All Questions
Tagged with statistics stochastic-processes
713
questions
0
votes
1
answer
49
views
A Gaussian process and a Rademacher proecss are sub-Gaussian
This is a question-and-answer just for me, but if you have alternate answers or comments, feel free to share them.
Let $(T,\rho)$ be a metric space and $\{X_t\}_{t\in T}$ be a stochastic process ...
- 1,565
4
votes
0
answers
54
views
When $X_t$ is conditionally normal distributed and has density $p_t$, how can we compute $\text E\left[\left\|\nabla\ln p_t(X_t)\right\|^2\right]$?
Let $d\in\mathbb N$ and $(X_t)_{t\ge0}$ be an $\mathbb R^d$-valued process. Assume $$\operatorname P\left[X_t\in\;\cdot\;\mid X_0\right]=\mathcal N(X_0,\Sigma_t)\tag1$$ for some covariance matrix $\...
- 13.9k
5
votes
2
answers
130
views
Ergodic series converge to the expectation?
Let $(X_i, Y_i)_{i\in\mathbb{N}}$ be a real-valued stochastic process. We say that $X$ is mean-ergodic, if $$\frac{1}{n}\sum_{i=1}^nX_i\to \mathbb{E}X_1$$ in probability as $n\to\infty$.
Let $S_n:=\{i\...
- 13.8k
2
votes
1
answer
104
views
Asymptotic Gambler's Ruin Probability with Unequal Gain/Loss with Zero-Mean Payoff Distribution
The gambler's ruin problem with unequal gain/loss with a payoff distribution whose support is a finite subset of $\mathbb Z$ is an old problem; for example, see Feller (1968, Vol.1, Section XIV.8) and ...
- 8,424
1
vote
1
answer
117
views
How to deduce an expression of a specific conditional expression
The problem occurs when reading Bombardini et al., 2023, "Did US Politicians Expect the China Shock?", American Economic Review, Vol.1, PP174-209.
The authors define $\xi_{it}$ to be a ...
- 3,120
2
votes
1
answer
119
views
Covariance Operator corresponding to multivariate covariance function
The usual definition of a covariance operator on $L_2(D)$ is:
$$
C : L_2(D) \to L_2(D), \qquad (C \psi)(x) = \int_D c(x,y) \psi(y) dy \qquad \forall x\in D, ~~\psi \in L_2(D),
$$
where $c(x,y): D \...
- 5,213
1
vote
0
answers
14
views
M/M/1 Queues : Exclusive Queue Length is not Markov
For a M/M/1 queue let $N_q(t) = (Q(t)-1)^{+}$ be the number of customers in the queue except the one being served. We have to show that $N_q(t)$ is not a continuous-time Markov chain. [src: Sidney ...
- 708
0
votes
1
answer
28
views
Variance of time integral of a function on an Ito process
I was struggling a bit with the time integral of an Ito process.
Say I have this:
$$\int^T_t \alpha\circ X_t ds$$
Where $X_t$ is an Ito process, and $\alpha$ is a continuous function. What can we say ...
- 3,766
0
votes
1
answer
56
views
Can addition of noise to dynamical system reduce estimation errors
I am using Kalman filter to estimate the states of a stochastic dynamical system which has very very small noise( consider zero ). The filter is not aware that the noise is zero. Implementation of KF ...
- 420
0
votes
0
answers
64
views
How to find joint pdf
The random arrival of $k$ phone calls within a time interval of length $t$ is described by the following pdf
$$f(t) = \frac{\lambda^k}{(k-1)!} t^{k-1} e^{-\lambda t}$$
where parameter $\lambda$ ...
- 1,330
0
votes
1
answer
43
views
Question on determining the posterior pdf
Can someone tell me how the pdf of noise (w) is equivalent to the conditional pdf of observations (x) given A, assuming noise is independent of A for the equation x[n]=A+w[n] where A is the mean (and ...
- 2,042
0
votes
0
answers
28
views
Genomic and sum of geometric random variables
In their paper The Maximum of independent Geometric Random Variables as the Time for Genomic Evolutionthe authors noted that if to consider the genomic word of L letters, than the measure of the time ...
- 430
1
vote
0
answers
47
views
Mean value of sqrt. root cox-ingersoll-ross process
Consider the so-called Cox-Ingersoll-Process model
\begin{equation}
dr_t=a(b-r_t)dt+\sigma \sqrt{r_t}dW_t
\end{equation}
It can be shown (Wikipedia), that the mean of this process is
\begin{...
- 155
2
votes
1
answer
99
views
Variance recursion formula in Galton-Watson process
Consider a Galton-Watson process with expected offspring $\mathbb{E}[\xi]=\mu<\infty$ and variance $\text{Var}(\xi)=\sigma^2<\infty$ where the offspring in generation $t\in\mathbb{N}$ is given ...
3
votes
1
answer
94
views
Existence of Malthusian parameter
Consider a continuous time point process $\eta(t)$ representing the number of points in the interval $[0, t]$. Let $\eta(\infty)$ be distributed as the total number of children of a particle. Define $\...
- 3,766