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Suppose $X(t), t \ge 0$ is a standard Brownian motion. I need to compute $\lim\limits_{n \longrightarrow \infty} \frac{X(n)}{n}$, where, $n = 1, 2, 3, \ldots$.

Since $X(t), t \ge 0$ is Gaussian, intuitively I can see that the when $n$ becomes very large, the ratio $\frac{X(n)}{n}$ should tend to zero. However, I do not have a rigorous argument to support my claim. I would appreciate any relevant ideas and comments.

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The sequence $\{X_k -X_ {k-1}\}$ is i.i.d and $X_n$ is the n-th partial sum of this sequence. The required (almost sure) limit is 0 by SLLN.

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