CONTI NUING FROM Chapter 1, general tools from the theory of L p spaces and probabilities are reviewed, up to martingale CLTs. This, together with the material of Chapter 2, provides us with a launch pad for CLTs for weighted sums of random variables, where those variables are initially m.d.'s and then short-memory linear processes. Next, the desire to obtain convergence statements for quadratic forms forces us to delve into the theory of integral operators. Certain classes of compact operators are studied, including Hilbert -Schmidt and nuclear ones. Both the final statements and some auxiliary results are important for later applications. For example, in Chapter 5 the gauge inequality is applied seven times.
In this chapter we deal with two types of L p spaces: on the segment [0, 1] or the square [0, 1] 2 , for approximation purposes, and on a probability space (V, F , P), for probabilistic results. To distinguish between these, the first two are denoted L p , as in the previous chapter, and the latter L p . If X is a random vector, the norm in L p , p , 1, is defined by kXk p ΒΌ (EkX(Γ)k p 2 ) 1=p [the norm of X at the left is in the space L p (V) and at the right is in the finite-dimensional space R dim X , with apologies for the confusion].
3.1 GENERAL INFORMATION
In this, some well-known facts from probability theory are reviewed.
3.1.1 Chebyshov Inequality
Lemma. If X is a random vector and kXk p , 1, p , 1, then P(kXk 2 ! 1) 1 Γp kXk p p for all 1 . 0:
Proof. There are several versions of this inequality, but all of them are based on the same idea. Using an obvious fact that 1 kXk 2 =1 on the set {kXk 2 ! 1}, we prove the