infinite number of tosses of a fair coin. The probability of heads equals tails. So carried to infinity does the fact of equal probabilities for head and tails make the particular random walk from these tosses go to zero.  In other words will the walk tend to zero distance at infinity because the flips obey the probabilities of the fair coin.

Let $X_n$ be an iid sequence of random variables with $$\mathbb P(X_1) = p = 1-\mathbb P(X_1) = 0. $$
Set $\bar X_n = \frac1n\sum_{i=1}^n X_i$. Then
$$\mu:=\mathbb E[\bar X_n] = \mathbb E\left[\frac1n \sum_{i=1}^n X_i\right] = \frac1n\cdot np = p. $$
The strong law of large numbers tells us that
$$\mathbb P\left(\lim_{n\to\infty} |\bar X_n-\mu| = 0\right). $$
Hence $\bar X_n$ converges to zero almost surely (and therefore in probability and in distribution), but $\lim_{n\to\infty} \mathbb E[\bar X_n] = p>0$ so $\bar X_n$ does not converge in $L^p$ for $p\geqslant 1$.
