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Let $\{U_k\}$ be a sequence of independent random variables, with each variable being uniformly distributed over the interval $[0,2]$, and let $X_n = U_1 U_2\cdots U_n$ for $n \geq 1$.

(a) Determine in which of the senses (a.s., m.s., p., d.) the sequence $\{X_n\}$ converges as $n\to\infty$, and identify the limit, if any. Justify your answers.

(b) Determine the value of the constant $\theta$ so that the sequence $\{Y_n\}$ defined by $Y_n = n^\theta \ln(X_n)$ converges in distribution as $n\to\infty$ to a nonzero limit.

  • 0
    Consider $Y_n$ with $\theta=0$. Is it a random walk? What do you know about the convergence of random walk?2011-10-13
  • 1
    Note that $X_n$ is a martingale.2011-10-13

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