# 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.

7,576
questions

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### Show that no two sets in the probability space with $\mathbb{P}(\{k\})=2^{-k!}$ are independent [closed]

Let $\mathcal{P}(\mathbb{N})$ denote the power set of $\mathbb{N}$.
Show that no two non-trivial sets in the probability space $(\mathbb{N},\mathcal{P}(\mathbb{N}),\mathbb{P})$ with $\mathbb{P}(\{k\})=...

-1
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2
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93
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### Cumulants of a sequence of variables with zero mean and variance

Can one prove for a sequence of positive random variable $X_{n}$ such that $\lim_{n\to \infty}E[x_{n}] = 0$ and $\lim_{n\to \infty}E[x_{n}x_{n}]= 0$ all the cumulants go to zero once $n\to \infty$ ?

0
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1
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62
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### Functional relationship between two quantities

Let $\mu \in \mathbb R^n$ and let $\Sigma$ be a positive-definite matrix of order $n \ge 2$. Fix $t \ge 0$ and define $\alpha(\mu,\Sigma,t) > 0$ by
$$
\alpha(\mu,\Sigma,t) := \sup_{\|w\| = 1}\frac{...

6
votes

1
answer

409
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### Max decoupling inequality

Let $X_1,\ldots,X_n$ be $\{0,1\}$-valued random variables drawn from some joint distribution. Let $\tilde X_1,\ldots,\tilde X_n$ be their independent version: $\mathbb{E}X_i=\mathbb{E}\tilde X_i$ for ...

3
votes

2
answers

125
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### Continuity of Radon transform w.r.t the angle

Let $f \in L^1(\mathbb R^n)$ (or in case it helps, actually a probability density on $\mathbb R^n$). Define the Radon transform $R[f]:S_{n-1} \times \mathbb R \to \mathbb R$ of $f$ by
$$
R[f](w,b) := ...

0
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0
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72
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### $L^p$ inequality for "positively correlated" random variables

Suppose that we have $m$ complex-valued random variables $\xi_1,\ldots,\xi_m$ and assume the following "positive correlation" property: for all non-negative integers $\alpha_1,\ldots,\...

3
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0
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85
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### Does smoothing a non-log-concave distribution make it more log-concave?

Suppose that $p$ is a density on $\mathbb{R}^d$ that is $C^2$ and nonzero everywhere, and such that the Hessian of its negative logarithm is lower bounded:
$$-\nabla^2 \ln p\succeq L$$
for some matrix ...

3
votes

0
answers

48
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### Random assignment problem under multinomial or Poisson distribution

We place $m$ balls at random (uniformly) inside $n^2$ urns arranged as a $n \times n$ square. Then we must choose $n$ urns, such that no two urns belong to the same row or column, with the objective ...

1
vote

1
answer

39
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### Estimation of Lévy measure of ID distribution

Suppose that the positive random variable $X$ is infinitely divisible and supported on $\mathbb R_+$. Due to Lévy-Khintchine, its moment generating function then writes :
$$M(t) = \mathbb E\left(e^{tX}...

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80
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### Stopping times for martingale

The nonnegative integer set is denoted by $\mathbb{Z}_+$.
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space and $\{\mathcal{F}_{n}\}_{n\in{\mathbb{Z}_+}}$ be an increasing sequence ...

1
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0
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91
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### On probability of coprimality of a list of numbers

We know $r$ randomly chosen integers are coprime with probability $\frac1{\zeta(r)}$.
Pick a bound $N$ and pick $k\lceil N^{\alpha}\rceil$ uniformly random integers in $[0,N^{\alpha+\beta}]$ where $\...

0
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0
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43
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### On the Markov property of a limit process

Let $E$ be a locally compact separable metric with countable base. We consider a sequence of Hunt processes $\{X^{(n)}\}_{n \in \mathbb{N}}$ on $E$. That is, each $X^{(n)}=(\{X_t^{(n)}\}_{t \in [0,\...

