How to Prove Forward Equation Holds for Continuous State Process Mathstackexchange
Questions tagged [pr.probability]
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
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-1 votes
0 answers
25 views
Inequality between probabilities
Consider $x_{s}$ which is a random variable from a standard normal distribution, and given values $w_{s}>0$ for $s=1,2,...,r,$ where $r$ is some finite number. We have that $$(\sum_{s=1}^rw_{s}x_{s}...
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0 votes
1 answer
39 views
Distribution of root with Poisson leaves
We have a pool of items, termed as item A, generated following a Poisson distribution. We use a pair of items A to produce an item B with success rate $r\in(0,1)$. My question is: is B Poisson ...
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-3 votes
0 answers
45 views
Asymptotic results for spectral edge of GOE matrix
Following this question:Asymptotic results for smallest gap of Gaussian random matrix. If we consider the normalized GOE matrix $A$ with ordering eigenvalues $\lambda_1\le \lambda_2\le \cdots \...
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2 votes
0 answers
55 views
Probabilistic optimization problem on set selection without replacement with combinatorial constraints
We are given a collection $\mathcal{S}$ of $n$ sets $S_1, S_2, \ldots, S_n$, such that for all $i\in\{1,2,\dots n\}$, we have $|S_i|\ge 1$, with the constraint $\sum_{i=1}^{n} |S_i|=2n$. In a ...
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Tight upper-bounds for the Gaussian width of intersection of intersection of hyper-ellipsoid and unit-ball
Let $\Lambda$ be a positive-definite matrix of size $n$ and let $R \ge 0$, which may depend on $n$. Consider the set $S := \{x \in \mathbb R^n \mid \|x\|_2 \le R,\,\|x\|_{\Lambda^{-1}} \le 1\}$ where $...
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2 votes
0 answers
53 views
Upper bound of conditional mutual information
Given random variables $X$,$Y$ with a joint distribution $P(X,Y)$ and another random variable $Z$, it is known that there are cases when the conditional mutual information $I(X;Y|Z)$ is greater than ...
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3 votes
0 answers
66 views
Does the existence of regular conditional measure follow from that of regular conditional distribution?
For more succinct description, I use the following abbreviations, i.e., RCPD: Regular conditional probability distribution. RCPM: Regular conditional probability measure. First are definitions 10.4....
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3 votes
1 answer
164 views
Push-forward of uniform measure and uniqueness
On a standard Borel (or Polish) space $X$, any probability measure $\mu$ is the push-forward of the uniform measure on $[0, 1]$ under some $f : [0, 1] \to X$. This $f$ is not unique in general. ...
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Percolation: at what length scale do we see it?
Consider classical bond percolation on $\mathbb{Z}^d$. Each edge is included with probability $p$ and deleted with probability $1-p$. As is well known, there is a $p_c(d) \in (0,1)$ such that $p>...
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2 votes
0 answers
78 views
Optimization problem on randomly selecting subintervals from a given interval with combinatorial constraints
We select uniformly at random $k$ pairwise disjoint intervals from a given interval $[0,s]$ with length respectively equal to $\ell_1, \ell_2, \ldots, \ell_k\ $, i.e., we select uniformly at random $k$...
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0 votes
0 answers
81 views
+50
The lower bound for the hitting time of $h(t)$
Let $A$ be an $n$ by $n$ GOE matrix. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \lambda_n$ with corresponding eigenvectors $v_1,\dots, v_n$. Let $X_t$ be the solution to the ...
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1 vote
0 answers
52 views
Mixed moments of traces
I've seen a host of results concerning computations for $$\mathbb{E} \left[ \operatorname{tr} A^{i_1}\cdots \operatorname{tr} A^{i_j} \,\overline{\operatorname{tr} A^{k_1} \cdots \operatorname{tr} A^{...
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1 vote
0 answers
48 views
Reference request: rates of weak convergence of Polish space-valued random variables
Let $(E,\mathscr{E})$ be a Polish space $E$ together with its Borel $\sigma$-algebra $\mathscr{E}$. Let $(\Omega, \mathscr{F},P)$ be a probability space and let $(X^{(n)})_{n \in \mathbb{N}}$ be a ...
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2 votes
1 answer
132 views
Minimum of random walks
Let $M$ independent and identical random walks that follow the chi-squared distribution, i.e. in each step, a $X^2_1$ random variable is added. I am interested in the minimum random walk at each step. ...
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Asymptotic results for smallest gap of Gaussian random matrix
For a symmetric Gaussian random matrix $G=\{G\}_{1\le i,j \le n}$ with iid $E[G_{ij}]=0$ and $E[G_{ij}^2]=1/n$ (it is normalized), ordering its eigenvalues $\lambda_1\le \lambda_2\le\cdots \lambda_n$. ...
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