Questions tagged [asymptotics]
Asymptotic behavior of functions, asymptotic series and related topics
947
questions
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Sum of reciprocal of Pochhamer symbols through multiples of a natural L
In a question in StackExchange (https://math.stackexchange.com/questions/4236635/sum-of-quotients-of-gamma-functions), I asked if there is a closed expression for the following sum related with a ...
1
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1
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418
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Asymptotic cone
Let $S$ be a subset in a real vector space $\mathbb{R}^n$. Define the asymptotic cone $S\infty:=\{y\in\mathbb{R}^n\mid\textrm{there exists a sequence }(y_k,\varepsilon_k)\in S\times\mathbb{R}^+\textrm{...
3
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0
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Probabilistic behavior of greedy point selection in the plane
Let $\mathcal{X} = X_1,\dots,X_n$ be a collection of independent, uniform samples in the unit square. Let $\mathcal{S}=\{X_1\}$, and consider the following process: for $i=2,\dots,n$, let $x^*$ be ...
6
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1
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Asymptotic estimate for an integral involving the squared modulus of the Riemann zeta function
For any fixed $\frac{1}{2} < \sigma < 1$, let
$$\int_0^T \frac{|\zeta(\sigma+it)|^2}{\sqrt{1+t^2}} \ dt = O(T^\theta), \qquad T \uparrow \infty. $$
It is clear that $\theta > 0$, since we ...
3
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0
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154
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On analogues of Weber's formula
Let $J_0(x)$ be the $0$-th Bessel function of the first kind. Weber's formula states that
$$
\int_0^{+\infty}e^{-x}J_0(2\sqrt{\alpha x})J_0(2\sqrt{\beta x})dx=e^{-\alpha-\beta}I_0(2\sqrt{\alpha\beta})....
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112
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Asymptotics of holonomic recurrences and the Birkhoff-Trjitzinsky method
While reading about asymptotics of holonomic recurrences, I had the following impressions that are related to the divulgation of the theory:
I don't know what is the current status of the divulgation ...
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Finding tighter upper bound of a complicated function
Define the following functions from $[0,1) \to \mathbb{R}$:
Let $f(x)=1-x^n$.
Let $r=1-\frac{1}{n(1-x)+1}$ and $u(x)=\sum_{i=1}^n\binom{n}{i}\frac{x^{n-i}(1-x)^i(1-r^i)}{(1-x^n)(1-r)i}$.
Let $g(x)=1-\...
3
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1
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144
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About the sequence $s_n:=f_{n,n} $ where $f_{0,n}=f_{n,0}= n^n$ and $f_{m,n} = f_{m-1,n}+ f_{m,n-1} + f_{m-1,n-1}$
Let the sequence:
$s_n:=f_{n,n} $ where $f_{0,n}=f_{n,0}= n^n$ and $f_{m,n} = f_{m-1,n}+ f_{m,n-1} + f_{m-1,n-1}$, for $mn>0$.
Computationally it seems that $\frac{s_{n+1}}{s_{n}} \approx e\cdot ...
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Asymptotics of $\sum \frac{d(n)}{n}$ with generating functions
We can determine the asymptotics of partial sums involving the divisor function accurately by means of, for example, the hyperbola method: $$\sum_{n\leq N}\frac{d(n)}{n}=\frac{1}{2}(\log(N))^{2}+2\...
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Asymptotic behavior of hypergeometric function ${}_{3}F_{2}(a,b-n,c-n;d-n,e-n;1)$ for $n\to\infty$
Suppose $a,b,c,d,e\in\mathbb{R}$ are such that $d+e-a-b-c>0$ and $d,e\notin\mathbb{Z}$. I would like to know whether it is possible to deduce an asymptotic formula for the sequence given by the ...
2
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1
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Approximation of $\Phi (p)$
I am trying to find the asymptotic behavior (with respect to N) of the integral $$ \frac{2}{\sqrt{\pi}}\int_0^\infty \varPhi^{N-2}(p)e^{-p^2}\ dp. $$ In Rényi and Sulanke's paper Uber die konvexe ...
2
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1
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Asymptotic bound of quotient of absolute and squared deviation from mean
The following fraction shows up when trying to show consistency of the OLS slope estimator in a simple linear regression on a log-log scale where the window of observation changes as the sample size $...
