# Questions tagged [limits-and-convergence]

Convergence of series, sequences and functions and different modes of convergence.

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### Does the limit of $x_n$, defined by $x_{n+1}=1/(m+1-nx_n)$ exist?

Let $m$ be positive integer, and consider the recursion $$x_{n+1}=\frac{1}{m+1-nx_n}.$$ Does the limit of $x_n$ exist? I'm guessing the limit doesn't exists for any $m$.
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### Limit of alternating sum of factorial moments which diverge

Consider the non-negative, integer valued random variable $X$, and its $i^{\text{th}}$ factorial moment $E_{i}[X]$. Then we have that $$P(X=0) = \sum _{i=0}^{\infty} \frac{(-1)^i E_{r}[X]}{ i!}$$ ...
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### Convergence of stopped stochastic processes

Let $(E,d)$ be a locally compact separable metric space. Let $\mathcal{D}=\mathcal{D}([0,\infty),E)$ denote the space of right continuous functions on $[0,\infty)$ having left limits and taking ...
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### On the distance to the stationary distribution

A Markov Chain $M$ has only one stationary distribution $q$. For distribution $p$, with $D_{TV}(p,Mp)=x$, can we bound $D_{TV}(p,q)$? Clearly, $x=0$ implies $D_{TV}(p,q)=0$. Does general bound hold? ...
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1 vote
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### Taming families of rate functions

$\newcommand\R{\mathbb R}$Let us say that a function $r\colon\R_+\to\R_+$ is a rate function if $r$ is nondecreasing and $r(x)\to\infty$ as $x\to\infty$. Let us say that a family $(r_j)_{j\in J}$ of ...
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### Rearrangement, conditional convergence, and "placid" permutations

This question came out of a conversation with my students about Riemann's rearrangement theorem, and the general problem of which permutations are "safe" w/r/t summing infinite series. It ...
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### Fokker-Planck: uniqueness and convergence to stationary distribution

Consider the Langevin equation ($N$-dimensional) with nonlinear drift term but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of ...
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### Equivalent formulation of uniform convergence in measure

If $(X,m)$ is a measure space, and $f_n, f : X \to \mathbb C$ are measurable functions, then it is known that $f_n \to f$ in measure if and only if every sub-sequence $(f_{n_i}) _{i \ge 0}$ contains a ...
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### Tricky (for me) limit

I've been trying to compute the following limit for a few hours. Let $f(\gamma, \beta)$ be defined as follows: $$f(\gamma, \beta)=\lim_{x \rightarrow \infty} (1-\gamma^{1/x})(\log(x))^{\beta}.$$ I am ...
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### Does non-parametric density estimation converge when cross-validation is used for model selection?

Suppose you have an infinite sequence of parametric probability density models, $\phi_i(\theta)$, with monotonically increasing parameter counts as $i$ increases, and a training sample of size $N$. ...
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### What's the convergence condition for the generating function formula of Legendre polynomials?

What is the convergence condition of the next infinite series about the Legendre polynomials $P_n(x)$? $$\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^\infty P_n(x)t^n$$ I know it is convergent at least ...
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### Mikusiński's approach to Bochner integrals; replace absolute by unconditional?

In the book The Bochner Integral, Mikusiński described an approach to Lebesgue and Bochner integrals via absolutely convergent series corresponding to step functions: Defn. Let $X$ be a Banach space. ...
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### Limit of a sum with binomial coefficients

Let $$A_k = \frac{\sum_{i=1}^ki{2k-i-1 \choose i-1}{i-1 \choose k-i}}{k{2k-1\choose k}}$$ $$B_k = \frac{\sum_{i=1}^ki{2k-i-2 \choose i-1}{i \choose k-i}}{k{2k-1\choose k}}$$ C_k = \frac{\sum_{i=1}^k(...
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