Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $X_i$ be a sequence of i.i.d. standard normal random variables. Let $Y_i=\sum_{k=1}^iX_k$ and $Z_i=\sum_{k=1}^iY_k$. I am interested in upper and lower bounds for $P(\sup_{1\leq i\leq m}|X_i|\leq c)$, $P(\sup_{1\leq i\leq m}|Y_i|\leq c)$ and $P(\sup_{1\leq i\leq m}|Z_i|\leq c)$. I managed to figure out the first two, and for the second one I got bounds of something like $1-\mbox{const}\times e^{-c^2/n}$ by considering $P(|B_t|\geq c)$ and using the reflection principle for $\tau=\inf_{t\leq m}\{t: \ |B_t|\geq c)\}$.

The problem is, the same trick doesn't quite work when figuring out the $P(\sup_{1\leq i\leq m}|Z_i|\leq c)$. In fact, the best I could do was write $Z_{n+1}=Z_n+B_{n+1}$ where $B$ is a standard Brownian motion. Maybe this can become a stochastic differential equation? But, it feels intractable unless I'm missing something. I was wondering how one might get good upper and lower bounds for the supremum over $Z_i$? Maybe I'm thinking too hard and there's an easier way which doesn't resort to Brownian motion.

Any help would be greatly appreciated!

share|improve this question
    
I assume you mean $Z_i = \sum_{k=1}^i Y_k$, not $Z_i = \sum_{k=1}^i Y_i$ –  Robert Israel Feb 28 '12 at 22:00
1  
Your bound for $P(\sup_{1\le i\le m} |Y_i| \le c)$ is not good as $m \to \infty$ for fixed $c$: it should go to $0$ exponentially in $m$. Note that if any $|X_k| > 2 c$, then $\max(|Y_{k-1}|, |Y_k|) > c$. Similarly, if any $|X_k| > 4c$, then $\max(|Z_{k-2}|,|Z_{k-1}|,|Z_k|) > c$. So these should also go to $0$ exponentially in $m$. –  Robert Israel Feb 28 '12 at 22:21
    
Corrected. Thanks! –  Alex R. Feb 29 '12 at 0:53

2 Answers 2

up vote 0 down vote accepted

Maybe you find this useful. This paper by Charles. E. Clark http://www.eecs.berkeley.edu/~alanmi/research/timing/papers/clark1961.pdf gives a way to approximate the maximum of a finite set of correlated normal variables $Y_1,\ldots,Y_n$ by a normal variable itself.

Namely, if $Y_1\sim N(\mu_1,\sigma_1)$, $Y_2\sim N(\mu_2,\sigma_2)$ are normal variables with correlation $\rho$ denote $$ a^2 = \sigma_1^2 + \sigma_2^2 -2\sigma_1\sigma_2\rho\ ,\\ \alpha = (\mu_1-\mu_2)/a\ . $$ The first and second order of the maximum $W = \max(Y_1,Y_2)$ are given by $$ \nu_1 = \mu_1\Phi(\alpha) + \mu_2\Phi(-\alpha) + a\varphi(\alpha)\ , \\ \nu_2 = (\mu_1^2+\sigma_1^2)\Phi(\alpha) + (\mu_1^2+\sigma_1^2)\Phi(-\alpha) + (\mu_1+\mu_2)a\varphi(\alpha)\ , $$ where $\Phi$ is the standard normal cdf and $\varphi$ the standard normal density. Using this it is possible to define a normal variable $\widetilde{W}\sim N(\nu_1,(\nu_2-\nu_1)^2)$ that approximates the maximum of $Y_1$ and $Y_2$. You can obtain then an approximation of $\max(Y_1,Y_2,Y_3)$ using the correlation $$ \rho(W,Y_3) = \sigma_1\rho_1\Phi(\alpha) + \sigma_2\rho_2\Phi(-\alpha)/(\nu_2 - \nu_1)^{1/2} $$ where $\rho_1=\rho(Y_1,Y_3)$ and $\rho_2=\rho(Y_2,Y_3)$. This can be done recursvely but the general recurrence is a little bit more tricky since you need the correlations $\rho(W_j,Y_i)$ for $i\geq j+2$ at every step being $$ W_j \approx \max(Y_1,\ldots,Y_{j+1})\ . $$ The approximation is good enough for many numerical purposes and it may be good as well to obtain the bounds that you require.

But I feel that maybe you are right and the answer is simpler than all that.

share|improve this answer

This may be overkill, but: Fernique's theorem says that for any Gaussian measure $\mu$ on a separable Banach space $(X, \|\cdot\|)$, there are constants $C,\epsilon$ such that $\mu(\{ x : \|x\| > t\}) \le Ce^{-\epsilon t^2}$. (You can find a proof at Theorem 4.10 of these lecture notes of mine.) $(Z_1, \dots, Z_m)$ is a Gaussian random vector (being a linear transformation of $(X_1, \dots, X_m)$), hence its law $\mu$ is a Gaussian measure on $\mathbb{R}^m$. If we let $X$ be $\mathbb{R}^m$ equipped with its $\ell^\infty$ norm, then Fernique's theorem says that $$P(\max_{1 \le i \le m} |Z_i| > t) \le C e^{-\epsilon t^2}.$$

If you want more explicit control over the constants, you may have to do more work.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.