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1
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Asymptotics of the integral of an oscillating function
I would like to know the asymptotics of the following sequences of integrals:
$$ I_n = \displaystyle { \int _0 ^{+ \infty}
\dfrac{t^n}{(t + i)^{n + 1}}
...
2
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0
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Uniform bound on a certain family of hypergeometric functions
We have the following problem, which we can't solve.
Let $a \in \mathbb{C}$ be fixed, with real part $1/2$ and imaginary part $\neq 0$. We consider parameters $n \in \mathbb{Z}$ and $k \in \mathbb{Z}_{...
13
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2
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435
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Asymptotics of a randomized Fibonacci sequence
Let $f(1)=f(2)=1$ and recursively define $f(n+1) = f(n) + f(i)$, where $i$ is chosen uniformly at random from $1,\ldots,n-1$. About how big should we expect $f(n)$ to be for $n$ large? We can examine ...
3
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2
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133
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Asymptotics of a sequences of integrals
I would like to know the asymptotics of the following sequences of integrals:
$$ I_n = \int _0
^{+ \infty}
e^{-t} \left ( \dfrac{t}{1 + t} \right )^n
\...
6
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1
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Reference request: Different definitions of Big O notation
This question might sound strange, but I would like to settle this problem once and for all.
For as long as I can remember, I was introduced to the Big O notation by this definition:
Def. 1: Let $f, g$...
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1
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250
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Asymptotic behavior of an ODE
Consider the following ODE eigenproblem of $y(x)$
\begin{equation}
y'' + [\varepsilon + b^2 x - (a + \frac{b^2}{2}x^2)^2 ] y=0
\end{equation}
with eigenvalue $\varepsilon$, real constants $a,b$. ...
3
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Is there an asymptotic bound between converging and diverging series? [closed]
Let us define for every $k\in\mathbb{N}$ and every large enough $x\in \mathbb{R}$,
$$
\log^{[k]}(x) =
\begin{cases}
\log^{[k-1]}(\log(x)) & k>0 \\
x & k=0
\end{cases}.
$$
It is well known, ...
2
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2
answers
133
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Asymptotic behavior of a Bessel function on a sequence on zeros with a shifted parameter of type
Let $J_\nu$ be a Bessel function of the first kind and let $\{\lambda_{n, \nu}\}_{n\ge 1}$ be a sequence of its zeroes. I claim that
$$
\inf_{n\ge 1}\bigg|\sqrt{\lambda_{n,\nu}} J_{\nu+1}(\lambda_{n,\...
3
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Will an integer combination of some number of copies of the set of powers of 2 and the set of powers of 3 always have natural density 0?
Consider a Diophantine equation of the form
$$(c_1 2^{x_1} + \dots + c_n 2^{x_n}) + (c_{n+1} 3^{x_{n+1}} + \dots + c_m 3^{x_m}) = y$$
where $x_1, \dots, x_m, y$ are our variables (here $x_1, \dots, ...
4
votes
1
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Estimate of $\frac{\int x^{2p}\,e^{-x^{2n}\,+\,\omega(x,y)}\;dx}{\int e^{-x^{2n}\,+\,\omega(x,y)}\;dx}$
For every $x,y\in\mathbb R$ let
$$ V(x,y) \,\equiv\, a\,x^{2n} + b\,y^{2m} - \omega(x,y)\,$$
where $a,b>0$, $n,m\in\mathbb N$, $n\geq m\geq1$, and $\omega$ is such that $\omega(x,y)/(x^{2n}+y^{2m})...
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Existence of function $f$ such that $f(x) \sim \sum_{j \in \mathbb{N}} x^{1 - \frac{1}{j}}$
Is there a function $f$ on $\mathbb{R}$ such that as $x \to 0$,
$$
f(x) = \sum_{j=0}^N x^{1 - \frac{1}{j}} + o(x^{1- \frac{1}{N}}),
$$
for every $N \in \mathbb{N}$?
Heuristically there shouldn't be ...
