# Expectation of exponential of a function of independent Rademacher r.v.'s involving the error function

Let $$Z,Z'\in\{-1,1\}^n$$ be two independent vectors of i.i.d. Rademacher r.v.'s, where $$1\leq n \leq d$$ are two integers ($$d\gg 1$$). I am trying to get an upper bound on $$\mathbb{E}_{ZZ'}\left[ \exp\left({k\left(e^{\sum_{j=1}^n \mathrm{Erf}(\varepsilon Z_j /\sqrt{d})} -1\right)\left(e^{\sum_{j'=1}^n \mathrm{Erf}(\varepsilon Z'_{j'} /\sqrt{d})} -1\right)}\right) \right] \tag{\dagger}$$ where $$t\geq 1$$, $$\varepsilon \in(0,1]$$, $$k\geq 1$$ is an integer, and $$\mathrm{Erf}$$ is the error function. In particular, I am interested in the dependence on $$n$$: since $$\mathrm{Erf}(x) \operatorname*{\sim}_{x\to\ 0} \frac{2}{\sqrt{\pi}}x$$ I expect that, for $$n = n(d) \ll \sqrt{d}$$, $$(\dagger)$$ should be roughly $$\mathbb{E}_{ZZ'}\left[ \exp\left({\frac{4}{\pi}\frac{\varepsilon^2 k}{d}\sum_{j,j'} Z_jZ'_{j'}}\right) \right] \leq e^{C \frac{\varepsilon^4 k^2}{d^2}\cdot n^2}$$ (the last bound for $$d$$ big enough). But I'm unclear if the above qualitative behavior still holds for $$n \gg \sqrt{d}$$: do we still get that $$n^2$$ dependence in the exponent?

Note: I am actually interested in the quantity $$\mathbb{E}_{ZZ'}\left[ \exp\left({k \left(\prod_{j=1}^n (1+\mathrm{Erf}(\varepsilon Z_j /\sqrt{d}))-1\right)\left(\prod_{j'=1}^n (1+\mathrm{Erf}(\varepsilon Z'_{j'} /\sqrt{d}))-1\right)}\right) \right] \tag{\ddagger}$$ (same questions), but $$(\dagger)$$ is similar and hopefully a bit cleaner to analyze.

• @MattF. Sure, why not. I made these changes, roughly. Nov 29, 2018 at 17:21
• Thanks! I had not noticed the double-exponential before Nov 29, 2018 at 17:35
• While we're suggesting simplifications---you might want to define $\delta := \varepsilon/\sqrt{d}$ to lower the number of parameters. (Also, this is probably dumb, but it seems like things get a lot nicer if you write the outer exponential as a Taylor series.) Dec 3, 2018 at 15:31
• @NoahStephens-Davidowitz I see the advantage of the reparamerization (although it makes the interpretetaion of the relation $n$ v. $d/\varepsilon^2$ less clear). But I'm curious about what you mean with the series expansion: this would bring powers of $\sum_{j,j'} Z_j Z'_{j'}$ to handle, wouldn't it? Dec 3, 2018 at 16:34
• @ClementC. Yes, but I guess you can apply independence to write this as $\sum_i \mathbb{E}[f_i(Z_1,\ldots, Z_n)]\mathbb{E}[f_i(Z_1',\ldots, Z_n')]$, which seems like a pretty big win to me. Again, this is probably stupid, but it's what I would try. Dec 3, 2018 at 20:42

Every $$Z_i = \pm 1$$, so you have $$\mathrm{Erf}(Z_i \delta) = Z_i \mathrm{Erf}(\delta) \equiv Z_i \tau$$, and the products become $$(1 + \tau)^b (1 - \tau)^{n - b}$$ where $$b$$ follows the Binomial distribution $$B(n, 1/2)$$, and the same for the primed quantities. The expectation is then $$\mathbb{E}_{bb'}\left[\exp\left({k \left((1 + \tau)^b (1 - \tau)^{n-b}-1\right)\left((1 + \tau)^{b'} (1 - \tau)^{n-b'}))-1\right)}\right) \right] .$$ Perhaps this allows you to get the kind of bound you are looking for. You have $$0 < \tau < 1$$, so you can bracket $$(1 + \tau)^b(1 - \tau)^{n-b}$$ by $$(1-\tau)^n$$ and $$(1+\tau)^n$$. Without having too good an idea about how to proceed, I would naively expect the expectation to behave like $$\exp\bigl(k \, (1+\tau)^{2n})\bigr)$$ as $$n$$ increases. A quick Mathematica experiment with $$k = 1.23$$, $$\delta = 0.1$$ seems to bear that out, but there could be all kinds of things going wrong here.
(Since you now have only two r.v.s to contend with, you can compute the expectation explicitly to get some additional insight, at least until the arithmetic overflows for large $$n$$.)