Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [rate-of-convergence]

For questions related to the rate of convergence (or the order of convergence). The rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit.

In numerical analysis, the rate (or order) of convergence of a convergent sequence is a quantity that represents how quickly the sequence approaches its limit. A sequence $(x_{n})$ that converges to $x^{*}$ is said to have order of convergence $q \geq 1$ and rate of convergence $\mu$ if

$\displaystyle \lim _{n\rightarrow \infty }{\frac {\left|x_{n+1}-x^{*}\right|}{\left|x_{n}-x^{*}\right|^{q}}}=\mu .$

For more, check out this link.

131 questions
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sublinear rate of convergence in mathematical optimization

In Wiki page, the sublinear convergence rate refers that for a sequence $\{x_k\}$ with limit $L$, \begin{align*} \limsup \frac{\|x_{k+1} - L\|} {\|x_k - L\|} = 1. \end{align*} In most convex optimization books, a sequence converges at a rate…
user1101010
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Convergence of a linear recurrence equation

Let $T \colon \mathbb{C}^n \to \mathbb{C}^n$ be a linear operator. Let $\{u_k\} \subset \mathbb{C}^n$ and $\{v_k\} \subset \mathbb{C}^n$ be two sequences of vectors. Suppose the spectral radius of $T$ is $\rho(T) = q <1$ but we don't know whether or…
user1101010
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Bernoulli numbers and $\pi^2$.

It is probably well-known that: $$ \lim_{n\to\infty}\frac{b_{2n}n^2}{b_{2n+2}}=-\pi^2, $$ where $b_n$ are the Bernoulli numbers. By a numerical experiment I have found that the quotient $$ \frac{b_{2n}(n+a_1)(n+a_2)}{b_{2n+2}} $$ with $a_1=1,…
user
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Comparing largest real roots analytically

Context I’m trying to prove that the largest real root of a polynomial $p_{2}\left ( x, \ell \right )$ is greater than that of $p_{1}\left ( x, \ell \right )$ for all positive integers $\ell> 1$, where both polynomials arise from computing the…
Dang Dang
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$E[Y_n^{-1}]$ converges at the same rate as $E[Y_n]^{-1}$ where $Y_n = \bigg(n\sum_{i=1}^n X_i^2 - \bigg(\sum_{i=1}^nX_i \bigg)^2\bigg)^k$

Let $X_1,X_2,\dots,X_n$ be i.i.d observations of a continuous random variable $X$. Let $Y_n$ be the sample variance: $$ Y_n = \bigg(n\sum_{i=1}^n X_i^2 - \bigg(\sum_{i=1}^nX_i \bigg)^2\bigg)^k. $$ Actually, for $k=1$, $Y_n$ is simply the sample…
sonicboom
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Convergence rate law of iterated logarithm for a Brownian motion

The law of iterated logarithm has the following implication for a standard Brownian motion $(W_t, t\geq 0)$, $$ \mathbb{P}\left(\limsup_{t\downarrow 0}\frac{W_t}{\sqrt{2t\ln\left(\ln\left(\frac{1}{t}\right)\right)}} = 1\right) = 1. $$ I wonder if…
Aguazz
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Rate of convergence in probability - log transform

Let $C>0$ and $(X_n)$ be a sequence of positive random variables. Assume that $$ |X_n - C| = o_p(r_n^{-1}) \iff r_n|X_n-C|=o_p(1) $$ for some fixed sequence $(r_n)$ with $r_n \to \infty$. What can we say about the rate of convergence of the…
John
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Limit of a sequence given by recurrence relation and convergence rate

Suppose we have a sequence $\{a_n\}_{n=0}^{\infty}$ which is generated by \begin{align*} a_{n+1} - \left(q+ \frac{A} {n+1} \right) a_n - \frac B n a_{n-1} = C, \end{align*} for $n \ge 1$, where $q, A, B, C$ are fixed positive constants and $0 < q <…
user1101010
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Rate of $L_1$ loss in estimating density on $[0,1]$

Let $f$ be a density on $[0,1]$ and let $X_1,X_2,\ldots$ be $\textit{iid}$ $f$-distributed. Also, let $f_n$ denote the kernel density estimator, i.e. $$f_n(x) = \frac{1}{nh_n} \sum_{i=1}^n K\left(\frac{x-X_i}{h_n}\right),$$ where $K$ is a kernel…
Yannik
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In practice, what does it mean for the Newton's method to converge quadratically (when it converges)?

I was studying about the Newton's method (and other root-finding methods) and apparently Newton's method converges quadratically (or more) when it does. Suppose that the sequence $\{x_k \}$ converges to the number $L$. We say that this sequence…
user168764
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What is the rate of convergence for a certain cubic iterated function system arising from Newton's method applied to a smooth cutoff?

A somewhat silly mathematical diversion was proposed to me by a friend, and I have reduced the question to the following one: Let $f(x)=e^{-\frac{1}{x^2}}$. Given some $x\in \mathbb{R}$ with $0< x < 1$, find asymptotics on the sequence of values…
npdotrand
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Is it true that $\mathbb{E}\left[\sum\limits_{k=1}^m \frac{\frac{1}{X_k}}{\sum_{j=1}^m \frac{1}{X_j}}\chi_{(r,+\infty)}(X_k)\right]\to0,m\to\infty?$

The following problem arose in the process of showing the convergence of a particular regression algorithms. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Suppose that $X,X_1,X_2,...:\Omega\to(0,+\infty)$ are $\mathbb{P}-$i.i.d.…
Bob
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Rate of convergence of a bisection-like algorithm

Fix a value $x\in(0,1)$ and recursively define the…
Simply Beautiful Art
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Rate of weak convergence of sin(nx)

Since $\sin(n\cdot)$ converges weakly to zero, we know that $$ \lim_{n\rightarrow\infty} \int_a^b g(x)\sin(nx)\mathrm{d}x = \int_a^b g(x)\cdot 0\,\mathrm{d}x = 0 $$ holds for all $g\in L^2([a,b])$. Is there a way to find an explicit formula for the…
user574084
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2 answers

Faster Convergence for the Smaller Values of the Riemann Zeta Function

I have a C++ program that uses the equation $$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$$ to calculate the Riemann zeta function. This equation converges fast for larger values, like 183, but converges much slower for smaller values, like 2. For…
esote
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