Local strong convexity around the minimizer

Question

I have the following problem where I want to show the local strong convexity based on some assumptions. Let $l(\theta, X)$ be a function with $\theta \in \mathbb{R}^d$ and $X$ a real-valued random variable. Define $M = E[l(\theta, X)]$. Let $\theta_0$ be the minimizer of $M$. Further suppose:

1. $l(\theta, X)$ is three times continuously differentiable.
2. $l(\theta, X)$ is $L(X)$-Lipschitz in a neighborhood of $\theta_0$.
3. $EL(X)^2<\infty$.

I want to show that in the neighborhood of $\theta_0$ we have that $$M(\theta) \geq M(\theta_0) + \lambda ||\theta- \theta_0||^2$$ for some $\lambda$.

I think the question is related to the following previous posts:

1. Is a smooth function convex near a local minimum?
2. Details on equivalent strong convexity