Final answer:
The question concerns the almost sure convergence of a sequence of exponentially distributed random variables as n approaches infinity. The properties of the exponential distribution and convergence criteria are considered, but available information is insufficient for a definitive answer on almost sure convergence.
Step-by-step explanation:
The question is asking whether a sequence of independent random variables Xn following an exponential distribution with mean 1/ln(n) converges almost surely to a finite random variable as n approaches infinity. To answer this, we need to analyze the characteristics of the exponential distribution and the concept of almost sure convergence.
In the exponential distribution with mean μ, the decay parameter is m, and the probability density function is given by f(x) = me-mx. As n increases, the means of the random variables become smaller, which could imply that Xn converges to zero. The memoryless property of the exponential distribution indicates that past information does not affect future probabilities.
However, to establish almost sure convergence, one would typically use tools such as the Borel-Cantelli lemmas, which are not detailed here. Given the properties of the exponential distribution and without the additional theorems, we cannot definitively say whether Xn converges almost surely based on the provided information.