# Questions tagged [limits-and-convergence]

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

772
questions

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### Solve divergent iteration to a certain number of decimal places [closed]

I have a quantity X which I have to take a percentage of, but then it adds to the quantity and the percentage has to be taken of the new result. This ends up in the divergent equation:
...

1
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0
answers

131
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+50

### Proof of convergence of sets

Consider the following sets:
\begin{align}
&A(y) \equiv \Bigl\{ x\in X: \Pr\biggl(\lim_{n \to \infty}d\bigl(p_n(y), [\ell(x,y), u(x,y) ] \bigr)= 0\biggr)=1 \Bigr\},\\
& A_n(y) \equiv \Bigl\{...

1
vote

1
answer

71
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### Mittag-Leffler expansions converging to bounded function

Is it true that
$$\lim_{N\to\infty}\left\langle\sum_{n=-N^2}^{N^2}\frac1{(Nx-n)^2}\right\rangle_N=\pi^2$$
for some suitable definition of "minima smoothing" such as $\langle f(x)\rangle_N\...

0
votes

0
answers

42
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### How to find A(i, d)?

Let $s(n)$ denote the digit sum of a natural number $n$. For $i, d\in \mathbb{N}$ define $$A(i, d) = \limsup_{m\to \infty}\frac{|\{n\leq m | s(n)\equiv i\mod d\}|}{m}.$$ Compute $A(i, d)$ for all $i, ...

1
vote

2
answers

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### Show convergence result

Consider the following sets:
$$
A = \Big\{ x\in X: \Pr\bigg(\lim_{n \to \infty}d\big(p_n, [\ell(x), u(x) ] \big)= 0\bigg)=1 \Big\},
$$
and
$$
A_n = \Big\{ x\in X: d\big(p_n, [\ell(x), u(x) ] \big)...

2
votes

1
answer

63
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### Lipschitz continuity of eigenprojections

This question has the same flavor of this and this questions, but asks for something stronger.
Assume that
$A$ is a symmetric $n \times n$ matrix,
$H$ is a $n \times n$ perturbation matrix.
Moreover ...

1
vote

1
answer

106
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### Limit of $F_{n}(\lfloor{nx}\rfloor)$ where $ F_{n}(k)=G_{n}(k)+H_{n}(k)F_{n}(k+1) $ and $F_{n}(n)=\mu.$

The following conjecture is inspired by asymptotic results in generalizations of the secretary problem.
CONJECTURE
Consider a sequence of functions {$F_n$} with $F_{n}:[0,n]\cap
\mathbb{Z}\rightarrow\...

0
votes

1
answer

293
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### Necessary conditions for convergence of convolution

In math.SE, I've asked a question about the convergence of convolution of two functions which have bilateral Laplace transform and also have disjoint Region Of Convergence (ROC) but the question didn'...

7
votes

1
answer

510
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### A variation on the Borel–Cantelli lemma theme

Let $X,X_0,X_1,\dots$ be nonnegative independent identically distributed (i.i.d.) random variables. Let
\begin{equation*}
E:=\bigcap_{n\ge0}B_n,
\end{equation*}
where
\begin{equation*}
B_n:=\...

2
votes

2
answers

295
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### Existence of the limit of periodic measures

Let $T: X \to X$ be a continuous map over a compact metric space. We say that a measure $\mu$ is $T$-invariant if $T_{\ast} \mu= \mu$. We denote by $M(X, T)$ the space of all $T$-invariant Borel ...

3
votes

2
answers

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### $\lim_{n \to \infty} \frac{2^n}{n} \left[ 1 - \sum_{k = 1}^{n-1} \frac{(1 - \lambda 2^{-n})^{2^k}}{2^k} \right]$

Find the limit
\begin{equation*}
\lim_{n \to \infty} \frac{2^n}{n} \left[
1 - \sum_{k = 1}^{n-1} \frac{(1 - \lambda 2^{-n})^{2^k}}{2^k}
\right]
\end{equation*}
where $\lambda > 0$.
My guess is that ...

2
votes

5
answers

895
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### Binomial series

I am interested in the limit $\frac{\sum_{k=0}^n \sqrt{k}\cdot\binom{n}{k}}{\sqrt{n}\cdot2^n}$ as $n$ goes to infinity. Any reference or argument?
In general what do we know about the asymptotic ...