2
votes

1
answer

118
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### Verify if array is orthogonal

This is a repost from the computer science stackexchange. The question has been offered a bounty, but received no answers. Therefore, I would like to ask this question here.
Orthogonal arrays often ...

0
votes

0
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22
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### L^2 approximation error in Gaussian Process Regression (finite data setting)

I am learning about Gaussian Process Regression. I would like to have some references or results regarding the distribution of the error between a given function, and the posterior obtained in ...

2
votes

0
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37
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### techniques in studying moments of shifted integral process $\mu(T_{a},T_{a}+t)$

We have a strictly increasing measure $\mu$ on $[0,\infty)$ given by $\mu(0,x):=\int_{0}^{x}e^{X(s)-\frac{1}{2}\ln1/\epsilon}ds$, where $X(s)$ is a mean zero Gaussian field with truncated log ...

7
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2
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185
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### On permanent of a square of a doubly stochastic matrix

Let $A = (a_{i,j})$ be a double stochastic matrix with positive entries. That is, all entries are positive real numbers, and each row and column sums to one. A permanent of a matrix $A = (a_{i,j})$ is ...

1
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0
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41
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### Counting overlaps of random intervals

We have a strictly increasing measure $\mu$ on $[0,\infty)$ given by $\mu(0,x):=\int_{0}^{x}e^{X(s)-\frac{1}{2}\ln1/\epsilon}ds$, where $X(s)$ is a mean zero Gaussian field with truncated log ...

1
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0
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122
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### A basic formula for the falling factorial

Whis is a question I asked on Math.SE, but didn't get any response.
Suppose we have a family $\mathfrak{A}$ of some subsets of $\Omega$, which is locally finite, i.e.
$$
X(\omega): = \sum_{A \in \...

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0
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29
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### How to use Itô's formula to show that $ K_N(s,t)-K(t,t)=\int_s^t[-U' K_N(u,t)+\left<\mathbf{J}x_u, x_t\right>]du+\frac{1}{N}\sum x_t^i(B_s^i-B_t^i) $?

I am reading a lecture note Dynamics for Spherical Models of Spin-Glass and Aging by Alice Guionnet. On page 124, it shows that for $s\ge t$,
$$
K_N(s,t)-K(t,t)=\int_s^t[-U' K_N(u,t)+\langle\mathbf{J}...

3
votes

2
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118
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### On finding an upper bound on the error of a sparse approximation

I posted this question on math.stackexchange earlier, but didn't see any response. So, I am posting it here, in case someone else has an answer.
Original question: https://math.stackexchange.com/...

0
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0
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### Vector Norms of Sub-Gaussian Matrix Multiplication?

Let $X, Y$ be $n\times n$ matricies with i.i.d. sub-Gaussian entries.
I am interested in tail bounds for $\lVert XY\rVert$, where $\lVert\cdot\rVert$ is a norm on $\mathbb{R}^{n^2}$, i.e. I want tail ...

4
votes

1
answer

134
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### Population growth with good and evil children - probability good outnumbers evil

Consider the following discrete-time population model. We start with a single "good" individual who reproduces asexually into $k$ children and dies in the process. At generation $t=2$, those ...

1
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1
answer

76
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### Hölder continuity of Radon transform of smooth function

Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by
$$
R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)...

1
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0
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46
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### Sobolev variant of Wasserstein space

Let $\mathcal{P}(\mathbb{R}^n)$ be the set of Borel probability measures on the Euclidean space $\mathbb{R}^n$ and consider thereof consisting of all probability measures $\mathbb{P}$ satisfying $\int\...

4
votes

1
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210
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### How to get $\lim_{N\to \infty} \sum_{i=1}^N e^{\lambda_i}u_i^2=\int e^{\lambda}d\sigma(\lambda)$?

I am reading the one lecture note Dynamics for Spherical Models of Spin-Glass and Aging.
On page 126. In the Sherrington-Kirkpatrick (SK) model, we suppose that there are $N$ people labeled as $[N]:=\{...

