# Questions tagged [limits-and-convergence]

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

643
questions

7
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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 $m$.

1
vote

0
answers

74
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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
$$
P(X=0) = \sum _{i=0}^{\infty} \frac{(-1)^i E_{r}[X]}{ i!}
$$
...

0
votes

0
answers

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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 ...

2
votes

1
answer

61
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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?
...

5
votes

0
answers

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### Very slow continued fraction convergence

Let $a(0)+b(0)/(a(1)+b(1)/(a(2)+b(2)/(a(3)+\dots)))$ be a continued fraction, and $p(n)/q(n)$ its $n$-th convergent. If it converges (i.e., $p(n)/q(n)$ tends to some limit
$S$ as $n\to\infty$), then $...

3
votes

0
answers

172
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### Some pending questions about $\sum_{p\leq\sqrt{n}}p=\pi(n)$

Here it was showed that $S(n)\sim \pi(n)$, where $S(n)=\sum_{p\leq\sqrt{n}}p$, $p$ refers to prime numbers, and $\pi(n)$ is the prime counting function. Here it was proved that $S(n)=\pi(n)$ for ...

2
votes

1
answer

89
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### Convergence in probability of a supremum

Let $A>0$ be fixed and consider $X_1,\ldots$ i.i.d. nonnegative random variables such that $E[1/X_1]<\infty$.
Is is true that $$\sup_{a\in \big (0,\frac A{\sqrt n} \big]} \sum_{i=1}^n 1_{X_i>...

4
votes

2
answers

127
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### Does the average of correlated Gaussian random variables with mean zero and different variances converge in probability to their mean?

Let $X_i\sim N(0,\sigma_i^2)$ and $\operatorname{Corr}(X_i,X_j)>0$. Is it possible to show that $$\frac{1}{N} \sum_{i=1}^N X_i \overset{p}\rightarrow E[X_i]=0.$$ Do you have a reference to a law of ...

0
votes

0
answers

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### Relevance of the deduction of similar theorems than Maier's theorem for other constellations of primes

A year ago I asked this question on Mathematics Stack Exchange with identifier 4245823 and same title Relevance of the deduction of similar theorems than Maier's theorem for other constellations of ...

0
votes

2
answers

108
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### Show that the set of strictly stationary, mean zero and finite variance stochastic processes is closed (or not)

Let $\mathcal{P}$ be the set of real-valued and strictly stationary processes with expectation zero and finite variance, i.e.:
\begin{equation}
\mathcal{P}:=\left\{ X = (X_t)_{t \in \mathbb{Z}} \, ...

3
votes

1
answer

279
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### Limit of the average of telescopic products

I am trying to show that
$$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\prod_{j=0}^{i}\frac{kn-j-k}{kn-j}=\frac{k^{k+1}-(k-1)^{k+1}}{(k+1)k^{k}}$$
for all $k\in\mathbb{N}$, $k\geq 4$.
I could verify the ...

2
votes

1
answer

74
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### Approximation of a stationary process by a sequence of ergodic and stationary sequence of stochastic processes

Let $X = [X_t : t \in \mathbb{Z}] \sim P$ and $Y = [Y_t : t \in \mathbb{Z}]\sim Q$ be two stochastic processes. Let's define the Mallows metric. Let $\mathcal{M}_m$ be the random vectors $(X,Y)$ ...

1
vote

0
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### Normal numbers and law of the iterated logarithm

If I remember correctly, for the binary digits of a real number in $[0,1]$, I was told that satisfying the law of the iterated logarithm (LIL) is stronger than being normal. That is, supposedly, some ...

1
vote

1
answer

61
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### Locallic maps given by series

Maps between real numbers are often defined by convergent series. For example, to define the exponential map, we can just prove that series $$\sum_{n = 0}^{\infty} \frac{x^n}{n!}$$ converges, which ...

5
votes

0
answers

101
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### Convergence of the best constant in the $s$-fractional $L^p$ Sobolev inequality

It is known that the fractional $L^p$ Sobolev inequality
$$
\|f\|_{L^{p^*_s}(\mathbb R^n)}^p \leq \sigma_{n,p,s} (1-s) \int_{\mathbb R^n}\int_{\mathbb R^n} \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} dx dy
$$
...

17
votes

2
answers

995
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### "Find $\lim_{n \to \infty}\frac{x_n}{\sqrt{n}}$ where $x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$" -where does this problem come from?

Recently, I encountered this problem:
"Given a sequence of positive number $(x_n)$ such that for all $n$,
$$x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$$
Find the limit $\lim_{n \rightarrow \infty} \...

0
votes

0
answers

92
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### If the Dirichlet series $L(z,\chi)$ diverges for $\sigma< 1$, does its alternating version converge for some $\sigma_0 < 1$, and conversely?

