Some progress on the problem I mentioned earlier, of density near the origin in central limits. I can now handle another fairly broad case, that of centrally symmetric distributions. In fact, for these distributions, a stronger statement about density near the origin can be made, applying to all bounded-radius balls rather than merely to sufficiently large ones:

Theorem: Let \( F \) be a centrally symmetric distribution with bounded support on \( \mathbb{R}^d \), and let \( X_i \) be independent draws from \( F. \) Then for any \( K\gt 0 \) and even \( n, \)

\[ P\left[\left|\sum_{0\le i\lt n} X_i \right| \le K\right] = \Omega(n^{-d/2}). \]

Proof: Consider \[ Y = \sum_{0\le i\lt n/2} X_i \] and \[ Z = \sum_{n/2\le i\lt n} -X_i. \] \( Y \) and \( Z \) are independent, and (by symmetry) identically distributed. By an appropriate multidimensional central limit theorem (applied to the affine hull of the support of \( F \)), with constant probability both \( Y \) and \( Z \) are bounded by \( O(\sqrt{n}) \). Partition the ball of radius \( O(\sqrt{n}) \) into \( O(n^{d/2}) \) subsets \( S_j \), each of diameter at most \( K \), and let \( p_j \) be the probability that \( Y \) (or \( Z \)) belongs to \( S_j \). If both \( Y \) and \( Z \) belong to the same \( S_j \), then they are within distance \( K \) of each other, so \[ P\left[\left|\sum_{0\le i\lt n} X_i \right| \le K\right] = P[|Y-Z|\le K] \ge \sum_j p_j^2. \] But the sum of squares of a set of variables, that together add to a fixed value, is minimized when each variable is equal, so

\[ \sum_j p_j^2 = \Omega(n^{-d/2}) \]

As a corollary, the original statement of the problem, that there exists \( K \) such that the probability of being within distance \( K \) of the origin is \( \Omega(n^{-d/2}) \) for all \( n \) (not just all even \( n \)) holds for these distributions as well. For, if \( n \) is even, it follows from the theorem above, while if \( n \) is odd, by the triangle inequality \[ \left|\sum_{0\le i\lt n} X_i \right| \le \left|\sum_{0\le i\lt n-1} X_i \right| + |X_{n-1}| \] and we can choose \( K \) to be any constant greater than the maximum possible value of \( |X_i| \). The example of a simple random walk on an integer lattice shows that this choice of \( K \) is the smallest possible.

ETA: The same argument seems to apply more generally to nonconstant radii, showing that \[ P\left[\left|\sum_{0\le i\lt n} X_i \right| \le r\right] = \Omega(r^d n^{-d/2}) \] whenever \( r = O(\sqrt{n}) \) and either \( n \) is even or \( r\gt (1+\epsilon)E[|X_i|] \) for some constant \( \epsilon\gt 0 \).