# Questions tagged [large-deviation-theory]

Use this tag for question on large deviations theory

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### Importance Sampling Expectation Estimation

Say we have: $dX_t = \epsilon dW_t$ where $\epsilon$ is small. and $X_0 = a$ where $0 < a < 1$. I am intresting in finding the following quantity : $p^{\epsilon} = P(|X_{t=1}| \geq 1)$. One ...
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### Unique Uniformily Viscosity Solution

Say we have the following pocess: $$dX^{\epsilon} = b(X^{\epsilon}(s))ds + \sqrt\epsilon \sigma(X^{\epsilon}(s))dW(s)$$ for $s \in [0, T]$ and $X^{\epsilon}(0) = x_0$ We want to estimate a quantity ...
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### Cumulant generating function and largest eigenvalue of operator

I am working on a recent paper (arXiv:1805.02887) about an application of large deviation theory to the statistical mechanics of active matter and am a bit bewildered by a result dropped in appendix F ...
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### Rate function in Large deviation theory

A function I : X → [0, ∞] is called a rate function, if (i) I ≡ ∞; (ii) I has compact level sets (in particular, it is lower semicontinuous). How can I calculate the rate function $I$ for a ...
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### Cramer's Rule when excluding the largest j samples

For a simple case, assume $X_i$ i.i.d and it has moment generating function defined everywhere, i.e for $t\in\mathbb{R}$, $\mathbb{E}(e^{tX_1}) < \infty$. Denote $\mu = \mathbb{E}(X_1)$, then one ...
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### projection of a random vector and angle?

I have a random vector $v$ whose expected squared projection length along a unit vector $\hat{u}$ is large, while the expected squared length is small, i.e. $\mathbf{E}[|<v,\hat{u}>|^2]$ is ...
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### Asymptotically tight bounds $P \left[1-\frac{1}{n} \le \frac{\sum_{i=1}^n Z_i^2 }{n} \le 1+\frac{1}{n} \right]$

I am looking for assymptotically tight bounds on \begin{align} P \left[1-\frac{1}{n} \le \frac{\sum_{i=1}^n Z_i^2 }{n} \le 1+\frac{1}{n} \right]=P \left[ \left| E[Z^2] - \frac{\sum_{i=1}^n Z_i^2 }{n} ...