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thatsmaths.com | ||
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almostsuremath.com
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| | | | | The martingale property is strong enough to ensure that, under relatively weak conditions, we are guaranteed convergence of the processes as time goes to infinity. In a previous post, I used Doob's upcrossing inequality to show that, with probability one, discrete-time martingales will converge at infinity under the extra condition of $latex {L^1}&fg=000000$-boundedness. Here, I... | |
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hbfs.wordpress.com
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| | | | | $latex n!$ (and its logarithm) keep showing up in the analysis of algorithm. Unfortunately, it's very often unwieldy, and we use approximations of $latex n!$ (or $latex \log n!$) to simplify things. Let's examine a few! First, we have the most known of these approximations, the famous "Stirling formula": $latex \displaystyle n!=\sqrt{2\pi{}n}\left(\frac{n}{e}\right)^n\left(1+\frac{1}{12n}+\frac{1}{288n^2}-\frac{139}{51840n^3}-\cdots\right)$, Where the terms... | |
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nickhar.wordpress.com
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| | | | | 1. Low-rank approximation of matrices Let $latex {A}&fg=000000$ be an arbitrary $latex {n \times m}&fg=000000$ matrix. We assume $latex {n \leq m}&fg=000000$. We consider the problem of approximating $latex {A}&fg=000000$ by a low-rank matrix. For example, we could seek to find a rank $latex {s}&fg=000000$ matrix $latex {B}&fg=000000$ minimizing $latex { \lVert A - B... | |
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djalil.chafai.net
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| | | The Girko circular law theorem states that if \( {X} \) is a \( {n\times n} \) random matrix with independent and identically distributed entries (i.i.d) of variance \( {1/n} \) then the empirical measure \[ \frac{1}{n}\sum_{i=1}^n\delta_{\lambda_i(X)} \] made with the eigenvalues of \( {X} \), converges, as the dimension \( {n} \) tends to infinity, to the uniform law... | ||
