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Results tagged with pr.probability
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user 56843
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
1
vote
0
answers
67
views
Can any point process be thinned into a homogenous Poisson point process?
Let $X$ be a stationary and isotropic point process. Under what sufficient conditions can $X$ be thinned into an (approximation of an) homogenous Poisson point process $Y$ of positive intensity?
What …
- 290
0
votes
Gaussian expectation of an exponentiated outer product
If a 'closed form' solution is allowed to be an infinite series then ...
Let $X_i$ be the random variable for the i-th row of $\mathbf{x}$ where
$$
X_i \sim N\left( \mu_i, \Sigma_i \right)
$$
We se …
- 290
0
votes
Accepted
Sum of a random number of identically distributed but dependent random variables?
I [think I have] proved the calculation for $\mu_Q$ and $\sigma^2_Q$ by applying the methods I used in a related problem.
I then [also think I have] proved that $Q$ is asymptotically normal (under ce …
- 290
3
votes
positive Harris recurrent, aperiodic, stationary Markov chain
In the following, all results are referenced from Bradley (2005).
Suppose that $X := (X_k, k \in \mathbb{Z})$ is a strictly stationary Markov chain. If $X$ is Harris recurrent and aperiodic then $X$ …
- 290
3
votes
Gaussian expectation of an exponentiated outer product
The following solves the algebra for element-wise calculations by brute-force, and extracts a 'matrix-form solution' of sorts.
For $i = j$
We seek
$$
E\left[ \exp\left( X_i^2 \right) \right] = \frac …
- 290
5
votes
1
answer
2k
views
Sum of a random number of identically distributed but dependent random variables?
Background
Let $X_t$ be the continuous time Markov process on the state space {Working, Broken} with failure rate $\alpha$ and repair rate $\beta$. By elementary calculations [1]
$$
\begin{align*}
…
- 290
6
votes
1
answer
845
views
Publishing an elementary proof of a less-general and less-useful version of a classic result?
Background
Let $X_t$ be a stochastic process on the state space {Working, Broken}. Let $U$ be the cumulative sojourn Working during an interval $[0,\tau]$ (the process's uptime). It is well-known [1 …
- 290