2
votes

0
answers

154
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### Question about the ergodic mean

This is a repost from this MathStackExchange question, where unfortunately I was not able to resolve this question.
I've read a thesis where there is an example on ergodic mean, where however there is ...

7
votes

3
answers

834
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### Using the Stone-Weierstrass theorem to solve an integral limit

The following question was posted on math stack exchange here but it got no answers
Let $c\in (1, +\infty)$ and $f \colon [0, c] \to \mathbb{R}$ be a continuous and monotonically increasing function ...

5
votes

1
answer

380
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### On the convergence of a martingale

Let $W$ be a standard one dimensional Brownian motion and let $A$ be the process defined by :
$$\forall \ t\geq 0: \quad A_t := \int_0^t\left(1 + e^{W_s}\right)\mathrm{d}s$$
and for $t\geq 0$, we ...

-1
votes

2
answers

81
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### Limits of integral series

Suppose we have the series of functions:
\begin{equation}
F(x)=\sum_{n=1}^{\infty} f_n(x)
\end{equation}
where convergence is uniform.
Additionally, consider the partial functions of the series:
\...

27
votes

5
answers

3k
views

### How to show a function converges to 1

Consider the following recurrence relation in two variables:
$$f(a, b) = \frac{a}{a+b} f(a-1,b)+ \frac{b}{a+b}f(a+1,b-1) $$
for positive integers $a$ and $b$,
with the boundary conditions $f(0,b)=0$ ...

3
votes

2
answers

231
views

### Definition of weak conditional convergence of random variables

I am looking for a definition of conditional convergence. Suppose that $X_1, X_2, \dots, X_n$ are $\mathbb R$-valued random variables with finite second moments, and $W_1, W_2, \dots, W_n$ are iid $\...

7
votes

1
answer

252
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### High dimensional Fekete's subadditive lemma: does $|\vec x_{n+m}|\leq |\vec x_n+\vec x_m|$ imply the convergence of $\{\vec x_n/n\}$?

Let $d\geq 1$ be a positive integer. If $\{\vec x_n\}_{n=1}^\infty$ is a sequence of $d$-dimensional vectors satisfying $$\lvert\vec x_{n+m}\rvert\leq \lvert\vec x_n+\vec x_m\rvert\qquad \text{for all ...

10
votes

1
answer

312
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### Does the mean ratio of the largest prime factor in prime gaps to the lower bound of the gap converge?

Posting in MO since this questions has been unanswered in MSE for 3 months.
Let $p_n$ be the $n$-th prime and $q_n$ be largest among all the prime factors of the composite numbers between $p_n$ and $...

0
votes

1
answer

130
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### Does weak $H^1$ convergence imply $L^2$ convergence when multiplied with an exponentially decaying function?

I'm trying to see if given a sequence $\{f_n\}_n\in H^1$ which converges weakly in $H^1$ to a function $f_*$, the $L^2$ norm $\|R^2f_n\|_{L^2}^2$ converges to $\|R^2f_*\|^2_{L^2}$, where $R$ is a ...

4
votes

0
answers

152
views

### Question about $n$ random points in a regular polygon, and a limiting probability

Suppose we choose $n$ uniformly random points in a disk, then draw the smallest circle that encloses all of those points. There is evidence suggesting that the probability that the enclosing circle is ...

0
votes

1
answer

240
views

### Nature of $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $ [closed]

I'm trying to determine the nature of this series $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $, but I'm not getting anywhere. I've tried using the Abel and trigonometric formulas, but I can't ...

2
votes

2
answers

197
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### Weak convergence of measures on continuous function spaces

Let $S$ be the unit sphere of $C[0,1], \|\cdot\|_{\infty})$, let $(B_{t})_{t}$ the brownian motion.
I would like to show that the measure $\mu_r$ defined on $\mathbb{B}(S)$ by
$\mu_r(A):=P\Big(\frac{...

2
votes

2
answers

188
views

### Limit of a integral whose integrand diverges under the limit

I am trying to simplify the following limit of integral where $\mu$ is given:
$$p(y) = \lim_{\sigma \to 0} \int_{\mathbb R} |x| \cdot \frac{1}{\sqrt{2\pi\sigma^2} } e^{-\frac{1}{2\sigma^2} (xy - \mu)^...

