Uniform bounded of Riemann-like sum and improper integral For any $h>0$, suppose $\{(y_i,y_{i+1}]\mid i\in \mathbb{Z}\}$ be a uniform partition of $\mathbb{R}$ with mesh size $h$. I am considering under what condition for a continuous transition density function $p(h,x,y)$ of a stochastic process, we can have
$$\sum_i p(h,x,\xi)h^3\le Mh^\alpha$$ 
uniformly in $h,x$. Let's take Gaussian density function as an example.
That is, given the Gaussian density function
$$p(h,x,y)=\frac{1}{2\pi\sqrt{h}}e^{-\frac{(x-y)^2}{2h}},$$
I am considering the following two questions:


*

*Whether there exists $M,h_0>0$ independent of $x$, such that  $h\in (0,h_0)$, we have
$$\sum_i p(h,x,\xi_i)h^3\le Mh^\alpha,\tag{1}$$ for some $\alpha>0$. 

*Whether the Riemann-like sum can be approximated by a corresponding improper integral in the sense that for any $\epsilon>0$, there exists $h_0$ independent of $x$, such that for any $0<h<h_0,x\in \mathbb{R}$, we have
$$|\sum_i p(h,x,\xi_i)h- \int_\mathbb{R}p(h,x,y)\,dy|<\epsilon .\tag{2}$$


Here $\xi_i\in (y_i,y_{i+1})$ satisfies $p(h,x,\xi_i)=\sup_{y\in [y_i,y_{i+1}]}p(h,x,y)$.

I am really sorry that I edited the question. I added more background information so that I can express my original question in a clearer way. 
Thanks for Kirvich Entracus's answer. Generalizing the idea in the answer, if 
$$p(h,x,y)\le \dfrac{M}{\sqrt{h}}$$
uniformly and such that for fixed $h$, there exists $\delta(h)>0$, such that  for any fixed $x$, if $|y-x|>\delta(h)$, we have $p(h,x,y)$ decreases when $|y-x|$increases, 
then we can have 
$$\sum_i p(h,x,\xi_i)h\le \dfrac{M}{\sqrt{h}} \delta(h)+1$$
uniformly for small $h$ by comparing it with the improper integral.

Looks not that complicated, if I understood the statements correctly.
For simplicity let me set $x=0$, $y_0=0$.
$i\geq 0\,\, \sum_{i\geq 0} p(h,0,\xi _i)h\leq \frac{h^\frac{1}{2}}{2\pi}+ \sum_{i\geq 0} q(h,0,\xi_i)h\leq\frac{h^\frac{1}{2}}{2\pi}+\int _{y\geq0}p(h,0,y)dy\leq \frac{h^\frac{1}{2}}{2\pi}+\frac{1}{2}\,\,\, where\,\, q(h,0,\xi_i)=inf\{p(h,0,y)|y\in (y_i,y_{i+1}]\}
\\
i<0\,\, \sum_{i< 0} p(h,0,\xi _i)h\leq \frac{h^\frac{1}{2}}{2\pi}+ \sum_{i< 0} q(h,0,\xi_i)h\leq\frac{h^\frac{1}{2}}{2\pi}+\int _{y<0}p(h,0,y)dy\leq\frac{h^\frac{1}{2}}{2\pi}+\frac{1}{2}\,\,\, where\,\, q(h,0,\xi_i)=inf\{p(h,0,y)|y\in (y_i,y_{i+1}]\}$
So for some constant $C>0$
$\sum_ip(h,0,y)h\leq Ch^{\frac{1}{2}}+1
\\
\sum_ip(h,0,y)h^3\leq (Ch^{\frac{1}{2}}+1)h^2$
So we can choose $(Ch_0^{\frac{1}{2}}+1)$ for M and $\alpha=2$.
When $x$ is not zero, slight modifications conclude the same result.
