# Questions tagged [asymptotics]

The asymptotics tag has no usage guidance.

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### The asymptotics of a vector sequence defined by a recursion relation

The sequence of vectors $(\mathbf{v}_0,\mathbf{v}_1,\mathbf{v}_2,\dots)$ obeys the recursion relation that
$A\mathbf{v}_j-\mathbf{v}_{j-1}=\sum_{k=0}^j diag(\mathbf{v}_k)B\mathbf{v}_{j-k}$,
where A ...

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### Asymptotics for a random set cover problem

Suppose you are given a positive integer $k$ and a probability distribution $f$ on the positive reals. I am interested in the limiting behavior of the following process as $n\to\infty$:
Create an ...

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### Product of independent random variables and tail deconvolution

Suppose $X, Y$ are two independent non-negative random variables. The conditions
(i) $\mathbb{P}(X > t) = \frac{C}{t^p} + o(t^{-p})$
(ii) $\mathbb{P}(Y > t) = o(t^{-q})$ for any $q > ...

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### Differentiating an integral that grows like log asymptotically

Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that
$$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$
...

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### Rate of convergence for difference between conditional and marginal probability

Suppose $X\sim \text{Bin}(2n,p)$ and $X_1,X_2\sim\text{Bin}(n,p)$ are independent, with $X_1+X_2=X$. I'm interested in the rate of convergence for the absolute difference
$$
\left\vert P(X>c|X_1\...

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### Are cyclic codes bounded by a continuous function?

In coding theory, we know that if you take the function
\begin{equation}
\alpha_q(\delta) := \limsup_{n \rightarrow \infty} \ \max \{ R(C) \mid \delta(C) \ge \delta \mid C \subseteq \mathbb{F}_q^...

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### Is there any solution that currently exists for the graph automorphism problem in the general case?

I was reading the Wikipedia pages on the graph automorphism, but I could not find any solution to the problem (Not even a brute force one). So, is it indeed true that no solutions exist for the ...

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### How did they come up with the MRRW bound?

Among the good asymptotic bounds in coding theory in the MRRW bound. It is obtained by using the linear programming problem of Delsarte's and providing a solution. The LP problem is
Suppose $C \...

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### Has this self-similar sequence the ratio $(\sqrt2+1)^2$?

This is inspired by a math.SE question, where an infinite sequence of pairwise distinct natural numbers $a_1=1, a_2, a_3, ...$ has been defined as follows:
$a_n$ is the smallest number such that $s_n:=...

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### a Kernel free asymptotic for a sampling operator

Let $\Pi=\left\{ t_{k}\right\} _{k\in\mathbb{Z}}$ a sequence of real numbers such that $-\infty<t_{k}<t_{k+1}<+\infty$ for every $k\in\mathbb{Z}$, $\lim_{k\rightarrow\pm\infty}t_{k}=\pm\infty$...

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### Friedlander-Iwaniec Flipping moduli

I am reading section 12 (Flipping Moduli) of the paper "The polynomial $X^2+Y^4$ captures its primes" by Friedlander and Iwaniec.
At page 997, just below equation (12.7) we start estimating the ...

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### Steepest descent integration in several dimensions

The method of steepest descent provides an asymptotic approximation for integrals of the form:
$$I = \int_C \exp(M f(z))\mathrm dz$$
for large positive $M$, where $f(z)$ is analytic in the region of ...

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### Asymptotic upper bound for partial binomial-like sum

I want to upper bound the quantity $$\sum_{i\le \alpha n} \binom{n}{i}\lambda^i$$, where ${\lambda>1}$, $0<\alpha<1$. It is not the same as partial sum of binomial coefficients. An asymptotic ...

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### Non-asymptotic tail bounds for $D_{\text{Hellinger}}(P\|\hat{P}_N)$

Let P be a distribution on a finite set of size $k$ and let $\hat{P}_N=(N_1/N,\ldots,N_k/N)$ be the empirical distribution (frequencies) from a samples of size $N$. Consider the Hellinger distance ...

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### Deriving condition to get correct asymptotic bound

Suppose that $X\sim \text{Bin}(n,\theta)$. Note that $X$ is the sum of $n$ $iid$ Bernoulli($\theta$) random variables. By the local limit theorem (Theorem 7 here) for the sum of discrete random ...