Here $\chi$ is a completely multiplicative function with $\chi(p)\in\{-1,+1\}$ for any prime $p$, but not necessarily a character. Also $s=\sigma + it$ as usual. The series corresponding to $L(s,\chi)$...

1
vote

1
answer

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### Does pointwise convergence yield the convergence under Skorokhod topology?

Let $D_+$ be the set of non-increasing functions $f: [0,T]\to [0,1]$ that are right-continuous. Let $(f_n)_{n\ge 1}\subset D_+$ be a sequence of continuous functions s.t. $\lim_{n\to\infty }f_n(t)$ ...

1
vote

1
answer

111
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### Interchange summation order in the limit of number of elements going to $\infty$

Considering the sum $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty} a_{ij}$, in general we are not allowed to interchange the summation order (i.e. pass to $\sum_{j=0}^{\infty}\sum_{i=0}^{\infty} a_{ij}$) but ...

2
votes

1
answer

63
views

### Law of large numbers for triangular arrays whose moments "look independent"

Let $(X_{n,k})_{k=1,\ldots,n}^{n\in\mathbb{N}}$ be a triangular array of random variables with finite moments of all orders, with no assumptions on their independence. Suppose that
$$
\mathbb{E}\left[\...

1
vote

0
answers

120
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### "Beautiful convergence" in Hejhal, The Selberg Trace Formula

A simple question:
Hejhal in his volumes on the Selberg Trace formula (particularly volume 2) uses the expression "converges beautifully".
I can't find the definition, even searching the ...

0
votes

0
answers

49
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### Dense subspace of square integrable functions on the complex disc

Denote by $L^{2}(D,(1-|z|^{2})^{a-1}|z|^{2b-2}dx dy)$ the set of square integrable functions on the complex disc $D= \lbrace z \in C, \; |z| <1 \rbrace$ with respect to the measure $(1-|z|^{2})^{a-...

1
vote

0
answers

38
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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 ...

8
votes

2
answers

268
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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 ...

0
votes

1
answer

83
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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 ...

0
votes

0
answers

41
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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 ...

3
votes

0
answers

73
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### On tangential approach regions for general power series converging on the unit disk

Notation and premises. Here it is a list of notations more or less explicitly used in the question:
If $z\in\Bbb C$ then $z = re(t)$ where $r\in \Bbb R_{\ge 0}$, $t\in [0,1]$ and $e(t)\triangleq \exp(...

2
votes

0
answers

71
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### Relations between different "propagation of chaos" type results?

My questions come from the paper Logarithmic Sobolev inequalities for some
nonlinear PDE’s written by F. Malrieu (May 2001). The basic set-up is that we have a $N$-particle system $(X^{i,N}_t)_{1\leq ...

-2
votes

1
answer

195
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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 ...

0
votes

0
answers

62
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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 $N$. ...

0
votes

0
answers

47
views

### 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 ...

2
votes

1
answer

100
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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. ...

1
vote

3
answers

228
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### Squeezing more convergence from the convergence in all $L^p$ spaces

Let $X$ be a space endowed with a finite measure $m$. Let $f_n : \to \mathbb C$ be measurable functions such that $|f_n| \le 1$ for all $n$ and $f_n \to 0$ in every space $L^p (X)$ with $p \in [1, \...

4
votes

1
answer

76
views

### Distance between trunctated random walk and its normal form

I have $$X_i \sim N(0,1), \quad S_n=X_1+\cdots+X_n,$$
$$ \mathscr{S}_n (t, \omega) := \frac{1}{ \sqrt{n} } \sum_{i=1}^{n} \left[ S_{i-1} (\omega ) + n \left( t - \frac{i-1}{n} \right) X_{i}(\omega) \...

2
votes

1
answer

232
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### One series converges iff the other converges

In Show that $\sum\limits_pa_p$ converges iff $\sum\limits_{n}\frac{a_n}{\log n}$ converges it is said that this sequence of partial sums converges
$$
\begin{split}
\sum_{1<n\leq N}\frac{a_{n}}{\...

0
votes

0
answers

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### Demonstrate a certain inequality related to the accompanying law theorem

Suppose I have a triangular array of r.v. $(X_{nj})_{1\leq j \leq k_n}$- $X_{nj} \sim \mu_{nj}$, $k_n \uparrow\infty$ - which are independent in each row and satisfies the uniformly asymptotically ...

3
votes

1
answer

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### For a random sequence $X_0, X_1, X_2, \ldots$ and $F_n$ the empirical CDF, does $F_n(X_0)$ converge to a uniform random variable?

Let $X_0, X_1, X_2, \ldots$ be a sequence of i.i.d. real-valued random variables on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with continuous CDF $F(x)$ and define a sequence of ...

