  For a random walk $S_n, n\geq 0$ in $\mathbb{Z}^d$, let $l(n,x)$ be its local
time at the site $x\in \mathbb{Z}^d$. Define the $\alpha$-fold self
intersection local time $L_n(\alpha) := \sum_{x} l(n,x)^{\alpha}$, and let
$L_n(\alpha|\epsilon, d)$ the corresponding quantity for $d$-dimensional simple
random walk. Without imposing any moment conditions, we show that the variances
of the local times $\mathop{var}(L_n(\alpha))$ of any genuinely $d$-dimensional
random walk are bounded above by the corresponding characteristics of the
simple symmetric random walk in $\mathbb{Z}^d$, i.e. $\mathop{var}(L_n(\alpha))
\leq C \mathop{var}[L_n(\alpha|\epsilon, d)]\sim K_{d,\alpha}v_{d,\alpha}(n)$.
In particular, variances of local times of all genuinely $d$-dimensional random
walks, $d\geq 4$, are similar to the $4$-dimensional symmetric case
$\mathop{var}(L_n(\alpha)) = O(n)$. On the other hand, in dimensions $d\leq 3$
the resemblance to the simple random walk $\liminf_{n\to \infty}
\mathop{var}(L_n(\alpha))/v_{d,\alpha}(n)>0$ implies that the jumps must have
zero mean and finite second moment.
