_0^1 F(t)\,dt$.$X$is of course itself a random variable. Questions: Does$X$exist? If so, what is the distribution of$...
Is anyone aware of a direct relationship between the residual of an exponential moving average and the Ornstein-Uhlenbeck process? For example, assume a series, $Y_{t}$, that follows a geometric ...