2
votes

1
answer

148
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### Are Chebyshev polynomials a Schauder basis of $\mathrm{Lip}[-1,1]$?

It is known that every Lipschitz function $f \colon [-1,1] \to \mathbb R$ can be expressed as a series in the Chebyshev polynomials $$f = \sum_{n = 0}^\infty a_n T_n $$ which is absolutely convergent ...

-1
votes

1
answer

146
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### Convergence to a constant or not? Reference request [closed]

Consider the function
$$f(n) = \log n /(n\ \log\theta(p_n)),$$
where $\theta$ is the first Chebyshev function and $p_n$ is the $n$-th prime. Does $f$ converge to a constant as $n$ grows to infinity, ...

1
vote

1
answer

101
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### A problem of the limit of $\frac{\sup_{0<\lvert y\rvert\leq \delta}\lvert f(x+y)-f(x)\rvert}{\delta^{a}}$

Suppose that $f$ is a continuous function on $[0,1]$. For $0<a<1$, if
$$ \varlimsup_{\delta \rightarrow 0} \frac{\sup_{0<\lvert y\rvert\leq \delta}\lvert f(x+y)-f(x)\rvert}{\delta^{a}} = \...

0
votes

0
answers

39
views

### Persistence of planar trajectory converging to a node / focus

I consider a planar system $\dot u =F(u,p)$ where p is a scalar parameter. Suppose that the flow $\phi^t(u_0; 0)$ from $u_0$ converges to a stable node / focus $x^{eq}_0$ for the parameter value $p=0$....

3
votes

1
answer

87
views

### Boundedness and convergence

If I know that $\Phi_\varepsilon$ is bounded in $L^{\infty}(\mathbb{R}^{2d})$ and that $\nabla \Phi_\varepsilon$ is bounded in $L^{\infty}(\mathbb{R}^{2d})$, is it true that $\nabla \Phi_\varepsilon \...

1
vote

0
answers

73
views

### $ \varlimsup_{x\rightarrow 0^+}\frac{f(x)}{x^a}=\varliminf_{x \rightarrow 0^+}\frac{f(x)}{x^a} $

$f(x)$ is continuous for $\forall x \geq 0$ and monotonically decrease. $f(0)=0$. $a>0$. Is it true
$$ \varlimsup_{x\rightarrow 0^+}\frac{f(x)}{x^a}=\varliminf_{x \rightarrow 0^+}\frac{f(x)}{x^a} $...

2
votes

1
answer

50
views

### Compare two limits related to Hölder condition

Suppose $f$ is a continuous function on $\mathbb{R}$. $0<a<1$. $B(x,r)$ is open ball centered at $x$ with radius $r$. Is it true that
$$ \varlimsup_{r\rightarrow 0} \frac{|f(x+r)-f(x)|}{|r|^\...

2
votes

2
answers

474
views

### About roots of polynomials [closed]

Let $n\in\mathbb N^*$, $P(x)=a_0+\dotsb+a_{n-1}x^{n-1}+x^n$ and $r_1,\dotsc,r_n\in\mathbb C$ the roots of $P$.
Is it true $\lim\limits_{\max(\lvert a_i\rvert,i=0\dotsc n-1)\rightarrow 0} \max(\lvert ...

1
vote

1
answer

281
views

### Lower bound and limit of a sum with binomial coefficients

Let
$$A_k = \sum_{i=1}^k i {3k-2i-1 \choose i-1} {2i-2 \choose k-i}$$
$$B_k = \sum_{i=1}^k i {3k-2i-2 \choose i-1} {2i-1 \choose k-i}$$
$$C_k = \sum_{i=1}^k (3k-2i-2) {3k-2i-3 \choose i-1} {2i\...

5
votes

4
answers

711
views

### 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(...

2
votes

2
answers

210
views

### Convergence almost everywhere of characteristic functions

Let $(\Phi_n)_n$ be the characteristic functions of probability measures $(\mu_n)_n$ and let $\Phi$ be the characteristic function of a probability measure $\mu$.
Do you know an example where $\Phi_n(...

1
vote

1
answer

61
views

### Convergence of the average weight of an infinite path through a weighted directed graph

Consider a directed graph $G = (V, E, w)$, where $V$ is the set of vertices, $E \subseteq V \times V$ is the set of directed edges (with self-loops allowed), and $w : E \to \mathbb{R}_+$ is a weight ...

4
votes

2
answers

220
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### $\lim\inf_{t\to \infty} f(x(t))=0\Longrightarrow \lim\inf_{t\to\infty} \|x(t)\|=0$

I am a PhD student and during my research I was presented to the claim that
For a positive definite function $f:\mathbb{R}\to \mathbb{R}$ continuous in $0$, with $0$ a stable point at $t=0$ for $x$, ...