0
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0
answers
94
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Asymptotic optimal sphericity
How quickly does maximum sphericity of polyhedra with $n$ faces approach 1 as $n→∞$? I can show that sphericity $1 - \frac{5 \sqrt{3} π}{27n} - O(n^{-3/2})$ is possible. Is this, especially $O(n^{-3/...
9
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356
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Being even or odd in the product expansion $\prod(1+x^k+x^{k+1})$
Consider the generating function of "partitions with distinct parts"
$$\sum_nQ(n)x^n=\prod_k(1+x^k).$$
It's known that
$$\left[\prod_k(1+x^k)\right] \mod 2=\prod_m(1-x^m)=\sum_{j\in\mathbb{Z}...
3
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0
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114
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Asymptotics of a combinatorial series
I am interested in the exact asymptotics of the following combinatorial series (which arises from the study of a Markov chain):
$$F(q):=\sum_{k \ge 1} \frac{q^{k^2}}{(q;q)_k^2}\quad \mbox{as } q \to 1^...
4
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2
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Using $\delta$-method to "estimate" undefined moments of a random variable?
I posted this over on MSE without much luck. Not sure if posting here is considered cross-posting but I can remove it if it is.
Let $X\sim\mathcal N(\sqrt 2,1/x^2)$. The expected value $\mathsf EX^{-1}...
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1
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Is the harmonic series worse than any summable series?
It is well-known that the harmonic series is not summable. In some sense this means that it takes a lot of rather large values.
We define the operator $F_{\varepsilon}: \ell^{\infty}(\mathbb N) \...
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1
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85
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Asymptotics of the right singular vectors as the number of rows diverge [duplicate]
Write $X_m \in \mathbb{R}^{m \times n}$ as a Gaussian ensemble, so that $(X_m)_{ij} \sim \mathcal{N}(0, 1)$ are independent and identically distributed. Assume that $m \geq n$. Write $X_m = U_m \...
4
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Product of all multinomial coefficients
At some moment I found that for computing a bound on a density, I need to compute (or find a good asymptotic) of the product of all multinomial coefficients, i.e.,
$$
\prod_{\substack{(\alpha_1,\ldots,...
3
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Randomized version of Turán's theorem II
$\newcommand{\om}{\omega}$Let $\om(G)$ denote the number of vertices in a largest clique of an (undirected) graph $G$ with the set $[n]:=\{1,\dots,n\}$ of vertices. Then
\begin{equation}
\om(G)\ge\...
5
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1
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206
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Randomized version of Turán's theorem
Turán's theorem says the following.
Take any natural $n$ and $r$. Suppose that
\begin{equation*}
|G|>\Big(1-\frac1r\Big)\frac{n^2}2, \tag{0}
\end{equation*}
where $|G|$ is the number of edges of ...
1
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Ratio limit results for restricted partition functions
This concerns difference/limit ratio results for special restricted partitions.
Let $r,a, b$ be nonnegative integers; define $p(r,a,b)$ to be the number of partitions of the integer $r$ using at most $...
4
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Is there an elementary subexponential upper bound on the size of the stable stems?
This is a question in stable homotopy theory which I will boil down to a pure combinatorics question. If you're not interested in the homotopy theory, feel free to skip to the end for the ...
0
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Perturbative approach starting from a probability distribution approximated form
I approximate a probability distribution $P_x(x)$ with a $P_x^{app}(x)$,
such that $P_x(x)-P_x^{app}(x) = O(\epsilon)$ uniformly in x, where epsilon is a small positive quantity.
Consider the generic ...
1
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1
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Critical point of saddle point equation
Consider the following integral:
\begin{equation}
\int \mathrm{d}\rho \frac{1}{\rho} e^{N f(\rho)}
\end{equation}
Where:
\begin{equation}
f(\rho)=\ln \rho-\frac{1}{2} \rho^{2}+\frac{1}{2 p w^{2}} \rho^...
9
votes
3
answers
691
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Asymptotic analysis of $x_{n+1} = \frac{x_n}{n^2} + \frac{n^2}{x_n} + 2$
Problem: Let $x_1 = 1$ and $x_{n+1} = \frac{x_n}{n^2} + \frac{n^2}{x_n} + 2, \ n\ge 1$.
Find the third term in the asymptotic expansion of $x_n$.
I have posted it in MSE six months ago without ...