0
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1
answer

128
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### Projecting a given point onto a random $2$-dimensional plane in more than $3$ dimensions

We are given $\mathbf{p}\in\mathbb{R}^d$, where $d\gg 1$. Let $\mathbf{v}$ be a point selected uniformly at random from the unit $(d-1)$-sphere $\mathcal{S}^{d-1}$ centered at the origin $\mathbf{0}\...

1
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0
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30
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### Handling sums of correlated random variables with a directed path structure

Recently, I've been seeing random variables with the following correlation structure based on directed paths on a graph. For example, there are $16$ directed paths "directed downwards from the ...

1
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1
answer

54
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### Non-independent Sub-gaussian variables and concentration

Let $g \in R^{d}$ have $iid$ Gaussian components. Let $a \in R^{d}$, and let $b \in R^{d}$. be arbitrary vectors.
Consider the random variable $Y_{g,g}:= \frac{1}{n}\langle g,a \rangle \langle g, b \...

0
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1
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48
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### Does the the equivalence of Total variation distance formulas assumes that the two distributions are symmetrical?

Does the the equivalence of Total variation distance formulas presented here(https://ece.iisc.ac.in/~parimal/2019/statphy/lecture-14.pdf) assumes that the two distributions are symmetrical ?

0
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0
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61
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### Converse to Cameron-Martin theorem

It is known by Cameron-Theorem that if $\mu$ is a centered Gaussian measure on Banach space $\mathcal B$, the equivalent mean-shift measures are exactly the mean-shift by the Cameron-Martin directions....

5
votes

1
answer

107
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### Second Skorokhod embedding in high dimensions

The first Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E X^2<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau$ is ...

0
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0
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29
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### Minimum eigenvalue of covariance matrix under probability constraint

Consider a random (column) vector $X\in \mathbb{R}^d$. I am interested in the quantity
$$
\Lambda(\alpha)=\inf_{E\in \mathcal{B}(\mathbb{R}^d), \ P(X\in E)\ge \alpha} \lambda_{\min}\left(E[XX'1\{X\in ...

2
votes

1
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68
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### Chung's law of the iterated logarithm for Brownian motion

I am looking for a reference that gives a detailed proof of Chung's law of the iterated logarithm for Brownian motion: $$\liminf_{u\to +\infty}\sqrt{\frac{\ln(\ln(u))}{u}}\sup_{r \in [0,u]}|X_r|=\frac{...

0
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0
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45
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### A sufficient condition for the decomposition of a bounded random vector

Given a matrix $A \in \mathbb{R}_{+}^{n\times m}$, where $n<m$ and rank($A$)=n. Define a set (a zonotope) $\mathcal{C}(A)=\{(x_1,x_2,\ldots,x_n)|\sum_{i=1}^m{\bf{a}}_ix_i,x_i \in [-1,1]\}$, where ${...

1
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0
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### A distribution $\pi \propto \exp(-f)$ satisfies log-Sobolev inequality, does $\exp(-af)$ also satisfy LSI?

Assume a distribution $\pi \propto e^{-f}$ satisfies log-Sobolev inequality (LSI)
$$\forall \rho \in P(\mathbb{R}^n), \quad KL(\rho\| \pi) \le \frac{1}{2\lambda} I(\rho \| \pi)$$
with LSI constant $\...

0
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0
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### Expectation of edge weights on the complete graph, Part 2

This question concerns the same basic set-up as my previous question: Expectation of edge weights on the complete graph
In that question an answer was given which shows that the expected value is as ...

2
votes

0
answers

50
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### Approximate logarithmic bound on expected maximum via central limit theorem

If $Z_i$ are standard normal, possibly dependent, one can show that
$$E\left[\max_{i=1,...,M} Z_i^2\right]\leq 3\ln M + 1.$$
I'm looking for a similar (asymptotic) bound for asymptotically normal ...

