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Results tagged with large-deviations
Search options answers only
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user 7691
4
votes
Accepted
A large noise limit
Let $\varphi$ be the standard normal density. Since
$P[W_1 \ge x] =(1+o(1))\varphi(x)/x$ as $ x \to \infty$ by [1], we obtain for fixed $\delta>0$ that as $\epsilon \to 0$,
$$P[W_1 \ge \epsilon^{-1}+ …
- 14.2k
2
votes
Sample average L1 convergence speed
Theorem 1 (Hsu and Robbins [HR])
Let $X_1,X_2,\ldots$ be i.i.d. random variables with finite mean $\mu$ and
finite variance. Then for all $\epsilon>0$,
$$
\sum_{n=1}^\infty P\bigl(|S_n-n\mu|\ge n \ep …
- 14.2k
3
votes
Large deviation of random walk
Let's assume that $p<1/2$, Otherwise the probability in question does not decay exponentially.
Then
$$ \max_{{1\leq i\leq n}}\Pr\left(S_i>\max_{1\leq i'\leq n}-S_{i'}\right)=
\Pr\left(S_1>\max_{1\le …
- 14.2k