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 ...