0
votes

1
answer

105
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### Functional CLT with an asymptotically small time change

This question was posted to MathSE but it seems like MathOverflow might be the more appropriate place for it.
Suppose I know that $(\frac{1}{\sqrt{m}}X(mt))_{0\leq t\leq 1}\xrightarrow[m\to\infty]{\...

1
vote

0
answers

87
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### Asymptotic distribution of L infinity norm of Gaussian random vector

Let $\mathbf{X}_n = (X_{n,1}, \ldots, X_{n,n})$ be a $n$-dimensional random vector with $N_n( \mathbf{0}_n, \boldsymbol{\Sigma}_n )$ distribution. The asymptotic distribution of the $L_\infty$-norm of ...

11
votes

2
answers

903
views

### Twice continuously differentiable implied by existence of limit

I have the following question. Let $f,g:\mathbb{R}\to\mathbb{R}$ be two continuous functions (vanishing at infinity) and assume that
$$
\frac{f(x+t)+f(x-t)-2f(x)}{t^2}\to g(x)
$$
for all $x\in X$ when ...

22
votes

1
answer

4k
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### A challenging (for me) limit calculation

How to calculate the following limit
$$
\lim_{n\to\infty}\sqrt{n}\underbrace{{}\sin(\sin(\sin(\sin(\cdots\sin(\frac{1}{\sqrt{n}})\cdots))))}_{n \text{ sin's}} \text{?}
$$
${}{}$

3
votes

1
answer

346
views

### On the convergence in total variation

$\newcommand\R{\mathbb R}$For a probability measure $\mu$ over $\R^2$ and a unit vector $u\in\R^2$, let $\mu^u$ denote the pushforward of $\mu$ under the projection map $\R^2\ni x\mapsto u\cdot x\in\R$...

1
vote

1
answer

98
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### How to prove this iterative convergence of trigonometric functions [closed]

Consider the set of sequences $ S=\{\{s_n\}_{n \ge 0}\mid s_n \in \{-1,+1\}\} $
For any set: $ s=\{s_n \}_{n \ge 0}\in S$, we define the sequence
$$ c_n=\sum_{k=0}^n \frac{s_0s_1 \dots s_k}{2^k}$$
Now,...

0
votes

1
answer

98
views

### If $a_1=1$ and $a_n=\sec (a_{n-1})$ then what does the proportion of positive terms approach, as $n\to\infty$?

Consider the sequence $a_1=1$ and $a_n=\sec (a_{n-1})$ for $n>1$.
What does the proportion of positive terms approach, as $n\to\infty$?
At first I thought the limiting proportion might be $\frac{...

2
votes

0
answers

54
views

### A lemma in the application of Lions's concentration compactness pricnciple in Hardy-Littlewood-Sobolev inequality

I'm encountering some problems when reading Lions' paper "the concentration-compactness principle in the calculus of variations. The limit case, Part 2".
The Hardy-Littlewood-Sobolev (HLS) ...

2
votes

1
answer

72
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### Conditions for absorption

Let $X$ be a Markov chain with countable state space $S$ and transition kernel $P$, and let $h \colon S \to [0,1]$ be a sub-harmonic or super-harmonic function. Assume that for all $\varepsilon >0$ ...

1
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0
answers

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### Asymptotics of ${2n \choose n+k} {2n \choose n}^{-1}$ when $k$ grows with $n$

The quotient $Q(n,k) := \frac{{2n \choose n+k}}{{2n \choose n}}$ clearly converges to one for $k \in \mathbb{N}$ fixed and $n \rightarrow \infty$. Simultaneously it converges to zero, if $k$ grows ...

3
votes

2
answers

248
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### On convergence of convex-concave functions

Let $(f_n)$ be a sequence of twice differentiable functions on $\mathbb R$ such that for each $n$ there exists some $x_n\in\mathbb{R}$ such that:
$f_n$ is strictly convex on $(-\infty,x_n)$,
$f_n$ is ...

1
vote

1
answer

215
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### Convergence of concave/convex function

Let assume that you have a sequence of twice differentiable functions $(f_n)_{n\in\mathbb{N}}\in\mathscr{C}^2(\mathbb{R})^{\mathbb{N}}$. Let suppose that for each $f_n$, it exists a $x_n\in\mathbb{R}$ ...

1
vote

1
answer

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### If a Hilbert space-valued mapping is norm-decreasing, can we make sense of the limit of sums consisting of projected-values?

