pr.probability – Reference request: Total variation between independent and dependent Bernoulli processes

Let $ X $ to be a random variable taking values ​​in $ {0,1 } ^ n $ with the next distribution. For each coordinate $ i $, we have $ p_i = P (X_i = 1) = c / sqrt n $, or $ c $ is a (very small) constant. Coordinates $ i $ and $ j have positive correlations decreasing exponentially in $ | i-j | $, with prefactor 1 $ / sqrt n $, in the following sense: writing $ p_ {i, j} = P (X_i = 1, , X_j = 1) $, we have
$$ 0 the p_ {i, j} – p_i p_j the exp (- | i-j |) / sqrt n. $$
It's a dependent Bernoulli process.
Let $ mu $ denotes the law of that.

Also, leave $ Y $ bean independent Bernoulli process with the same marginals: $ P (Y_i = 1) = p_i $ and the coordinates are independent.
Let $ nu $ denotes the law of that.

I want to limit the total variation distance $ | mu- nu | _ text {TV} $.
In particular, I want to show that the television distance decreases with $ c $, that is, by taking $ c to $ 0 given $ text {TV} to 0 $.


I know the Chen-Stein method to address such questions, but it seems to me more appropriate when the probabilities $ p_i $ are in order $ 1 / n, and so there is a number of $ 1 $s in the independent case (and the method shows that it is the same for dependent cases, under certain conditions). Perhaps Stein's method can be applied more generally?

In addition, the purpose of this question is not to obtain an accurate answer from anyone, but rather a reference or suggested approach method. The above is a simplified version of my current problem, but I feel that if I can understand the above, I will be able to convert it to my specific case.

linear algebra – non-negative irreducible matrices with random (correlated or independent) nonzero entries

allows $ M $ to be a non-negative irreducible matrix. According to Perron-Frobenius' theorem, the maximum eigenvalue of $ M $, $ lambda $, is positive and equal to its spectral radius $ rho (M) $.

Suppose now the matrix $ M $ it is not deterministic and its non-zero elements are equal to the random variables $ tanh (x_i) $ with $ x_i sim N (m> 0, sigma ^ 2) $. However, the null elements are the same deterministic zeros as before. My question is: what will happen to the expected value of the maximum eigenvalue if $ x_i $The s are correlated in case they are independent.

My observation is that the existence of a positive correlation between non-zero inputs increases the expected maximum eigenvalue compared to the case where the inputs are independent. But I can not justify this experience.

probability – Assume that $ A $ and $ B $ are independent events. For a $ C $ event such as $ P (C)> $ 0, prove that $ A $ event donated $ C $

assume $ A $ and $ B $ are independent events. For an event $ C $ such as $ P (C)> $ 0 , prove that the event of $ A $ given $ C $ is independent of the event of $ B $ given $ C $

We have A and B are independent so $ P (AB) = P (A) cdot P (B) $

We must show that $ P ((A mid C) cap (B mid C)) = P (A mid C) cdot P (B mid C) $

My procedure was like that
$$ P ((A mid C) cap (B mid C)) = P ((A mid C) mid (B mid C)) cdot P (B mid C) $$
$$ = frac {P (AB mid C)} {P (B mid C)} $$

I played until I got this
$$ frac {P (AC)} {P (C)} cdot frac {P (B mid AC)} {P (B mid C)} $$
Now the first part gives us $ P (A mid C) $ . I could not get from the second part the missing part that is $ P (B mid C) $.

Is my procedure correct? If so, how can I find the second part?

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canon 5d mark iv – Are the autofocus settings independent of the selected case?

Canon's AI Servo AF autofocus behavior can be changed by choosing one of six presets and can be further customized by adjusting the Tracking the sensitivity, Acceleration / deceleration tracking and Automatic AF point switching settings.

If I set the three parameters to the same value in two different cases, do they both behave the same way?

for example. would have Case 1 to be equal to Case 6 if I put Tracking the sensitivity, Acceleration / deceleration tracking and Automatic AF point switching exactly the same values ​​in both cases or will they behave differently?

rooting – What independent context will allow anything to execute it?

I know using root you can chcon a file / directory. What is the context that will allow any other context to execute this file / directory? I'm trying to run something without root, but unless seenforcing = false, this is not executed. I guess sepolicy prevents the execution.

Thank you.

can I publish a work from an independent website in the queue of publications to the list of queues of drupal.org?

I have heard that it is illegal to publish a paid job application in the queue of Drupal.org publications. is it true?

I sent an email to the manager, but they are busy doing paid work

What do you prefer and why an independent translator or translation service provider?

It can be difficult to choose between a translation agency and an independent translator. Both options have advantages and disadvantages, but what do you choose?

xorg – Dual monitor, independent workspace

I've been looking for a while but I can not find a solution.

I'm using Xubuntu 16.04 and Xfce 4.12.

When I connect a second monitor, my workspace is extended to both monitors (this is the default behavior). What I want instead is to display a different workspace on the second monitor (the rightmost workspace or any fixed workspace, both options would work well for me ).

Is it possible with Xfce or should I install another desktop environment?

Thank you in advance.

Diophantine approximation – independent numbers of more than $ mathbb {Q} $ but not BA? numbers that are not a base for a number field but are BA?

Has anyone ever discovered a vector of real algebraic numbers? $ (a_1, …, a_k) $ such as $ 1, a_1, …, a_k $ are linearly independent on $ mathbb {Q} $ and as $ (a_1, …, a_k) $ is do not "bad approximation"?

Has anyone ever discovered a vector of real algebraic numbers? $ (a_1, …, a_k) $ such as $$[mathbb{Q}(a_1,…,a_k):mathbb{Q}]> k + 1 $$ but as $ (a_1, …, a_k) $ is "bad approximation"?