2
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Entropy per site of quantum spin chain
It’s fork lore that von Neumann entropy (and free energy) grows linearly with respect to the size of a quantum system. Is there a rigorous demonstration in the toy model of a quantum spin chain with (...
2
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2
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303
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Asymptotic of an improper integral
I found myself stuck with an "elementary" claim in some article. A simplified version of the problem is:
Let $p : [0,1]\to [0,1]$ be a continuous and non decreasing function such that $p(0)=...
1
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0
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75
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Approximating the partial sum of remainders function
This is a question related to the one I posted here, but I have found some more interesting and general results and thought here might be a better place to ask.
Let $R_{k,N}$ denote the remainder of ...
0
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115
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Can the Bessel functions tend to a plane wave?
Can the Bessel functions tend to a plane wave?
If I have this function:
$$
y(u)= c_1J_{-\sqrt{b}/2}(e^{2u}/6)+c_2J_{\sqrt{b}/2}(e^{2u}/6)+c_1J_{-i\sqrt{b}/2}(e^{2u}/6)+c_2J_{i\sqrt{b}/2}(e^{2u}/6)
$$
...
9
votes
1
answer
548
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Two-term recurrence relation
We consider the following system of recurrence relations for $n \in \mathbb Z$ and $\vert \lambda \vert=1$ with $\lambda \in \mathbb{C}$
$$a_{n+1} = \lambda a_{n-1}+ \lambda^* a_n + \lambda^* n b_n $$
...
2
votes
0
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125
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Asking for results on critical points and similar properties of solutions of nonlinear Volterra integral equations - Physically coherent solutions
I have a system of nonlinear Volterra integral equations of form
$$x(t)=x_0+\int_0^t K(t,s)F(x(s))ds$$
and I am interested on the critical points of $x(t)$, I mean maximum, minimum, increasing and ...
0
votes
1
answer
173
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Asymptotic behavior of the Student's t-quantile function of Student's t-cumulative distribution function
Let's denote
$F_{t_u}^{-1}(x)$ the quantile function of the Student's t-distribution $t_u$ with $u$ degrees of freedom and
$F_{t_v}(x)$ the cumulative distribution function of the t-distribution $t_v$...
1
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0
answers
92
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Convergence result on Cornish Fisher expansion of binomial distribution
Since it is known that Cornish Fisher expansion of quantiles does not have guaranteed convergence for all distribution, I wonder specifically if any convergence result is known in literature for CF ...
2
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0
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323
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For the following class of matrices, are the determinants invariant under permutations?
I want to ask a question regarding the invariance of determinants under permutation. The following matrix is the one I want to discuss here. (It's just a symmetric block tridiagonal matrix with non-...
0
votes
1
answer
176
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lower bound for sum of the n factors of the inclusion exclusion principle
Suppose the following relation is established:
$P\Bigl(A\cup B \cup C\cup D\Bigr) < P\Bigl(E\cup F\cup G\Bigr)$
based on boole's inequality, for each of the above probabilities we can have the ...
1
vote
1
answer
96
views
Tail bounds on random series in Hilbert space
Tail bounds on random series in Hilbert space
Let $X_n$, $n \in \mathbb {N}$, be independent $\pm 1$ symmetric random variables, and $a_n$,
$n \in \mathbb {N}$, be a sequence in a Hilbert space $H$ ...
5
votes
1
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242
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Asymptotics of ratios of polynomially recursive sequences
A sequence $a_n$ is said to be polynomially recursive (P-recursive) if it satisfies:
$$p^{[r]}(n)a_{n+r}+\cdots+p^{[1]}(n)a_{n+1}+\cdots + p^{[0]}(n)a_n=0$$
where $p^{[i]}(t)\in \mathbb{Q}[t]$ are ...
6
votes
1
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341
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Prove or disprove that $\sup_{n\in\mathbb{N}}\left|\sum_{\substack{d|n \\d<Q}}\mu(d)\right|\sim\pi(Q)$
To begin, let us set
$$A_Q(n):=\sum_{d|n \\ d<Q}\mu(d)$$
If we fix $Q$ and let $n$ vary, we get a very surprising amount of cancellation. For instance, the trivial bound
\begin{align*}
\mathbb{E}_{...