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

        -2
        votes
        0answers
        32 views

        Induction in Ratio Test [on hold]

        Here we have the series ((2^n)(n!))/(5x8x11.....(3n+2)) going from n=1 to infinity. I think I have to use induction, but basically, I'm not sure how to find the limit of this problem using the ratio ...
        0
        votes
        0answers
        39 views

        Asymptotic distribution of $n\mathbb E_{\hat{P}_n}[g(Z;\theta)]^T\operatorname{Cov}_{\hat{P}_n}[g(Z;\theta)]^{-1}\mathbb E_{\hat{P}_n}[g(Z;\theta)]$

        Setup This question is a followup on this question. I'm interested in the asymptotic distribution of certain quadratic forms. So, let $Z$ be a $p$-dimensional random vector with (unknown) ...
        5
        votes
        1answer
        183 views

        Asymptotic behavior of a certain trigonometric partial sum

        Let $a<0$ and $b>0$ be real numbers such that $a<-2b$. Let $n>1$ be a positive integer and consider the following partial sum: $$ f(n) = \frac{1}{(n+1)^2}\sum_{i=1}^{n}\sum_{j=1}^{n} (-1)^{...
        -2
        votes
        2answers
        108 views

        What is the limit of this integral as $n$ approaches infinity for integer $k\geq 0$ and real $m\geq 1$? [closed]

        $\int_{0}^{1}u^k\cot{\frac{\pi(1-u)}{m}}\sin{\frac{2\pi n(1-u)}{m}}\,du$
        0
        votes
        0answers
        19 views

        Optimization in Function Space

        Let $H = W_q[0,1]$ be a Sobolev space, that is, for $f \in H$, f has $q-1$ derivatives, and $f^{q-1}$ is absolutely continuous, and $f^{(q)} \in L^2[0,1]$. $H$ has the inner product, for $f,g \in H$, ...
        1
        vote
        1answer
        116 views

        Asymptotic distribution of $\mathbb E_{\hat{P}_n}[Z]^T\operatorname{Cov}_{\hat{P}_n}[Z]^{-1}\mathbb E_{\hat{P}_n}[Z]$

        Under very general conditions on the random $p$-dimensional vector $Z$, what can be said about the asymptotic distribution of the (random) scalar quantity $R_n := \mathbb E_{\hat{P}_n}[Z]^T\...
        3
        votes
        0answers
        88 views

        Asymptotic behavior of row sums in 2-d array of random variables

        Set-up. Let $f : \mathbb{N} \to \mathbb{N}$ be increasing. For each $m \in [0,1]$, consider an infinite two-dimensional array of random variables, where row $n$ has $f(n)$ variables: $B^m_{1,1}$ $B^...
        6
        votes
        1answer
        162 views

        Weak*-convergence of signed measures

        Let $X$ be a compact Hausdorff space and let $M(X)$ denote the space of signed measures that is naturally dual to $C(X)$, the space of continuous functions on $X$. I am interested whether the ...
        0
        votes
        0answers
        46 views

        $L_1$ convergence for a product of indicator functions

        Let $X_1,X_2,\ldots$ be a sequence of identically distributed random variables and let $A\subset\mathbb{R}$ be some set such that $P(X_1\in A)<1$. I have a product of indicator functions $$ \lim_{N\...
        -2
        votes
        1answer
        101 views

        Weak convergent $+$ strongly convergent subsequence $\Rightarrow$ strong convergence? [closed]

        Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ converges weakly to $f\in X$, and also has a strongly convergent ...
        1
        vote
        0answers
        41 views

        A convergence condition on tempered representation

        Assume $\pi$ is a tempered representation of $GL_n(\mathbb{Q}_p)$. $N_n$ is a maximal unipotent subgroup of $GL_n$, and $\xi$ is a non-degenerate character of $N_n(\mathbb{Q}_p)$. Let $\Pi$ be the ...
        5
        votes
        0answers
        68 views

        Positive splitting of Sobolev convergence

        Let $f,g,h \in H^1(\mathbb{R}^n)$ be non-negative Sobolev functions wuch that $f^2 = g^2 + h^2$. Let also $\{f_k\} \subseteq H^1(\mathbb{R}^n)$ be non-negative Sobolev functions such that $f_k \to f$ ...
        2
        votes
        0answers
        58 views

        Convergence of SDEs

        Suppose that $\{a_n(x)\}_{n \in \mathbb{N}}$ is a sequence of real-valued Lipschitz functions with domain $\mathbb{R}^d$, which converges $m$-a.e. to a Lipschitz function $a$. Suppose that $b$ is a ...
        2
        votes
        0answers
        67 views

        Convergence Based on Recurrence Relation

        I am studying a sequence based on the following recurrence: $$X[t] = \sqrt{\alpha X[t-1]^2+(X[t-1]^2-\alpha X[t-2]^2)\frac{(2-X[t-1])^2}{X[t-1]^2}}$$ $$X[0]=0$$ $$X[1]>0$$ $$\alpha \in (0,1)$$ I ...
        2
        votes
        2answers
        62 views

        Divergence rate of geometric sum of random variables

        Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that $$ 0<\lim_{\beta\rightarrow 1}(1-\...

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