Let $H$ be some separable Hilbert space with a given orthonormal basis $\{ e_n \}$. Write the projection onto the subspace spanned by first $N$ basis elements to be $P_N$.\
Now,let $g(t) : [0,1] \to H$...

2
votes

1
answer

148
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### Continuity of an upper semi-continuous function over periodic points

Let $f: X \to \mathbb{R}$ be an upper semi-continuous function on $X$, which is a compact subspace of a vector space. Let sequence $x_n, n \in \mathbb{N}$, with positive elements - periodic: there ...

0
votes

1
answer

430
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### A complex question related to a certain convergence of Lévy measures

Consider the sequence of stochastic processes $(X_n, n \geq 1)$, where $X_n = (X_{t;n})_{t\in \mathbb Z}$ and:
\begin{equation}\label{I}\tag{SP}
X_{t;n} = \sum_{j=0}^\infty \theta_{jn} \varepsilon_{t-...

0
votes

0
answers

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### Characterizing functions that are limits of integrable lower-bounded functions

Let $X$ be a separable Hausdorff topological space, endowed with a positive finite Borel regular measure. Consider those (trivially measurable) functions $f : X \to \mathbb R$ such that their ...

0
votes

0
answers

67
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### Error bound for stochastic gradient descent method

To solve an optimization problem $\min_x G(x)$ using standard stochastic gradient descent method, we let $x_0$ be the initial point and $x_k$ be the $k$-th point such that
\begin{equation}
x_k = x_{k-...

4
votes

1
answer

460
views

### How "correct" is Knuth's fast addition $(a,b) \mapsto (a \oplus b) \oplus ((a\land b) \ll 1)$?

Donald Knuth suggested a bitwise approximation for addition on the non-negative integers that is very fast on common processors:
$(a,b)\mapsto (a\oplus b) \oplus ((a\land b) \ll 1)$,
where $a,b$ are ...

3
votes

1
answer

584
views

### Sequence of $L^2$ functions converging to zero weakly s.t. $|f_n|^2$ converges to 1 weak-star?

I am trying to construct a sequence $\{f_n\} \in L^2([0,1])$ with $f_n \geq 0$ a.e. such that $f_n \to 0$ weakly in $L^2$ (meaning $\int f_n dx \to 0$ for all $f \in L^2([0,1]))$ and such that for all ...

1
vote

1
answer

70
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### Asymptotic scaling of mean and variance for the norm of random vector (non gaussian components)

The norm of a vector whoose components $X_i$ are normally distributed follows the Non-central chi distribution and it can be shown that, increasing the number of components $k$ (i.e. the dimension of ...

0
votes

0
answers

68
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### Meromorphic extension of a limit function

Suppose $f_j(z)$, $j=1,2,..$ is a sequence of meromorphic functions on the complex plane $\mathbb{C}$. With a common set of all poles given by $S = \{-i,-2i, -3i,..\}$.
Assume that each of them is ...

1
vote

0
answers

29
views

### Can we modify this extended pseudometric such that its convergence is equivalent to that in measure?

Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple ...

4
votes

1
answer

199
views

### Show that $\frac{1}{2 \pi i} \oint_{\mathbb{S}^1} \frac{1-\hat{f}(\xi)}{1-\xi}\cdot \frac{\mathrm{d} \xi}{\xi^{n+1}} \to 0$ as $n \to \infty$

Let $f = (f_0,f_1,\ldots,f_n,\ldots) \in \mathcal{P}(\mathbb N)$ be a probability distribution on $\mathbb N$ and denote by $$\hat{f}(z) = \sum_{n\geq 0} z^n f_n$$ for its probability generating ...

1
vote

1
answer

76
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### Supremum or upper bound of bivariate function involving logarithms and combinatorial coefficients or the gamma function over a region of the integers

This is a repost from MSE because I got no answers there.
I have been trying to find the supremum of this bivariate function over a specific region. However, the expressions that I get are horrible. I ...

1
vote

0
answers

47
views

### Convergence of random variables based on shifts of a markov chain

Suppose we have a discrete time (not necessarily stationary) Markov chain $X=(X_0,X_1,X_2,\dots)$ on $(\Omega, F)$. We assume $X$ is Harris ergodic with an invariant distribution.
Suppose we have a ...