On the Strong Law of Large Numbers in Spaces of Compact Sets
Abstract
<jats:p> Let <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">{\mathfrak Y}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="fraktur">Y</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> be the space of all nonempty compact subsets of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">{\mathbb R}^d</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and let <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">{\cal K}({\mathfrak Y})</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">K</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="fraktur">Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> be the space of all nonempty compact subsets of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">{\mathfrak Y}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="fraktur">Y</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . For a random set with values in <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">{\cal K}({\mathfrak Y})</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="script">K</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="fraktur">Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , after defining the expectation, we establish a version of the strong law of large numbers. Some related results concerning the case of nonempty compact convex subsets of a Banach space <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">{\mathbb E}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="double-struck">E</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> are included </jats:p>