MathDB

For problem 11 , i couldn’t find the correct translation , so i just posted the hungarian version . If anyone could translate it ,i would be very thankful . [tip=see hungarian]Az

X

́es

Y

valo ́s ́ert ́eku ̋ v ́eletlen v ́altoz ́ok maxim ́alkorrel ́acio ́ja az

f(X)

́es

g(Y )

v ́altoz ́ok korrela ́cio ́j ́anak szupr ́emuma az olyan

f

́es

g

Borel m ́erheto ̋,

\mathbb{R} \to \mathbb{R}

fu ̈ggv ́enyeken, amelyekre

f(X)

́es

g(Y)

v ́eges sz ́ora ́su ́. Legyen U a

[0,2\pi]

interval- lumon egyenletes eloszl ́asu ́ val ́osz ́ınu ̋s ́egi v ́altozo ́, valamint n ́es m pozit ́ıv eg ́eszek. Sz ́am ́ıtsuk ki

\sin(nU)

́es

\sin(mU)

maxim ́alkorrela ́ci ́oja ́t. Edit: [hide=Translation thanks to @tintarn] The maximal correlation of two random variables

X

and

Y

is defined to be the supremum of the correlations of

f(X)

and

g(Y)

where

f,g:\mathbb{R} \to \mathbb{R}

are measurable functions such that

f(X)

and

g(Y)

is (almost surely?) finite. Let

U

be the uniformly distributed random variable on

[0,2\pi]

and let

m,n

be positive integers. Compute the maximal correlation of

\sin(nU)

and

\sin(mU)

. (Remark: It seems that to make sense we should require that

E[f(X)]

and

E[g(Y)]

as well as

E[f(X)^2]

and

E[g(Y)^2]

are finite. In fact, we may then w.l.o.g. assume that

E[f(X)]=E[g(Y)]=0

and

E[f(Y)^2]=E[g(Y)^2]=1

Problem Statement