0
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0
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39
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### Total variation convergence of eigenvalues in the bulk of the GUE?

Let $\lambda_{k(n)}$, $k(n) / n \to a \in (0,1)$, be an eigenvalue in the bulk
of the $n \times n$ Gaussian Unitary Ensemble (GUE) normalized so that its spectral measure converges to the semi-circle ...

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0
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### Open problems in derivatives, options and finance [closed]

I would like to ask what kind of problems remain open in finance. In particular, those concerning pricing derivatives, computing the risk, etc. It could be really of great help if you can recommend to ...

1
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1
answer

131
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### Expectation of edge weights on the complete graph

Let $n,k \geq 3$ be positive integers with $n$ much larger than $k$ and consider a random assignment of weights to the edges of the complete graph $K_n$. On each vertex of $K_n$ we attach a random ...

2
votes

1
answer

87
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### Probability density of a hyperplane for a Gaussian distribution

I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution
$$P[\textbf{x}\in S]
=\int_{\textbf{x}\in S}
\det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$...

4
votes

1
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70
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### Distance between trunctated random walk and its normal form

I have $$X_i \sim N(0,1), \quad S_n=X_1+\cdots+X_n,$$
$$ \mathscr{S}_n (t, \omega) := \frac{1}{ \sqrt{n} } \sum_{i=1}^{n} \left[ S_{i-1} (\omega ) + n \left( t - \frac{i-1}{n} \right) X_{i}(\omega) \...

3
votes

3
answers

465
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### How close are two Gaussian random variables?

Given two Gaussian random variables A and B with (mean, standard deviation) of (a,s) and (b,m) respectively, is there a scalar w in [0,1] that indicates how close A and B are?

1
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0
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68
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### On the closedness of a certain subset of $\mathbb R$

Let $\mu$ be a probability measure on measurable space $X=\mathbb R^n$ (euclidean), and let $F$ be a family of $\mu$-measurable functions $X \mapsto \mathbb R$ which are uniformly bounded, i.e $b:=\...

0
votes

0
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61
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### Extending proofs from Lebesgue measure to non-atomic (probability) measures [closed]

On $\mathbb R^n$, is there a relationship between non-atomic probability measures and the Lebesgue measures ?
What kinds of results about non-atomic measures have the same proof as for Lebesgue ...

2
votes

1
answer

62
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### Compactness of the integral of a set-valued function

Let $X$ be a compact space (e.g. a compact subset of $\mathbb R^n$) and $P$ be a probability measure on $X$. Let $A$ be a compact subset of some $\mathbb R^d$. Finally, let $F$ be the collection of $P$...

1
vote

1
answer

54
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### Independent Sums and Orlicz Norms

Let $X_{i}$ be a collection of iid random variables of cardinality $n$, and let $S_{n}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}X_{i}.$
Let $|| X||:=\inf_{B}\{E[\exp(X/B)-1]\leq 1\}$. This is the so-called sub-...

3
votes

0
answers

33
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### Continuation : Uniqueness of the solution to some SDE with discontinuous coefficient

Consider the SDE below
$$X_t=X_0+\int_0^t b(s)ds+\int_0^t\frac{dW_s}{1+m(s){\bf 1}_{\{b(s)>0\}}},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$
where $X_0>0$ is square integrable, $b:\mathbb R_+\...

0
votes

0
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29
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### How to recalculate the weights for an event that happens multiple times and requires all outcumes to be unique?

I think it's easiest to explain with an example.
I have a weighted probability list
A : 0.15
B : 0.15
C : 0.15
D : 0.1
E : 0.1
F : 0.1
G : 0.1
H : 0.075
I : 0.075
...

7
votes

2
answers

182
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### Does entropy of the random walk control the return probability

Given an infinite connected graph $G$ of bounded degree with vertex set $X$, let $P_x^n$ the time $n$ distribution of the simple random walk started at the vertex $x$ (so $P^n_x(y)$ is the probability ...