Frequently Asked Questions (FAQ) for End Cap Display

The answer was provided to me by Eric Cator. Here is his formulation which is almost the same. Let $Z$ be a Gaussian random vector $mathcalN(0,I_n)$ for some positive integer $n$. Let $q>0$, $FsubsetmathbbR^n$ be a cone, and let $F^o$ be its polar cone. Then,

$$mathbbPleft(|Zmu|>q|Zmuin Fright)leq mathbbPleft(|Z|>q|Zin Fright), quad forall muin F^o.$$

Proof:We rewrite the conditional probability in polar coordinates. Let $SomegainmathbbR^n| ; |omega|1$ be the unit sphere in $mathbbR^n$. Denote $nu$ the surface measure on $S$. We have

$$mathbbPleft(|Zmu|>q|Zmuin Fright) fracint_rq^inftyint_omegain S cap Ft^n-1e^-frac12|tw-mu|^2nu(domega)dtint_r0^inftyint_omegain S cap Fs^n-1e^-frac12|sw-mu|^2nu(domega)ds $$

This means that the conditional density of $|Zmu|$ provided that $Zmuin F$ is given by

$$g_mu(t) fract^n-1e^-frac12t^2int_omegain S cap Fe^tw^Tmunu(domega)int_r0^inftyint_omegain S cap Fs^n-1e^-frac12s^2sw^Tmunu(domega)ds.$$

Denote $A(mu)$ the normalization constant in the previous display, that is

$$A(mu) int_r0^inftyint_omegain S cap Fs^n-1e^-frac12s^2sw^Tmunu(domega)ds.$$

Define

$$G_mu(t) fracg_0(t)g_mu(t) fracA(mu)A(0)fracint_omegain S cap Fnu(domega)int_omegain S cap Fe^tomega^Tmunu(domega).$$

Since $muin F^o$, then

$$forallomegain S cap F, quad omega^Tmu leq 0.$$

Thus, function $tmapsto G_mu(t)$ is nondecreasing over $(0,infty)$ whatever the value of $muin F^o$.Let $h$ be some measurable nondecreasing function defined on $mathbbR_$. Note that for any couple of nonnegative real numbers $(r_1,r_2)$, since both $G_mu(t)$ and $h(t)$ are nondecreasing functions in $t$ over $(0,infty)$, we have

beginequation

(h(r_1)-h(r_2))(G_mu(r_1)-G_mu(r_2))geq 0 qquad (1)

endequation

Let $R_1$ and $R_2$ be two i.i.d. random variables with a common density defined on $(0,infty)$. We have, due to (1),

$$mathbbEleftleft(h(R_1)-h(R_2)right)left(G(R_1)-G(R_2)right)rightgeq 0.$$

We deduce that

beginequation

mathbbElefth(R_1)G_mu(R_1)rightgeq mathbbElefth(R_1)rightmathbbEleftG_mu(R_1)right.

qquad (2)

endequation

Assume now that $R_1$ has the density $g_mu$, and denote $mathbbE_mu$ (resp. $mathbbE_0$) the expectation under $g_mu$ (resp. $g_0$). We have now

beginalign*

mathbbE_mulefth(R_1)G_mu(R_1)right & mathbbE_0lefth(R_1)right

mathbbE_muleftG_mu(R)right & 1.

endalign*

The second line in the previous display comes from the fact that $g_0$ is a density over $(0,infty)$. Hence, due to (2), we may write

beginequation

mathbbE_mulefth(R)right leq mathbbE_0lefth(R)right.

qquad (3)

endequation

Set $h(t) 1_t>q$ (the indicator function of the set $(q,infty)$). Function $h$ is nondecreasing over $(0,infty)$, therefore we could apply (3) on it and get

$$fracint_rq^inftyint_omegain S cap Ft^n-1e^-frac12|tw^Tmu|^2nu(domega)dtint_r0^inftyint_omegain S cap Fs^n-1e^-frac12|sw^Tmu|^2nu(domega)ds leq fracint_rq^inftyint_omegain S cap Ft^n-1nu(domega)dtint_r0^inftyint_omegain S cap Fs^n-1nu(domega)ds. $$

This is exactly what we claim.

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Greece** Serious answers please!When and how do you think we will get over this crisis?

There is no simple answer to your question. I will answer it in general terms, in terms that will not only apply to Greece but in every country on earth. I will cite examples and will strive not to be biased. At the moment, there are several non-prosperous countries that have big amounts of natural resources. But one cannot classify them as prosperous since they have big percentages of people who live in poverty and faces social and economic unrest. Compared to Greece they have far bigger challenges. These are Russia, Nigeria, Brazil, South Africa and Venezuela, among others. Russia has huge oil reserves, big farmlands and other minerals; so are Nigeria and Venezuela. South Africa has the worlds largest diamond reserves and lots of farmland and wild life. Brazil has plenty of cattle and huge farms and other resources. Yet these countries are not necessarily considered prosperous. As a matter of fact they find themselves in a constant struggle to maintain national stability. On the other end of the scale are countries like Finland, Singapore, Switzerland and Austria that enjoy tremendous stability and prosperity. I choosed them as examples because they lack the kind of resources of the previously mentioned countries. They seem to have in common THREE OUTSTANDING QUALITIES (caps for emphasis) which stokes the fire of their high standards of quality of life for the majority of its people. 1) These countries display a great deal of honesty in government and in their overall society. Other countries seem to be constantly mired in scandal and corruption. Crime is less, interest groups and individuals who operate above and outside the law are fairly non-existent. 2) The 2nd characteristic of these prosperous societies is that they are functioning democracies. We can pay lip service and ballot-casting every few years or so, and many less prosperous countries boast of being democracies. But are they functioning democracies? In The Bahamas, for example, there is an oversized, overbearing government, with almost dictatorial powers being ascribed to the office of the prime minister. 3) Lastly, the people of prosperous countries mentioned above put a premium on quality education for all...and the result shows in the productivity of their people. The Bahamas as an example again here, though it devotes a large portion of its national budget to education, the results shows an educated citizenry which are far deficient. So what should Greece do? Greece, like most countries--- has unsustainable commitments on retirement incomes. A staged increase in retirement and pension ages, as has been done in Australia, is a reasonably credible way of constraining future growth in expenditure without too much adverse impact in the short run. An increase in the ratio of tax revenue to national income should be realized over time. That suggests a commitment to raise the VAT rate gradually over several years, with the extra revenue being reserved to debt service. On the income tax side, the most credible would be to raise revenue by a vigorous action against tax avoidance and evasion. Unlike the case of standard government bonds, there is no need to preserve access to this kind of borrowing in the future. As a matter of fact action to ensure that no future government can borrow in this hidden fashion is recommended. These deals should be renegotiated to reduce the burden they impose, with the threat of repudiation/default in the absence of a satisfactory agreement. And lastly, Greece has to find another prominent and illustrious statesman as well as a charismatic leader...like Eleftherios Venizelos. You asked a serious question and asked for a serious answer...so there it is.