stat612-notes0-mod - Statistics 612: L p spaces, metrics on...

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Unformatted text preview: Statistics 612: L p spaces, metrics on spaces of probabilites, and connections to estimation Moulinath Banerjee March 17, 2009 1 L p spaces and Hilbert spaces We first formally define L p spaces. Consider a measure space ( X , A , μ ), where μ is a ( σ-finite) measure. Let F be the set of all real–valued measurable functions defined on X . The space L p ( μ ) comprises that subset of functions of F that have finite p ’th moments (here p ≥ 1). In other words: L p ( μ ) = { f ∈ F : Z | f ( x ) | p d μ ( x ) < ∞} . When the measure μ is a probability P , we obtain the class of all random variables on the probability space, ( X , A , P ) that have finite p ’th moments. This is a normed linear space over R with the norm (length) of the vector f (this is called the L p norm) being given by f p = Z | f ( x ) | p d μ ( x ) ¶ 1 /p . The above norm induces a metric d where d ( f, g ) = f- g p . Note that d ( f, g ) = 0 if and only if f = g a.e. μ , in which case we identify f with g . The L p norm, like all worthy norms, satisfies the triangle inequality: f + g p ≤ f p + g p ; this is precisely Minkowski’s inequality. For random variables X, Y defined on the same probability space and having finite p ’th moments, Minkowski’s inequality states: E ( | X + Y | p ) 1 /p ≤ E ( | X | p ) 1 /p + E ( | Y | p ) 1 /p . Minkowski’s inequality is a consequence of H¨older’s inequality which states that for measurable real-valued functions f, g defined on X , we have: | Z f ( x ) g ( x ) d μ ( x ) |≤ Z | f ( x ) | p dμ ( x ) ¶ 1 /p Z | g ( x ) | q dμ ( x ) ¶ 1 /q . 1 The space L p ( μ ) is a Banach space for p ≥ 1 – this is a normed linear space that is complete – i.e. in which every Cauchy sequence has a limit. The notion of continuity for real valued functions defined on L p ( μ ) is a natural extension of the usual one for Euclidean spaces. A sequence of functions g n converges to a function g in L p ( μ ) if g n- g p → 0. A real-valued function ψ defined on L p ( μ ) is said to be continuous if ψ ( g n ) converges to ψ ( g ) as a sequence of real numbers whenever g n converges to g in L p ( μ ). Let’s concretize to a specific example. Let X be the space of positive integers N , A be the power set of N and μ be counting measure that assigns the cardinality of a subset of N as its measure. The space L p ( μ ) is then the space of all real valued sequences { x 1 , x 2 , . . . } that are p ’th power summable – i.e. ∑ ∞ i =1 | x i | p < ∞ . Clearly all sequences that have only finitely many non-zero entries satisfy this condition. This space is referred to as the l p space. The l p spaces are infinite-dimensional spaces – i.e. these spaces do not have a finite basis....
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This note was uploaded on 04/14/2010 for the course STATS 612 taught by Professor Moulib during the Winter '08 term at University of Michigan.

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stat612-notes0-mod - Statistics 612: L p spaces, metrics on...

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