Let U1, U2, . . . be a sequence of independent random variables, with each variable being uniformly distributed over the interval [0, 2], and let Xn = U1U2 Un for n ? 1.

(a) Determine in which of the senses (a.s., m.s., p., d.) the sequence (Xn) converges as n ? ?, and identify the limit, if any. Justify your answers.

(b) Determine the value of the constant ? so that the sequence (Yn) defined by Yn = n ? ln(Xn) converges in distribution as n ? ? to a nonzero limit.

2.17 Convergence of a product Let U1,U2,. .. be a sequence of independent random variables, with each variable being uniformly distributed over the inter- val [0,2], and let X\" = U1U2 .. -U.n for 'n, 2 1. (a) Determine in which of the senses (as, m.s., p., d.) the sequence (Xn) con verges as it :~ 00, and identify the limit, if any. Justify your answers. (b) Determine the value of the constant I9 so that the sequence (YR) dened by Y\" = n9 111(Xn) converges in distribution as n > 00 to a nonzero limit