# All Questions

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### Independence is preserved by joint weak convergence

Suppose a sequence of random vectors $(X_n,Y_n)$ converges jointly to some $(X,Y)$ in the weak topology. Question: If $X_n$ and $Y_n$ are independent for all $n$, are also $X$ and $Y$ independent? ...
80 views
+100

### Inscribing a sphere in an elliptical cone

Suppose you're given the cone whose curved surface is given by $(r - V)^T Q (r - V) = 0$ and its flat surface is $z = 0$ Question: How would you inscribe a sphere in the cone, such that it is ...
1 vote
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### Question about proof of Kolmogorov inequality for Bernoulli random variables

The question is about one inequality which shows in Kolmogorov's paper (inequality (3.1)) but is not proved. The inequality says that, if we assume $Y_1,Y_2,\ldots$ are i.i.d. Bernoulli random ...
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+250

### Concerns about the definition of Hawkes process

In the lecture notes I am reading about Point process, when we introduced the Hawkes process several expressions are given and I have some difficulty to understand properly what is the $Z_t$ (defined ...
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### Can this be done? Split Pascal's triangle (without the $1$s) with a straight line into two regions of equal sums.

Consider Pascal's triangle with $n$ rows, without the $1$s, with each number corresponding to a vertex on a pyramid of equilateral triangles, as shown below with example $n=5$. Can the triangle be ...
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### How to evaluate $\sum_{i=1}^n i^{2 i}$?

Let, $$\mathcal{S}(n) = \sum_{i=1}^n i^{2 i}$$ for $n \in \mathbb{N}$ I will be completely honest. When I was returning from my physics tuition center, and suddenly this popped up in my head from ...
1 vote
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### Estimate true probabilities from weighted sampling without replacement

Suppose we have a random event with $n$ outcomes, with (unknown) true probabilities $p_1, p_2, \dots, p_n$. We have performed a study of this event, sampling it in batches of $k < n$. From this we ...
97 views
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### A question about divisor function and primes

$\sigma_1(n)$ denotes the divisor function which sums the divisors of a natural number $n > 1$. We calculate the remainder of the division of $\sigma(n+\sigma(n+1+\sigma(n+2)))$ by $n+1$. $r$ is ...
1 vote
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+150

### Are finitist systems the ones with a proof-theoretic ordinal of at most $\omega^\omega$?

The proof-theoretic ordinal of $\mathsf{EFA}$ and $\mathsf{RCA}_0^*$ are $\omega^3$ and the one of $\mathsf{PRA}$, $\mathsf{I\Sigma1}$, $\mathsf{RCA}_0$, etc. is $\omega^\omega$. See https://ncatlab....
1 vote
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### Show the convergence of a sequence of random variables

Consider some discrete random variables $X_1,X_2,...$ each with support $A$. Take another sequence of random variables $Y_1,Y_2,\dots$ Given $x\in A$, assume: A1: \$\Pr(X_t=x| X_{1},\dots, X_{t-1}, Y_1,...
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### Why is only the first (highest) term of the divisor in polynomial long division used to divide?

There is one small matter that has always stumped me with polynomial long division. In the example from the Wikipedia on Polynomial long division, why is the equation only divided by the first/highest ...