Let $\mu$ be a probability measure with finite support on integers or the real line with the property that $\mu( 0) \le p$ for a fixed $0<p <1$. Let $S\_n$ denote the random walk starting at $0$, where each step has distribution $\mu$. Denote by $p\_n$ the probability that $S\_n=0$, so $p=p\_1$.


Is there a function $f(n)$ depending only on $p$ (and not on the entire distribution of $\mu$) such that $p\_n \le f(n)$ and $f(n)$ tends to zero as $n \to \infty$?


If $f(n)$ can also depend on $\mu$, then this is a very classical problem with precise asymptotic expressions for $p\_n$. However, all these results seem to depend on other parameters of $\mu$. I would appreciate any comments or pointers to papers and books with similar results.



Uniform upper bounds for the return probability of random walks on $ \mathbb{R}$
The actual (negative) answer was given in fedja's comment.


Since fedja said he would wait for somebody to come up with a reference for the relaxed statement, here we go:


Let $X$ be a random variable (r.v.) with distribution $\mu$. By (say) inequality (2.5) of Chapter III in [Petrov's book](https://link.springer.com/book/10.1007/978-3-642-65809-9), for some universal real constant $C>0$ and all real $t>0$
$$P(S\_n=0)\le C/\sqrt{n(1-Q(t))},$$
where
$$Q(t):=\sup\_{x\in\mathbb R}P(x\le X\le x+t).$$
Letting now $t\downarrow0$, we get
$$P(S\_n=0)\le f(n):=C/\sqrt{n(1-q)},$$
where
$$q:=Q(0)=\sup\_{x\in\mathbb R}P(X=x).$$
Assuming now that $q<1$, we get $f(n)\to0$ as $n\to\infty$.


