All Questions
Tagged with statistics stochastic-processes
712
questions
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What are necessary and sufficient conditions for the Bellman Equation to be solvable?
I am studying Markov Rewaed Processes right now, and I wish to gain a deeper understanding of the Bellman equation's relationship with them.
I learned the Bellman equation in the following form:
$v = ...
- 1
6
votes
1
answer
94
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Show that $X_{t}:=\alpha X_{t-1}+\epsilon_{t}$ is strictly stationary for $|\alpha|<1$ and $\epsilon_{t}$ i.i.d$~\sim N(0,\sigma^{2})$.
The title can be shortened to "prove that $AR(1)$ processes are strictly stationary when $|\alpha|<1$". This has been discussed many times on MSE and Cross Validated, but I found no ...
- 13.2k
2
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299
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Rough path expected signature vs cumulant-generating function / characteristic function
What is the point of using rough path expected signature to characterize the law of а stochastic process when the cumulant generating function is known ($\log\mathbb{E}[e^{i\theta X(t)}]$)?
Since an ...
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Approximation of a new kernel by a linear combination of previous kernels
From the reference by Knutsen, page 25, Kernel linear independence test is explained
Knutsen, Sverre. "Gaussian processes for online system identification and control of a quadrotor." (2019)...
- 83.9k
9
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1
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278
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Expected difference between the largest and second largest observations in a sample of i.i.d. normal variables
Let $X_1,\dots,X_n$ be an i.i.d. sample from the standard normal distribution. Let
\begin{align}
\mu_n = \mathbb{E}[X_{(n)} - X_{(n-1)}],
\end{align}
be the expected difference between the largest ...
- 239k
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43
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Expected value of dirac measure
Let $x_n$ be a series in $\mathbb R^d$. When does $\mu_n = \delta_{x_n}$ converge weakly?
This is my attempt to answer this. Weak convergence means:
$$\int f \, d \mu_n \rightarrow \int f \, d \mu$$ ...
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2
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How to take the variance of a second order expansion? $\text{Var}\left[aX+bY+cXY+mX^2+nY^2\right]$
How to take the variance of a second order expansion? $\text{Var}\left[aX+bY+cXY+mX^2+nY^2\right]$
Let say we have 5 real-valued constant parameters $\{a,\ b,\ c,\ m,\ n\}$, and two random variables $...
- 170k
2
votes
1
answer
249
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How far are the Mode and the Median of the Log-Normal distribution from behaving as Linear functions?
How far are the Mode and the Median of the Log-Normal distribution from behaving as Linear functions?
Intro_______________
Recently I made a question where later I figure out I was requiring that the ...
- 1,558
3
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1
answer
98
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Magical relationship between Exponential distribution and Poisson process
Consider i.i.d. random variables $X_1,X_2,\ldots,X_n$ satisfying exponential distribution $\operatorname{Exp}(1)$. Let $Y=X_1+X_2+\ldots+X_n$. We know that the p.d.f. of $Y$ is the Gamma distribution
$...
- 2,325
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Let be $Z_1, Z_2, \cdots, Z_n$ independent random variables with mean $0$ and variance $\sigma^2 < \infty$. Let $X_n=Z_1+Z_2+\cdots, Z_n$.
Let be $Z_1, Z_2, \cdots, Z_n$ independent random variables with mean $0$ and variance $\sigma^2 < \infty$. Let $X_n=Z_1+Z_2+\cdots, Z_n$. I'm trying to prove that $\mathbb{E}(X_nX_m)=\min(n,m)\...
- 3,047
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66
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Sample Space of such a random variable
Suppose there is a random experiment in which a person is asked to flip a coin $3$ times. The coin has $2$ sides (numbers and pictures). In the sample space of the random experiment, $3$ random ...
- 13.2k
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30
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Moment Generating Function of $\bar{M}−\bar{N}$
What is the Mean value of $\bar{M}−\bar{N}$; Moment Generating Function of $\bar{M}−\bar{N}$; and Variance of $\bar{M}−\bar{N}.$ Given $M_1,M_2,\dots,M_n$ is a random sample of size $p$ from the Gamma ...
4
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1
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232
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Estimating the value of $\sigma$ for Brownian motion
Let $X_t=\sigma W_t$ be a stochastic process, where $W_t$ is the Wiener process and $\sigma$ is an unknown parameter.
I want a formula to estimate the value of $\sigma$ (which could not be found in ...
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Consider the stochastic process $\{X_t\}$ such that $\mathbb{P}(X_0=1)=\mathbb{P}(X_0=-1)=1/2$. $X_t$ changes sign in Poisson times.
Consider the stochastic process $\{X_t\}$ such that $\mathbb{P}(X_0=1)=\mathbb{P}(X_0=-1)=1/2$. $X_t$ changes sign in Poisson times, that is, the probability of $k$ changes of sign in a time interval ...
- 61
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1
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53
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Suppose $g$ is a periodic function with period $k$. Let the stochastic process be defined as $X_t=g(t+T)$, where $T\sim U(0,k)$.
Suppose $g$ is a periodic function with period $k$. Let the stochastic process be defined as $X_t=g(t+T)$, where $T\sim U(0,k)$. I'm trying to prove that $\{X_t:t \geq0\}$ is weak-sense stationary. I ...
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