# Questions tagged [rate-of-convergence]

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### Order of convergence of iterative methods

Consider the fixed point iteration $$x_{n+1}=\phi(x_n), n\geq 0,\ \text{with}\ \phi(x)=3+(x-3)^3, \ x\in(2.5, 3.5)$$ and the initial approximation $x_0=3.25$. Then the order of convergence of the ...
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### The rate of convergence of the remainder of the power series for the Polylog function

Let $0<p<1$ be a positive real number strictly smaller than one and $q>0$ be a positive real number. Consider the series $$\mathsf{Li}_{-q}(p) = \sum_{\ell=1}^{+\infty}\ell^{q}p^{\ell}$$ ...
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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 ...
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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: $$Y_n = \bigg(n\sum_{i=1}^n X_i^2 - \bigg(\sum_{i=1}^nX_i \bigg)^2\bigg)^k.$$ ...
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### Continuous mapping theorem and convergence in $L_p$

My question kind of generalizes the question in Analogue of continuous mapping theorem for convergence in $L_2$ and it is related to the answer by Nate Eldredge. Edited to give more details: Suppose ...
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41 views

### Calculating the rate of convergence from a plot of a limit.

The rate of convergence represents how quickly a sequence approaches its limit. $$\lim _{n \rightarrow \infty} \frac{\left|x_{n+1}-x^{*}\right|}{\left|x_{n}-x^{*}\right|^{q}}=\mu$$ Here, the rate of ...
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### Convergence rate about a limit concerning the Poisson CDF.

The CDF of a Poisson distribution with rate parameter $\lambda$ is $$P(n;\lambda)=\sum_{k=0}^n \frac{\lambda^ke^{-\lambda}}{k!}.$$ As $n$ goes to infinity, the CDF would certainly approach 1. Now, ...
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### the limit of Poisson CDF with diverging rate parameter

The CDF of a Poisson distribution with rate parameter $\lambda$ is $$P(n;\lambda)=\sum_{k=0}^n \frac{\lambda^ke^{-\lambda}}{k!}.$$ As $n$ goes to infinity, the CDF would certainly approach 1. Now, ...
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1answer
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### What does first and second order convergence mean [with minimal equations] for optimization methods?

It can be shown that Newton's method has second order convergence provided some criteria is satisfied, and gradient descent has first order convergence, but what does order of convergence mean here? ...
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### Numerical Analysis: Heun's method

so I did the loglog plot of the maximum error vs the time step when using Heun's method. I see that for very small step sizes the order gets disrupted. Is there a specific reason for that? I also did ...
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### Different way of expressing rate of convergence for a sequence

Suppose that we have a sequence given by an iterative method: $$x_{n+1} = \phi(x_n)$$ where this sequence converges to a fixed point $\alpha$. Also, suppose that the sequence has the following rate ...
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1answer
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